Online Appendix

A Data

We follow Fernandez-Villaverde et al (2015) and use the following macroeconomic time series for the estimationand calculation of data moments

1 Output is real GDP (GDPC1)

2 Consumption is real personal consumption expenditures (PCECC96)

3 Investment is real gross private domestic investment (GPDIC96)

4 Civilian Noninstitutional Population (CNP16OV quarterly averages)

5 Inflation is GDP deflator (GDPDEF)

6 Hourly real wage is compensation per hour in the business sector (HCOMPBS) divided by the GDP deflator(GDPDEF)

7 Hours per capita are measured by hours of all persons in the business sector (HOABS)

Data for the period from 1970Q1 to 2016Q4 is from the St Louis Fedrsquos FRED database (mnemonics are inparentheses)

Government spending is government consumption and gross investment both from NIPA To construct theaggregate effective tax rates we follow Leeper et al (2010) (see also Appendix B in Fernandez-Villaverde et al 2015)and use national account information (NIPA) Specifically the average capital tax rate is calculated as

τk =τpCI + CT + PRT

CI + PRT

where CI denotes taxes on capital income CT denotes taxes on corporate income (NIPA Table 31 line 5) and PRTdenotes property taxes (NIPA Table 33 line 8) We define CI = PRI2 + RI + CP + NI where the first term ishalf of the proprietorrsquos (PRI NIPA Table 112 line 9) and the latter three terms are respectively rental income(RI NIPA Table 112 line 12) corporate profits (CP NIPA Table 112 line 13) and interest income (NI NIPATable 112 line 18) Following Jones (2002) the average personal income tax τp is computed as PIT

WSA+PRI+CI The

numerator is federal state and local taxes on personal income (PIT NIPA Table 32 line 3 plus NIPA Table 33line 4) The denominator is given by wage and salary accruals (WSA NIPA Table 112 line 3)

The Treasury yield data are from Gurkaynak et al (2007) (data are available for download on the website http

wwwfederalreservegovPubsfeds2006200628feds200628xls) Real term structure data are obtained bysplicing together real yields from Chernov and Mueller (2012) and Gurkaynak et al (2010) In particular the datafrom Chernov and Mueller (2012) span 1971Q3 to 2002Q4 We merge these data with those from Gurkaynak et al(2010) Throughout we remove data for 2003 due to a high illiquidity premium For the same liquidity reason wealso consider a shorter sample that excludes the financial crisis The relative (il)liquidity of TIPS from their inceptionuntil 2003 (when the Treasury reaffirmed its commitment to the TIPS program) and in the aftermath of the LehmanBrothers bankruptcy in late 2008 (which resulted in its considerable TIPS inventory being released into the market)have been discussed by Sack and Elsasser (2004) and Campbell et al (2009) among others

The PCs in Table ndash are constructed from the observed yields with maturities of 3 months one through fiveyears and ten-years The PCs are standardized to have unit standard deviation

The empirical analysis in Table uses quarterly returns to match the quarterly frequency of fiscal variablesExcess bond returns are measured by the return to a portfolio of Treasury bonds with maturities between 1-2 2-3

1

4-5 5 and 10 years and more than 10 years Only non callable non flower notes and bonds are included in theportfolios The portfolio returns are an equal-weighted average of the unadjusted holding period return for each bondin the portfolios Quarter-end to quarter-end excess returns are constructed by compounding simple returns to theportfolios then subtracting the compounded return to the shortest-maturity portfolio which contains bonds withmaturities less than 6 months Excess returns to the aggregate stock market are constructed in the same way usingthe CRSP value-weighted index Finally the five value-weighted quintile portfolios sorted on their book-to-marketratio (see Fama and French 1992) are from Kenneth Frenchrsquos website

B Solution and Estimation

The model is tractable enough to employ the estimation methodology recently proposed by Andreasen Fernandez-Villaverde and Rubio-Ramırez (2017) We proceed as follows

Model Solution First we solve the model To this end we induce stationarity by eliminating trending variableswith appropriate transformations The desired policy functions that characterize the equilibrium are then obtainedby employing a third-order perturbation approximation We require at least a third-order approximation to generatevariation in risk premia A standard approach to efficiently compute a higher-order approximation to DSGE modelswith a yield curve exploits the fact that bond prices beyond the policy rate do not affect equilibrium allocations andprices We take advantage of this property by solving the model in a two-step procedure in a first step we solve themodel without bond prices exceeding one period in a second step all remaining bond prices with maturities up toten years are computed recursively based on

Q(k)t = Et

[M$tt+1Q

(kminus1)t+1

]

where M$tt+1 = Mtt+1

1Πt+1

denotes the nominal stochastic discount factor and Mtt+1 denotes the real stochastic

discount factor We let k = 2 40 quarters The nominal yield curve with continuous compounding is then givenby y

(k)t = minus 1

klogQ

(k)t We also compute the real term structure based on

Q(k)treal = Et

[Mtt+1Q

(kminus1)t+1real

]

This two-step procedure reduces the size of the simultaneous equation systems to be solved and therefore substan-tially reduces the computational burden of the approximation

Analytical Model Moments Second we derive analytical closed-form expressions for first and second unconditionalmoments of the nonlinear pruned state-space of the model To ensure stable sample paths (and existence of finiteunconditional moments) we adopt the pruned state-space system for non-linear DSGE models suggested by Andreasenet al (2017) Intuitively pruning means we are going to omit terms of higher-order effects than the consideredapproximation order (third-order in our case) when the system is iterated forward in time1 Provided the linearizedsolution is stable Andreasen et al (2017) derive closed-form expressions for first and second unconditional momentsof the pruned state-space of the DSGE This allows us to efficiently compute the unconditional moments for ourDSGE model solved up to third-order2

Estimation Methodology Finally we estimate a subset of model parameters via generalized method of moments(GMM) In our estimation we use the first and second unconditional moments of the following quarterly macroeco-nomic and financial time series (i) log output growth ∆yt (henceforth ∆ denotes the temporal difference operator)

1For details on the pruning method see Sims et al (2008) for second-order and Andreasen et al (2017) forhigher-order approximations to the solutions of DSGE models

2Although we solve the model by a third-order perturbation we verified that our model moments are similarwhen we use a higher-order approximation and no pruning In particular we checked that our results do not changewhen we use a fifth order solution to our DSGE model To obtain a fifth order solution we use the tensor approachproposed by Levintal (2017)

2

(ii) log investment growth ∆invt (iii) log consumption growth ∆ct (iv) inflation πt (v) the one quarter nominal

interest rate rt (vi) the ten year nominal interest rate y(40)t and (vii) the slope of the term structure y

(40)t minus rt All

series are stored in datat which is of dimension 7 times 13 Our sample goes from 1970Q1 to 2016Q4 hence the matrixdata which is of dimension 7 times 188 We then define the following vector

qt =

datatdiag (datatdataprimet)vech (datatdataprimet)

Moreover let θθθ be a vector that contains the structural parameters Our GMM estimator is then given by

θθθGMM = argminθθθisinΘΘΘ

(1

T

Tsumt=1

qqqt minusm (θθθ)

)primeWWW

(1

T

Tsumt=1

qqqt minusm (θθθ)

)(B1)

Here W is a positive definite weighting matrix and m (θθθ) is a vector that contains the model-implied unconditionalmoments computed in closed-form as described above We use the conventional two-step implementation of GMM

by letting WWWT = diag(SSSminus1

)in a preliminary first step to obtain θθθstep 1 where SSS denotes the long-run variance

of 1T

sumTt=1 qqqt when re-centered around its sample mean Our final estimates θθθstep 2 are obtained using the optimal

weighting matrix WWWT = SSSminus1

θθθstep 1 where SSSθθθstep 1 denotes the long-run variance of our moments re-centered around

m(θθθstep 1

) The long-run variances in both steps are estimated by the Newey-West estimator using 10 lags but our

results are robust to using more lags

To summarize the estimation procedure implemented by GMM is as follows

bull 1 Step Let WWWT = diag(SSSminus1

)and obtain θθθstep 1 from B1

bull 2 Step Use θθθstep 1 to compute WWWT = SSSminus1

θθθstep 1 and obtain θθθstep 2 from B1

C Solving the Benchmark Model

C1 Households with Epstein-Zin Preference

The agentrsquos optimization problem is

max V (Ct Nst ) =

(1minus β)U(Ct N

st )1minusψ + βEt

[V 1minusγt+1

] 1minusψ1minusγ

11minusψ

st Et

[infinsums=0

M$tt+sPt+sCt+s

]le Et

[infinsums=0

M$tt+s(Wt+sPt+sN

st+s minus Pt+sTt+s + Pt+sΨt+s)

]

where

Ct =

[int 1

0

Ct(j)θminus1θ dj

] θθminus1

and

U(Ct Nst ) =

[C1minusψt

1minus ψ minus ϕ1minusψt

Nst

1+ω

1 + ω

] 11minusψ

3We have also repeated our procedure adding to the first and second moments used in the baseline estimationthe first and fifth autocovariances to capture the persistence in the data Our point estimates do not significantlychange and the conclusion from model-implied moments remain qualitatively the same

3

The first order conditions are

partVtpartCt

[V 1minusψt

] 11minusψminus1

1minus ψ (1minus β)Cminusψt minus λM$ttPt = 0 (C1)

partVtpartNs

t

[V 1minusψt

] 11minusψminus1

1minus ψ (1minus β)(minusϕ1minusψt Ns

tω) + λM$

ttWtPt = 0 (C2)

partVtpartCt+1

[V 1minusψt

] 11minusψminus1

1minus ψ β

(1minus ψ1minus γ

)Et[V 1minusγt+1

] 1minusψ1minusγ minus1

(1minus γ)V minusγt+1

partVt+1

partCt+1minus λM$

tt+1Pt+1 = 0 (C3)

FurthermorepartVt+1

partCt+1=

1

1minus ψ

[V 1minusψt+1

] 11minusψminus1

(1minus β)Cminusψt+1 (C4)

Finally combining (C1) (C3) and (C4) I obtain the intertemporal consumption optimality condition

λ(1minus ψ)

V ψt (1minus β)=CminusψtPt

= β

(Cminusψt+1

Pt+1

)(V ψminusγt+1

M$tt+1

)Et

[V

11minusγt+1

] γminusψ1minusγ

To get the nominal pricing kernel I solve for M$tt+1

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγ

C2 Aggregation

There is a continuum of intermediate goods firms j isin [0 1] producing differentiated output Yt(j) at price Pt(j)There is a representative final good producer that bundles the intermediate good into a final good via the aggregator

Y aggrt =

(int 1

0

Yt(j)ηminus1η dj

) ηηminus1

where η gt 1 is the elasticity of substitution among goods Following profit maximization by the final good producerthe first order condition gives the demand curve for each intermediate good

Yt(j) =

(Pt(j)

Pt

)minusηY aggrt (C5)

and the aggregate price index is

P 1minusηt =

int 1

0

Pt(j)1minusηdj

Integrating Equation (C5) over j to get the aggregation equation of output

Yt =

int 1

0

Yt(j)dj =

int 1

0

(Pt(j)

Pt

)minusηdj︸ ︷︷ ︸

Lpt

Y aggrt

4

where Lpt is the distortionary from price dispersion To deal with the integral we can use the property of Calvo(1983) such that only a α fraction of firms each period can optimally set their price to P lowastt

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj =

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusη (Ptminus1

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

(Ptminus1

Pt

)minusη int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusη int 1

0

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The resulting price dispersion is

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The aggregate price index can be calculated in a similar fashion

P 1minusηt =

int 1

0

Pt(j)1minusηdj =

int 1minusα

0

P lowastt1minusη

dj +

int 1

1minusαPtminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ α

int 1

0

Ptminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ αP 1minusηtminus1

which can be rewritten in the following price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

Finally aggregated output is

Yt = LptYaggrt

with market clearing condition

Y aggrt = Ct + Invt +Govt

C3 Loglinearized Phillips Curve

To linearize Ft and Jt we apply Taylor series expansion to the expectation terms in the following steps for

Equation () First define Υt = logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

] Then

Ft = 1 + αEt[Mnomtt+1

(Yt+1

Yt

)Πηt+1Ft+1

]Feft = 1 + αΥe

log Et[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]f + ft = log(1 + αΥeΥt)

= log(1 + αΥeΥ) +αΥeΥ

1 + αΥeΥ︸ ︷︷ ︸constf

(Υt minusΥ)

5

Notice a variable without a time subscript implies the non-stochastic steady state of the variable In steady state f= log(1 + αΥeΥ) so

ft = constfΥt minus constfΥ

= constf logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]minus constfΥ

= constf Et [mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1]

+1

2vart (mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1)

minus constfΥ

in which the last equality relies on the lognormality assumption

For Jt define Φt = logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

] then the same procedure

as above gives us the loglinearized Equation ()

jt

= constjΦt minus constjΦ

= constj logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

]minus constjΦ

= constjEt[mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

]+

1

2vart

(mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

)minusconstjΦ

where constj = αΦeΦ

1+αΦeΦ

C4 The System of Equations for the Model with Growth

We have a system of thirty-three equations resulting from equilibrium conditions first order conditions and policyrulesPricing kernel

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγValue function

Vt =

(1minus β)

(Ct

1minusψ

1minus ψ minus ϕ1minusψt

Nst

1+ω

1 + ω

)+ βEt

[V 1minusγt+1

] 1minusψ1minusγ

11minusψ

Fiscal rule

Taxt = τt + τkt RktKtminus1

τt = ρbDtminus1(t) + ρgGovt

Wage setting of the agent

Wt = ϕ(1minusψ)t Cψt N

stω

Production function

Yt = ZtKκtminus1(AtN

dt )1minusκ

6

Capital accumulation

Kt = ((1minus δ) + Φt)Ktminus1

Capital adjustment cost

Φt = b1 +b2

(1minus 1ζ)

(InvtKtminus1

)1minus1ζ

Φprimet = b2

(InvtKtminus1

)minus1ζ

Return on investment

1 = Et[Mtt+1RIt+1]

RIt qinvtminus1 = (1minus τkt )RKt + qinvt

(1minus δ + Φt minus Φprimet

InvtKtminus1

)1 = qinvt Φprimet

Aggregate labor supply and demand

Nst = Nd

t

Yt = LptYaggrt

Market clearing condition

Y aggrt = Ct + Invt +Govt

Government budget constraint

Dtminus1(t) = Taxt minusGovt + P realt Dt(t+ 1)

Capital labor ratio

Wt =(1minus κ)

κRKt

Ktminus1

Ndt

Optimal price setting[1

1minus α

(1minus α

(1

Πt

)(1minusη))] 1

(1minusη)

Ft =νκminusκ(1minus κ)minus(1minusκ)RKt

κW

(1minusκ)t Jt

ZtA1minusκt

Ft = 1 + αEt[Mnomtt+1

(Y aggrt+1

Y aggrt

)Πηt+1Ft+1

]Jt = 1 + αEt

[Mnomtt+1

(ZtZt+1

)(AtAt+1

)1minusκ(RKt+1

RKt

)κ(Wt+1

Wt

)(1minusκ)(Y aggrt+1

Y aggrt

)Π

(1+η)t+1 Jt+1

]Price dispersion

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

Price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

7

Nominal pricing kernel

Mnomtminus1t =

Mtminus1t

Πt(C6)

Euler equation

1

R(1)t

= Et[Mnomtt+1] (C7)

Real bond price

P realt = Et[Mtt+1] (C8)

Taylor rule

R(1)t

R=

(R

(1)tminus1

R

)ρr (Πt

Πlowast

)(1minusρr)ρπ(

Y aggrt AtY aggrtminus1 Atminus1

)(1minusρr)ρx

eut (C9)

where gt ut and zt are exogenous shocks to government spending monetary policy and productivity respectively

gt+1 = (1minus φg)θg + φggt + φgd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φgy log

(Y aggrt

Y aggr

)+ eσgt+1εgt+1

σgt+1 = (1minus φσg )θσg + φσgσgt + σσg εgσt+1

τkt+1 = (1minus φτk )θτk + φτkτkt + φτkd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φτky log

(Y aggrt

Y aggr

)+ e

στkt+1ετkt+1

στkt+1 = (1minus φστk

)θστk

+ φστkστkt + σσ

τkεσt+1

zt+1 = φzzt + eσzt+1εzt+1

σzt+1 = (1minus φzσ)θzσ + φzσσzt + σzσεzσt+1

∆at+1 = (1minus φa)ga + φa∆at + σaεat+1

ut+1 = σuεut+1

Finally balanced growth is achieved by specifying ϕt to be cointegrated with At as in Colacito Croce Ho andHoward (2017) in the following recursive process

log

(ϕtAt

)= φϕ logϕ+ (1minus φϕ)ga minus (1minus φϕ)

[∆at minus log

(ϕtminus1

Atminus1

)]

φϕ is calibrated to be 01

D VAR Analysis

All VARs are estimated in levels with two lags of each variable an intercept term and a time trend None ofthe results changes if we use a VAR with four lags an intercept term but no time trend Also in the empiricalanalysis we proxy for the nominal price level with GDP deflator proxying for the nominal price level with the BLSconsumer price index delivers almost identical results Similarly replacing the 5-year yield with the 10-year yielddelivers identical results Finally to improve precision we impose a Minnesota prior (see Hamilton 1994 p 360) onthe estimation and compute confidence bands by drawing from the posterior

As a preliminary check we investigate the forecast of inflation implied by our VAR Inflation is a key variable inour model and in the data covariances between shocks to current and expected inflation and bond prices determinethe sign and magnitude of bond risk premia Moreover for our analysis of the variance ratio in Section to be

8

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

(ii) log investment growth ∆invt (iii) log consumption growth ∆ct (iv) inflation πt (v) the one quarter nominal

interest rate rt (vi) the ten year nominal interest rate y(40)t and (vii) the slope of the term structure y

(40)t minus rt All

series are stored in datat which is of dimension 7 times 13 Our sample goes from 1970Q1 to 2016Q4 hence the matrixdata which is of dimension 7 times 188 We then define the following vector

qt =

datatdiag (datatdataprimet)vech (datatdataprimet)

Moreover let θθθ be a vector that contains the structural parameters Our GMM estimator is then given by

θθθGMM = argminθθθisinΘΘΘ

(1

T

Tsumt=1

qqqt minusm (θθθ)

)primeWWW

(1

T

Tsumt=1

qqqt minusm (θθθ)

)(B1)

Here W is a positive definite weighting matrix and m (θθθ) is a vector that contains the model-implied unconditionalmoments computed in closed-form as described above We use the conventional two-step implementation of GMM

by letting WWWT = diag(SSSminus1

)in a preliminary first step to obtain θθθstep 1 where SSS denotes the long-run variance

of 1T

sumTt=1 qqqt when re-centered around its sample mean Our final estimates θθθstep 2 are obtained using the optimal

weighting matrix WWWT = SSSminus1

θθθstep 1 where SSSθθθstep 1 denotes the long-run variance of our moments re-centered around

m(θθθstep 1

) The long-run variances in both steps are estimated by the Newey-West estimator using 10 lags but our

results are robust to using more lags

To summarize the estimation procedure implemented by GMM is as follows

bull 1 Step Let WWWT = diag(SSSminus1

)and obtain θθθstep 1 from B1

bull 2 Step Use θθθstep 1 to compute WWWT = SSSminus1

θθθstep 1 and obtain θθθstep 2 from B1

C Solving the Benchmark Model

C1 Households with Epstein-Zin Preference

The agentrsquos optimization problem is

max V (Ct Nst ) =

(1minus β)U(Ct N

st )1minusψ + βEt

[V 1minusγt+1

] 1minusψ1minusγ

11minusψ

st Et

[infinsums=0

M$tt+sPt+sCt+s

]le Et

[infinsums=0

M$tt+s(Wt+sPt+sN

st+s minus Pt+sTt+s + Pt+sΨt+s)

]

where

Ct =

[int 1

0

Ct(j)θminus1θ dj

] θθminus1

and

U(Ct Nst ) =

[C1minusψt

1minus ψ minus ϕ1minusψt

Nst

1+ω

1 + ω

] 11minusψ

3We have also repeated our procedure adding to the first and second moments used in the baseline estimationthe first and fifth autocovariances to capture the persistence in the data Our point estimates do not significantlychange and the conclusion from model-implied moments remain qualitatively the same

3

The first order conditions are

partVtpartCt

[V 1minusψt

] 11minusψminus1

1minus ψ (1minus β)Cminusψt minus λM$ttPt = 0 (C1)

partVtpartNs

t

[V 1minusψt

] 11minusψminus1

1minus ψ (1minus β)(minusϕ1minusψt Ns

tω) + λM$

ttWtPt = 0 (C2)

partVtpartCt+1

[V 1minusψt

] 11minusψminus1

1minus ψ β

(1minus ψ1minus γ

)Et[V 1minusγt+1

] 1minusψ1minusγ minus1

(1minus γ)V minusγt+1

partVt+1

partCt+1minus λM$

tt+1Pt+1 = 0 (C3)

FurthermorepartVt+1

partCt+1=

1

1minus ψ

[V 1minusψt+1

] 11minusψminus1

(1minus β)Cminusψt+1 (C4)

Finally combining (C1) (C3) and (C4) I obtain the intertemporal consumption optimality condition

λ(1minus ψ)

V ψt (1minus β)=CminusψtPt

= β

(Cminusψt+1

Pt+1

)(V ψminusγt+1

M$tt+1

)Et

[V

11minusγt+1

] γminusψ1minusγ

To get the nominal pricing kernel I solve for M$tt+1

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγ

C2 Aggregation

There is a continuum of intermediate goods firms j isin [0 1] producing differentiated output Yt(j) at price Pt(j)There is a representative final good producer that bundles the intermediate good into a final good via the aggregator

Y aggrt =

(int 1

0

Yt(j)ηminus1η dj

) ηηminus1

where η gt 1 is the elasticity of substitution among goods Following profit maximization by the final good producerthe first order condition gives the demand curve for each intermediate good

Yt(j) =

(Pt(j)

Pt

)minusηY aggrt (C5)

and the aggregate price index is

P 1minusηt =

int 1

0

Pt(j)1minusηdj

Integrating Equation (C5) over j to get the aggregation equation of output

Yt =

int 1

0

Yt(j)dj =

int 1

0

(Pt(j)

Pt

)minusηdj︸ ︷︷ ︸

Lpt

Y aggrt

4

where Lpt is the distortionary from price dispersion To deal with the integral we can use the property of Calvo(1983) such that only a α fraction of firms each period can optimally set their price to P lowastt

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj =

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusη (Ptminus1

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

(Ptminus1

Pt

)minusη int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusη int 1

0

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The resulting price dispersion is

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The aggregate price index can be calculated in a similar fashion

P 1minusηt =

int 1

0

Pt(j)1minusηdj =

int 1minusα

0

P lowastt1minusη

dj +

int 1

1minusαPtminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ α

int 1

0

Ptminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ αP 1minusηtminus1

which can be rewritten in the following price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

Finally aggregated output is

Yt = LptYaggrt

with market clearing condition

Y aggrt = Ct + Invt +Govt

C3 Loglinearized Phillips Curve

To linearize Ft and Jt we apply Taylor series expansion to the expectation terms in the following steps for

Equation () First define Υt = logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

] Then

Ft = 1 + αEt[Mnomtt+1

(Yt+1

Yt

)Πηt+1Ft+1

]Feft = 1 + αΥe

log Et[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]f + ft = log(1 + αΥeΥt)

= log(1 + αΥeΥ) +αΥeΥ

1 + αΥeΥ︸ ︷︷ ︸constf

(Υt minusΥ)

5

Notice a variable without a time subscript implies the non-stochastic steady state of the variable In steady state f= log(1 + αΥeΥ) so

ft = constfΥt minus constfΥ

= constf logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]minus constfΥ

= constf Et [mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1]

+1

2vart (mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1)

minus constfΥ

in which the last equality relies on the lognormality assumption

For Jt define Φt = logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

] then the same procedure

as above gives us the loglinearized Equation ()

jt

= constjΦt minus constjΦ

= constj logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

]minus constjΦ

= constjEt[mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

]+

1

2vart

(mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

)minusconstjΦ

where constj = αΦeΦ

1+αΦeΦ

C4 The System of Equations for the Model with Growth

We have a system of thirty-three equations resulting from equilibrium conditions first order conditions and policyrulesPricing kernel

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγValue function

Vt =

(1minus β)

(Ct

1minusψ

1minus ψ minus ϕ1minusψt

Nst

1+ω

1 + ω

)+ βEt

[V 1minusγt+1

] 1minusψ1minusγ

11minusψ

Fiscal rule

Taxt = τt + τkt RktKtminus1

τt = ρbDtminus1(t) + ρgGovt

Wage setting of the agent

Wt = ϕ(1minusψ)t Cψt N

stω

Production function

Yt = ZtKκtminus1(AtN

dt )1minusκ

6

Capital accumulation

Kt = ((1minus δ) + Φt)Ktminus1

Capital adjustment cost

Φt = b1 +b2

(1minus 1ζ)

(InvtKtminus1

)1minus1ζ

Φprimet = b2

(InvtKtminus1

)minus1ζ

Return on investment

1 = Et[Mtt+1RIt+1]

RIt qinvtminus1 = (1minus τkt )RKt + qinvt

(1minus δ + Φt minus Φprimet

InvtKtminus1

)1 = qinvt Φprimet

Aggregate labor supply and demand

Nst = Nd

t

Yt = LptYaggrt

Market clearing condition

Y aggrt = Ct + Invt +Govt

Government budget constraint

Dtminus1(t) = Taxt minusGovt + P realt Dt(t+ 1)

Capital labor ratio

Wt =(1minus κ)

κRKt

Ktminus1

Ndt

Optimal price setting[1

1minus α

(1minus α

(1

Πt

)(1minusη))] 1

(1minusη)

Ft =νκminusκ(1minus κ)minus(1minusκ)RKt

κW

(1minusκ)t Jt

ZtA1minusκt

Ft = 1 + αEt[Mnomtt+1

(Y aggrt+1

Y aggrt

)Πηt+1Ft+1

]Jt = 1 + αEt

[Mnomtt+1

(ZtZt+1

)(AtAt+1

)1minusκ(RKt+1

RKt

)κ(Wt+1

Wt

)(1minusκ)(Y aggrt+1

Y aggrt

)Π

(1+η)t+1 Jt+1

]Price dispersion

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

Price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

7

Nominal pricing kernel

Mnomtminus1t =

Mtminus1t

Πt(C6)

Euler equation

1

R(1)t

= Et[Mnomtt+1] (C7)

Real bond price

P realt = Et[Mtt+1] (C8)

Taylor rule

R(1)t

R=

(R

(1)tminus1

R

)ρr (Πt

Πlowast

)(1minusρr)ρπ(

Y aggrt AtY aggrtminus1 Atminus1

)(1minusρr)ρx

eut (C9)

where gt ut and zt are exogenous shocks to government spending monetary policy and productivity respectively

gt+1 = (1minus φg)θg + φggt + φgd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φgy log

(Y aggrt

Y aggr

)+ eσgt+1εgt+1

σgt+1 = (1minus φσg )θσg + φσgσgt + σσg εgσt+1

τkt+1 = (1minus φτk )θτk + φτkτkt + φτkd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φτky log

(Y aggrt

Y aggr

)+ e

στkt+1ετkt+1

στkt+1 = (1minus φστk

)θστk

+ φστkστkt + σσ

τkεσt+1

zt+1 = φzzt + eσzt+1εzt+1

σzt+1 = (1minus φzσ)θzσ + φzσσzt + σzσεzσt+1

∆at+1 = (1minus φa)ga + φa∆at + σaεat+1

ut+1 = σuεut+1

Finally balanced growth is achieved by specifying ϕt to be cointegrated with At as in Colacito Croce Ho andHoward (2017) in the following recursive process

log

(ϕtAt

)= φϕ logϕ+ (1minus φϕ)ga minus (1minus φϕ)

[∆at minus log

(ϕtminus1

Atminus1

)]

φϕ is calibrated to be 01

D VAR Analysis

All VARs are estimated in levels with two lags of each variable an intercept term and a time trend None ofthe results changes if we use a VAR with four lags an intercept term but no time trend Also in the empiricalanalysis we proxy for the nominal price level with GDP deflator proxying for the nominal price level with the BLSconsumer price index delivers almost identical results Similarly replacing the 5-year yield with the 10-year yielddelivers identical results Finally to improve precision we impose a Minnesota prior (see Hamilton 1994 p 360) onthe estimation and compute confidence bands by drawing from the posterior

As a preliminary check we investigate the forecast of inflation implied by our VAR Inflation is a key variable inour model and in the data covariances between shocks to current and expected inflation and bond prices determinethe sign and magnitude of bond risk premia Moreover for our analysis of the variance ratio in Section to be

8

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

The first order conditions are

partVtpartCt

[V 1minusψt

] 11minusψminus1

1minus ψ (1minus β)Cminusψt minus λM$ttPt = 0 (C1)

partVtpartNs

t

[V 1minusψt

] 11minusψminus1

1minus ψ (1minus β)(minusϕ1minusψt Ns

tω) + λM$

ttWtPt = 0 (C2)

partVtpartCt+1

[V 1minusψt

] 11minusψminus1

1minus ψ β

(1minus ψ1minus γ

)Et[V 1minusγt+1

] 1minusψ1minusγ minus1

(1minus γ)V minusγt+1

partVt+1

partCt+1minus λM$

tt+1Pt+1 = 0 (C3)

FurthermorepartVt+1

partCt+1=

1

1minus ψ

[V 1minusψt+1

] 11minusψminus1

(1minus β)Cminusψt+1 (C4)

Finally combining (C1) (C3) and (C4) I obtain the intertemporal consumption optimality condition

λ(1minus ψ)

V ψt (1minus β)=CminusψtPt

= β

(Cminusψt+1

Pt+1

)(V ψminusγt+1

M$tt+1

)Et

[V

11minusγt+1

] γminusψ1minusγ

To get the nominal pricing kernel I solve for M$tt+1

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγ

C2 Aggregation

There is a continuum of intermediate goods firms j isin [0 1] producing differentiated output Yt(j) at price Pt(j)There is a representative final good producer that bundles the intermediate good into a final good via the aggregator

Y aggrt =

(int 1

0

Yt(j)ηminus1η dj

) ηηminus1

where η gt 1 is the elasticity of substitution among goods Following profit maximization by the final good producerthe first order condition gives the demand curve for each intermediate good

Yt(j) =

(Pt(j)

Pt

)minusηY aggrt (C5)

and the aggregate price index is

P 1minusηt =

int 1

0

Pt(j)1minusηdj

Integrating Equation (C5) over j to get the aggregation equation of output

Yt =

int 1

0

Yt(j)dj =

int 1

0

(Pt(j)

Pt

)minusηdj︸ ︷︷ ︸

Lpt

Y aggrt

4

where Lpt is the distortionary from price dispersion To deal with the integral we can use the property of Calvo(1983) such that only a α fraction of firms each period can optimally set their price to P lowastt

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj =

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusη (Ptminus1

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

(Ptminus1

Pt

)minusη int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusη int 1

0

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The resulting price dispersion is

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The aggregate price index can be calculated in a similar fashion

P 1minusηt =

int 1

0

Pt(j)1minusηdj =

int 1minusα

0

P lowastt1minusη

dj +

int 1

1minusαPtminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ α

int 1

0

Ptminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ αP 1minusηtminus1

which can be rewritten in the following price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

Finally aggregated output is

Yt = LptYaggrt

with market clearing condition

Y aggrt = Ct + Invt +Govt

C3 Loglinearized Phillips Curve

To linearize Ft and Jt we apply Taylor series expansion to the expectation terms in the following steps for

Equation () First define Υt = logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

] Then

Ft = 1 + αEt[Mnomtt+1

(Yt+1

Yt

)Πηt+1Ft+1

]Feft = 1 + αΥe

log Et[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]f + ft = log(1 + αΥeΥt)

= log(1 + αΥeΥ) +αΥeΥ

1 + αΥeΥ︸ ︷︷ ︸constf

(Υt minusΥ)

5

Notice a variable without a time subscript implies the non-stochastic steady state of the variable In steady state f= log(1 + αΥeΥ) so

ft = constfΥt minus constfΥ

= constf logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]minus constfΥ

= constf Et [mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1]

+1

2vart (mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1)

minus constfΥ

in which the last equality relies on the lognormality assumption

For Jt define Φt = logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

] then the same procedure

as above gives us the loglinearized Equation ()

jt

= constjΦt minus constjΦ

= constj logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

]minus constjΦ

= constjEt[mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

]+

1

2vart

(mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

)minusconstjΦ

where constj = αΦeΦ

1+αΦeΦ

C4 The System of Equations for the Model with Growth

We have a system of thirty-three equations resulting from equilibrium conditions first order conditions and policyrulesPricing kernel

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγValue function

Vt =

(1minus β)

(Ct

1minusψ

1minus ψ minus ϕ1minusψt

Nst

1+ω

1 + ω

)+ βEt

[V 1minusγt+1

] 1minusψ1minusγ

11minusψ

Fiscal rule

Taxt = τt + τkt RktKtminus1

τt = ρbDtminus1(t) + ρgGovt

Wage setting of the agent

Wt = ϕ(1minusψ)t Cψt N

stω

Production function

Yt = ZtKκtminus1(AtN

dt )1minusκ

6

Capital accumulation

Kt = ((1minus δ) + Φt)Ktminus1

Capital adjustment cost

Φt = b1 +b2

(1minus 1ζ)

(InvtKtminus1

)1minus1ζ

Φprimet = b2

(InvtKtminus1

)minus1ζ

Return on investment

1 = Et[Mtt+1RIt+1]

RIt qinvtminus1 = (1minus τkt )RKt + qinvt

(1minus δ + Φt minus Φprimet

InvtKtminus1

)1 = qinvt Φprimet

Aggregate labor supply and demand

Nst = Nd

t

Yt = LptYaggrt

Market clearing condition

Y aggrt = Ct + Invt +Govt

Government budget constraint

Dtminus1(t) = Taxt minusGovt + P realt Dt(t+ 1)

Capital labor ratio

Wt =(1minus κ)

κRKt

Ktminus1

Ndt

Optimal price setting[1

1minus α

(1minus α

(1

Πt

)(1minusη))] 1

(1minusη)

Ft =νκminusκ(1minus κ)minus(1minusκ)RKt

κW

(1minusκ)t Jt

ZtA1minusκt

Ft = 1 + αEt[Mnomtt+1

(Y aggrt+1

Y aggrt

)Πηt+1Ft+1

]Jt = 1 + αEt

[Mnomtt+1

(ZtZt+1

)(AtAt+1

)1minusκ(RKt+1

RKt

)κ(Wt+1

Wt

)(1minusκ)(Y aggrt+1

Y aggrt

)Π

(1+η)t+1 Jt+1

]Price dispersion

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

Price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

7

Nominal pricing kernel

Mnomtminus1t =

Mtminus1t

Πt(C6)

Euler equation

1

R(1)t

= Et[Mnomtt+1] (C7)

Real bond price

P realt = Et[Mtt+1] (C8)

Taylor rule

R(1)t

R=

(R

(1)tminus1

R

)ρr (Πt

Πlowast

)(1minusρr)ρπ(

Y aggrt AtY aggrtminus1 Atminus1

)(1minusρr)ρx

eut (C9)

where gt ut and zt are exogenous shocks to government spending monetary policy and productivity respectively

gt+1 = (1minus φg)θg + φggt + φgd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φgy log

(Y aggrt

Y aggr

)+ eσgt+1εgt+1

σgt+1 = (1minus φσg )θσg + φσgσgt + σσg εgσt+1

τkt+1 = (1minus φτk )θτk + φτkτkt + φτkd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φτky log

(Y aggrt

Y aggr

)+ e

στkt+1ετkt+1

στkt+1 = (1minus φστk

)θστk

+ φστkστkt + σσ

τkεσt+1

zt+1 = φzzt + eσzt+1εzt+1

σzt+1 = (1minus φzσ)θzσ + φzσσzt + σzσεzσt+1

∆at+1 = (1minus φa)ga + φa∆at + σaεat+1

ut+1 = σuεut+1

Finally balanced growth is achieved by specifying ϕt to be cointegrated with At as in Colacito Croce Ho andHoward (2017) in the following recursive process

log

(ϕtAt

)= φϕ logϕ+ (1minus φϕ)ga minus (1minus φϕ)

[∆at minus log

(ϕtminus1

Atminus1

)]

φϕ is calibrated to be 01

D VAR Analysis

All VARs are estimated in levels with two lags of each variable an intercept term and a time trend None ofthe results changes if we use a VAR with four lags an intercept term but no time trend Also in the empiricalanalysis we proxy for the nominal price level with GDP deflator proxying for the nominal price level with the BLSconsumer price index delivers almost identical results Similarly replacing the 5-year yield with the 10-year yielddelivers identical results Finally to improve precision we impose a Minnesota prior (see Hamilton 1994 p 360) onthe estimation and compute confidence bands by drawing from the posterior

As a preliminary check we investigate the forecast of inflation implied by our VAR Inflation is a key variable inour model and in the data covariances between shocks to current and expected inflation and bond prices determinethe sign and magnitude of bond risk premia Moreover for our analysis of the variance ratio in Section to be

8

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

where Lpt is the distortionary from price dispersion To deal with the integral we can use the property of Calvo(1983) such that only a α fraction of firms each period can optimally set their price to P lowastt

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj =

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusη (Ptminus1

Pt

)minusηdj

=

int 1minusα

0

(P lowasttPt

)minusηdj +

(Ptminus1

Pt

)minusη int 1

1minusα

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusη int 1

0

(Ptminus1(j)

Ptminus1

)minusηdj

= (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The resulting price dispersion is

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

The aggregate price index can be calculated in a similar fashion

P 1minusηt =

int 1

0

Pt(j)1minusηdj =

int 1minusα

0

P lowastt1minusη

dj +

int 1

1minusαPtminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ α

int 1

0

Ptminus1(j)1minusηdj

= (1minus α)P lowastt1minusη

+ αP 1minusηtminus1

which can be rewritten in the following price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

Finally aggregated output is

Yt = LptYaggrt

with market clearing condition

Y aggrt = Ct + Invt +Govt

C3 Loglinearized Phillips Curve

To linearize Ft and Jt we apply Taylor series expansion to the expectation terms in the following steps for

Equation () First define Υt = logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

] Then

Ft = 1 + αEt[Mnomtt+1

(Yt+1

Yt

)Πηt+1Ft+1

]Feft = 1 + αΥe

log Et[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]f + ft = log(1 + αΥeΥt)

= log(1 + αΥeΥ) +αΥeΥ

1 + αΥeΥ︸ ︷︷ ︸constf

(Υt minusΥ)

5

Notice a variable without a time subscript implies the non-stochastic steady state of the variable In steady state f= log(1 + αΥeΥ) so

ft = constfΥt minus constfΥ

= constf logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]minus constfΥ

= constf Et [mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1]

+1

2vart (mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1)

minus constfΥ

in which the last equality relies on the lognormality assumption

For Jt define Φt = logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

] then the same procedure

as above gives us the loglinearized Equation ()

jt

= constjΦt minus constjΦ

= constj logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

]minus constjΦ

= constjEt[mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

]+

1

2vart

(mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

)minusconstjΦ

where constj = αΦeΦ

1+αΦeΦ

C4 The System of Equations for the Model with Growth

We have a system of thirty-three equations resulting from equilibrium conditions first order conditions and policyrulesPricing kernel

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγValue function

Vt =

(1minus β)

(Ct

1minusψ

1minus ψ minus ϕ1minusψt

Nst

1+ω

1 + ω

)+ βEt

[V 1minusγt+1

] 1minusψ1minusγ

11minusψ

Fiscal rule

Taxt = τt + τkt RktKtminus1

τt = ρbDtminus1(t) + ρgGovt

Wage setting of the agent

Wt = ϕ(1minusψ)t Cψt N

stω

Production function

Yt = ZtKκtminus1(AtN

dt )1minusκ

6

Capital accumulation

Kt = ((1minus δ) + Φt)Ktminus1

Capital adjustment cost

Φt = b1 +b2

(1minus 1ζ)

(InvtKtminus1

)1minus1ζ

Φprimet = b2

(InvtKtminus1

)minus1ζ

Return on investment

1 = Et[Mtt+1RIt+1]

RIt qinvtminus1 = (1minus τkt )RKt + qinvt

(1minus δ + Φt minus Φprimet

InvtKtminus1

)1 = qinvt Φprimet

Aggregate labor supply and demand

Nst = Nd

t

Yt = LptYaggrt

Market clearing condition

Y aggrt = Ct + Invt +Govt

Government budget constraint

Dtminus1(t) = Taxt minusGovt + P realt Dt(t+ 1)

Capital labor ratio

Wt =(1minus κ)

κRKt

Ktminus1

Ndt

Optimal price setting[1

1minus α

(1minus α

(1

Πt

)(1minusη))] 1

(1minusη)

Ft =νκminusκ(1minus κ)minus(1minusκ)RKt

κW

(1minusκ)t Jt

ZtA1minusκt

Ft = 1 + αEt[Mnomtt+1

(Y aggrt+1

Y aggrt

)Πηt+1Ft+1

]Jt = 1 + αEt

[Mnomtt+1

(ZtZt+1

)(AtAt+1

)1minusκ(RKt+1

RKt

)κ(Wt+1

Wt

)(1minusκ)(Y aggrt+1

Y aggrt

)Π

(1+η)t+1 Jt+1

]Price dispersion

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

Price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

7

Nominal pricing kernel

Mnomtminus1t =

Mtminus1t

Πt(C6)

Euler equation

1

R(1)t

= Et[Mnomtt+1] (C7)

Real bond price

P realt = Et[Mtt+1] (C8)

Taylor rule

R(1)t

R=

(R

(1)tminus1

R

)ρr (Πt

Πlowast

)(1minusρr)ρπ(

Y aggrt AtY aggrtminus1 Atminus1

)(1minusρr)ρx

eut (C9)

where gt ut and zt are exogenous shocks to government spending monetary policy and productivity respectively

gt+1 = (1minus φg)θg + φggt + φgd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φgy log

(Y aggrt

Y aggr

)+ eσgt+1εgt+1

σgt+1 = (1minus φσg )θσg + φσgσgt + σσg εgσt+1

τkt+1 = (1minus φτk )θτk + φτkτkt + φτkd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φτky log

(Y aggrt

Y aggr

)+ e

στkt+1ετkt+1

στkt+1 = (1minus φστk

)θστk

+ φστkστkt + σσ

τkεσt+1

zt+1 = φzzt + eσzt+1εzt+1

σzt+1 = (1minus φzσ)θzσ + φzσσzt + σzσεzσt+1

∆at+1 = (1minus φa)ga + φa∆at + σaεat+1

ut+1 = σuεut+1

Finally balanced growth is achieved by specifying ϕt to be cointegrated with At as in Colacito Croce Ho andHoward (2017) in the following recursive process

log

(ϕtAt

)= φϕ logϕ+ (1minus φϕ)ga minus (1minus φϕ)

[∆at minus log

(ϕtminus1

Atminus1

)]

φϕ is calibrated to be 01

D VAR Analysis

All VARs are estimated in levels with two lags of each variable an intercept term and a time trend None ofthe results changes if we use a VAR with four lags an intercept term but no time trend Also in the empiricalanalysis we proxy for the nominal price level with GDP deflator proxying for the nominal price level with the BLSconsumer price index delivers almost identical results Similarly replacing the 5-year yield with the 10-year yielddelivers identical results Finally to improve precision we impose a Minnesota prior (see Hamilton 1994 p 360) onthe estimation and compute confidence bands by drawing from the posterior

As a preliminary check we investigate the forecast of inflation implied by our VAR Inflation is a key variable inour model and in the data covariances between shocks to current and expected inflation and bond prices determinethe sign and magnitude of bond risk premia Moreover for our analysis of the variance ratio in Section to be

8

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Notice a variable without a time subscript implies the non-stochastic steady state of the variable In steady state f= log(1 + αΥeΥ) so

ft = constfΥt minus constfΥ

= constf logEt[emtt+1+∆yt+1+∆at+1+(ηminus1)πt+1+ft+1

]minus constfΥ

= constf Et [mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1]

+1

2vart (mtt+1 + ∆yt+1 + ∆at+1 + (η minus 1)πt+1 + ft+1)

minus constfΥ

in which the last equality relies on the lognormality assumption

For Jt define Φt = logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

] then the same procedure

as above gives us the loglinearized Equation ()

jt

= constjΦt minus constjΦ

= constj logEt[emtt+1minus∆zt+1+κ∆rKt+1+(1minusκ)∆wt+1+∆yt+1+∆at+1+ηπt+1+jt+1

]minus constjΦ

= constjEt[mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

]+

1

2vart

(mtt+1 minus∆zt+1 + κ∆rKt+1 + (1minus κ)∆wt+1 + ∆yt+1 + ∆at+1 + ηπt+1 + jt+1

)minusconstjΦ

where constj = αΦeΦ

1+αΦeΦ

C4 The System of Equations for the Model with Growth

We have a system of thirty-three equations resulting from equilibrium conditions first order conditions and policyrulesPricing kernel

M$tt+1 = β

(Ct+1

Ct

)minusψ (Pt+1

Pt

)minus1[

Vt+1

Et[V1minusγt+1 ]

11minusγ

]ψminusγValue function

Vt =

(1minus β)

(Ct

1minusψ

1minus ψ minus ϕ1minusψt

Nst

1+ω

1 + ω

)+ βEt

[V 1minusγt+1

] 1minusψ1minusγ

11minusψ

Fiscal rule

Taxt = τt + τkt RktKtminus1

τt = ρbDtminus1(t) + ρgGovt

Wage setting of the agent

Wt = ϕ(1minusψ)t Cψt N

stω

Production function

Yt = ZtKκtminus1(AtN

dt )1minusκ

6

Capital accumulation

Kt = ((1minus δ) + Φt)Ktminus1

Capital adjustment cost

Φt = b1 +b2

(1minus 1ζ)

(InvtKtminus1

)1minus1ζ

Φprimet = b2

(InvtKtminus1

)minus1ζ

Return on investment

1 = Et[Mtt+1RIt+1]

RIt qinvtminus1 = (1minus τkt )RKt + qinvt

(1minus δ + Φt minus Φprimet

InvtKtminus1

)1 = qinvt Φprimet

Aggregate labor supply and demand

Nst = Nd

t

Yt = LptYaggrt

Market clearing condition

Y aggrt = Ct + Invt +Govt

Government budget constraint

Dtminus1(t) = Taxt minusGovt + P realt Dt(t+ 1)

Capital labor ratio

Wt =(1minus κ)

κRKt

Ktminus1

Ndt

Optimal price setting[1

1minus α

(1minus α

(1

Πt

)(1minusη))] 1

(1minusη)

Ft =νκminusκ(1minus κ)minus(1minusκ)RKt

κW

(1minusκ)t Jt

ZtA1minusκt

Ft = 1 + αEt[Mnomtt+1

(Y aggrt+1

Y aggrt

)Πηt+1Ft+1

]Jt = 1 + αEt

[Mnomtt+1

(ZtZt+1

)(AtAt+1

)1minusκ(RKt+1

RKt

)κ(Wt+1

Wt

)(1minusκ)(Y aggrt+1

Y aggrt

)Π

(1+η)t+1 Jt+1

]Price dispersion

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

Price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

7

Nominal pricing kernel

Mnomtminus1t =

Mtminus1t

Πt(C6)

Euler equation

1

R(1)t

= Et[Mnomtt+1] (C7)

Real bond price

P realt = Et[Mtt+1] (C8)

Taylor rule

R(1)t

R=

(R

(1)tminus1

R

)ρr (Πt

Πlowast

)(1minusρr)ρπ(

Y aggrt AtY aggrtminus1 Atminus1

)(1minusρr)ρx

eut (C9)

where gt ut and zt are exogenous shocks to government spending monetary policy and productivity respectively

gt+1 = (1minus φg)θg + φggt + φgd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φgy log

(Y aggrt

Y aggr

)+ eσgt+1εgt+1

σgt+1 = (1minus φσg )θσg + φσgσgt + σσg εgσt+1

τkt+1 = (1minus φτk )θτk + φτkτkt + φτkd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φτky log

(Y aggrt

Y aggr

)+ e

στkt+1ετkt+1

στkt+1 = (1minus φστk

)θστk

+ φστkστkt + σσ

τkεσt+1

zt+1 = φzzt + eσzt+1εzt+1

σzt+1 = (1minus φzσ)θzσ + φzσσzt + σzσεzσt+1

∆at+1 = (1minus φa)ga + φa∆at + σaεat+1

ut+1 = σuεut+1

Finally balanced growth is achieved by specifying ϕt to be cointegrated with At as in Colacito Croce Ho andHoward (2017) in the following recursive process

log

(ϕtAt

)= φϕ logϕ+ (1minus φϕ)ga minus (1minus φϕ)

[∆at minus log

(ϕtminus1

Atminus1

)]

φϕ is calibrated to be 01

D VAR Analysis

All VARs are estimated in levels with two lags of each variable an intercept term and a time trend None ofthe results changes if we use a VAR with four lags an intercept term but no time trend Also in the empiricalanalysis we proxy for the nominal price level with GDP deflator proxying for the nominal price level with the BLSconsumer price index delivers almost identical results Similarly replacing the 5-year yield with the 10-year yielddelivers identical results Finally to improve precision we impose a Minnesota prior (see Hamilton 1994 p 360) onthe estimation and compute confidence bands by drawing from the posterior

As a preliminary check we investigate the forecast of inflation implied by our VAR Inflation is a key variable inour model and in the data covariances between shocks to current and expected inflation and bond prices determinethe sign and magnitude of bond risk premia Moreover for our analysis of the variance ratio in Section to be

8

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Capital accumulation

Kt = ((1minus δ) + Φt)Ktminus1

Capital adjustment cost

Φt = b1 +b2

(1minus 1ζ)

(InvtKtminus1

)1minus1ζ

Φprimet = b2

(InvtKtminus1

)minus1ζ

Return on investment

1 = Et[Mtt+1RIt+1]

RIt qinvtminus1 = (1minus τkt )RKt + qinvt

(1minus δ + Φt minus Φprimet

InvtKtminus1

)1 = qinvt Φprimet

Aggregate labor supply and demand

Nst = Nd

t

Yt = LptYaggrt

Market clearing condition

Y aggrt = Ct + Invt +Govt

Government budget constraint

Dtminus1(t) = Taxt minusGovt + P realt Dt(t+ 1)

Capital labor ratio

Wt =(1minus κ)

κRKt

Ktminus1

Ndt

Optimal price setting[1

1minus α

(1minus α

(1

Πt

)(1minusη))] 1

(1minusη)

Ft =νκminusκ(1minus κ)minus(1minusκ)RKt

κW

(1minusκ)t Jt

ZtA1minusκt

Ft = 1 + αEt[Mnomtt+1

(Y aggrt+1

Y aggrt

)Πηt+1Ft+1

]Jt = 1 + αEt

[Mnomtt+1

(ZtZt+1

)(AtAt+1

)1minusκ(RKt+1

RKt

)κ(Wt+1

Wt

)(1minusκ)(Y aggrt+1

Y aggrt

)Π

(1+η)t+1 Jt+1

]Price dispersion

Lpt =

int 1

0

(Pt(j)

Pt

)minusηdj = (1minus α)

(P lowasttPt

)minusη+ α

(Ptminus1

Pt

)minusηLptminus1

Price aggregator

1 = (1minus α)

(P lowasttPt

)1minusη

+ α

(Ptminus1

Pt

)1minusη

7

Nominal pricing kernel

Mnomtminus1t =

Mtminus1t

Πt(C6)

Euler equation

1

R(1)t

= Et[Mnomtt+1] (C7)

Real bond price

P realt = Et[Mtt+1] (C8)

Taylor rule

R(1)t

R=

(R

(1)tminus1

R

)ρr (Πt

Πlowast

)(1minusρr)ρπ(

Y aggrt AtY aggrtminus1 Atminus1

)(1minusρr)ρx

eut (C9)

where gt ut and zt are exogenous shocks to government spending monetary policy and productivity respectively

gt+1 = (1minus φg)θg + φggt + φgd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φgy log

(Y aggrt

Y aggr

)+ eσgt+1εgt+1

σgt+1 = (1minus φσg )θσg + φσgσgt + σσg εgσt+1

τkt+1 = (1minus φτk )θτk + φτkτkt + φτkd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φτky log

(Y aggrt

Y aggr

)+ e

στkt+1ετkt+1

στkt+1 = (1minus φστk

)θστk

+ φστkστkt + σσ

τkεσt+1

zt+1 = φzzt + eσzt+1εzt+1

σzt+1 = (1minus φzσ)θzσ + φzσσzt + σzσεzσt+1

∆at+1 = (1minus φa)ga + φa∆at + σaεat+1

ut+1 = σuεut+1

Finally balanced growth is achieved by specifying ϕt to be cointegrated with At as in Colacito Croce Ho andHoward (2017) in the following recursive process

log

(ϕtAt

)= φϕ logϕ+ (1minus φϕ)ga minus (1minus φϕ)

[∆at minus log

(ϕtminus1

Atminus1

)]

φϕ is calibrated to be 01

D VAR Analysis

All VARs are estimated in levels with two lags of each variable an intercept term and a time trend None ofthe results changes if we use a VAR with four lags an intercept term but no time trend Also in the empiricalanalysis we proxy for the nominal price level with GDP deflator proxying for the nominal price level with the BLSconsumer price index delivers almost identical results Similarly replacing the 5-year yield with the 10-year yielddelivers identical results Finally to improve precision we impose a Minnesota prior (see Hamilton 1994 p 360) onthe estimation and compute confidence bands by drawing from the posterior

As a preliminary check we investigate the forecast of inflation implied by our VAR Inflation is a key variable inour model and in the data covariances between shocks to current and expected inflation and bond prices determinethe sign and magnitude of bond risk premia Moreover for our analysis of the variance ratio in Section to be

8

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Nominal pricing kernel

Mnomtminus1t =

Mtminus1t

Πt(C6)

Euler equation

1

R(1)t

= Et[Mnomtt+1] (C7)

Real bond price

P realt = Et[Mtt+1] (C8)

Taylor rule

R(1)t

R=

(R

(1)tminus1

R

)ρr (Πt

Πlowast

)(1minusρr)ρπ(

Y aggrt AtY aggrtminus1 Atminus1

)(1minusρr)ρx

eut (C9)

where gt ut and zt are exogenous shocks to government spending monetary policy and productivity respectively

gt+1 = (1minus φg)θg + φggt + φgd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φgy log

(Y aggrt

Y aggr

)+ eσgt+1εgt+1

σgt+1 = (1minus φσg )θσg + φσgσgt + σσg εgσt+1

τkt+1 = (1minus φτk )θτk + φτkτkt + φτkd

(Dt(t+ 1)

Y aggrt

minus D

Y aggr

)+ φτky log

(Y aggrt

Y aggr

)+ e

στkt+1ετkt+1

στkt+1 = (1minus φστk

)θστk

+ φστkστkt + σσ

τkεσt+1

zt+1 = φzzt + eσzt+1εzt+1

σzt+1 = (1minus φzσ)θzσ + φzσσzt + σzσεzσt+1

∆at+1 = (1minus φa)ga + φa∆at + σaεat+1

ut+1 = σuεut+1

Finally balanced growth is achieved by specifying ϕt to be cointegrated with At as in Colacito Croce Ho andHoward (2017) in the following recursive process

log

(ϕtAt

)= φϕ logϕ+ (1minus φϕ)ga minus (1minus φϕ)

[∆at minus log

(ϕtminus1

Atminus1

)]

φϕ is calibrated to be 01

D VAR Analysis

All VARs are estimated in levels with two lags of each variable an intercept term and a time trend None ofthe results changes if we use a VAR with four lags an intercept term but no time trend Also in the empiricalanalysis we proxy for the nominal price level with GDP deflator proxying for the nominal price level with the BLSconsumer price index delivers almost identical results Similarly replacing the 5-year yield with the 10-year yielddelivers identical results Finally to improve precision we impose a Minnesota prior (see Hamilton 1994 p 360) onthe estimation and compute confidence bands by drawing from the posterior

As a preliminary check we investigate the forecast of inflation implied by our VAR Inflation is a key variable inour model and in the data covariances between shocks to current and expected inflation and bond prices determinethe sign and magnitude of bond risk premia Moreover for our analysis of the variance ratio in Section to be

8

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

valid it is important to verify that inflation forecasts implied by our dynamic VAR model are accurate in the sensethat they capture investor inflation expectations Figure D1 shows that this is indeed the case The dashed lineare forecasts of GDP (Panel A) and CPI (Panel B) inflation taken from the Philadelphia Fed Survey of ProfessionalForecasters (SPF) The solid line reports estimated inflation forecasts from the VAR model The figure documentsthat survey- and model-based forecasts of GDP and CPI inflation closely track each other

Figure D1 1- and 10-year ahead inflation forecasts from surveys and VAR model

1971 1976 1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(a) GDP deflator

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

9

10

Annu

aliz

ed p

erce

nt

(b) CPI

1982 1987 1993 1998 2004 2009 20150

1

2

3

4

5

6

7

8

Annu

aliz

ed p

erce

nt

Model-impliedSPF Expected CPI inflation

(c) CPI

The figure displays expected inflation over 1- Panels (a) and (b) and 10-years Panel (c) from the empirical VAR

(blue solid line) and from the SPF forecasts (red dashed line) The model underlying the solid line is the eight

variables VAR with two lags described in this Appendix The model uses GDP inflation in Panel (a) and CPI

inflation in Panels (b) and (c) These two series are contrasted with forecasts as of date t in the horizontal axis of

average GDP inflation Panel (a) and average CPI inflation Panels (b) and (c) CPI forecasts are unavailable prior

to 1981Q3 The SPF forecasts are not used in model estimation

To estimate the dynamic causal effects of level shocks to fiscal policy (government spending and capital income taxrates) we combine the structural VAR (SVAR) estimators with Instrumental Variable (IV) techniques Following theterminology in Stock and Watson (2017) we refer to this methodology as the SVAR-IV This method was introducedby Stock (2008) and has been used by Stock and Watson (2012) Mertens and Ravn (2013) Gertler and Karadi(2015) Ramey and Zubairy (2018) and a growing list of other researchers See also Ramey (2016) for a review Theintuition behind this approach is to find external instruments that are (1) contemporaneously correlated with the

9

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

structural policy shocks of interest (aka relevance condition) (2) contemporaneously uncorrelated with the otherstructural shocks (exogeneity condition) We refer to Mertens and Ravn (2013) Montiel Olea et al (2016) and Stockand Watson (2017) for a detailed econometric description of the SVAR-IV approach

Our instrument for government spending is the one-quarter ahead forecast revision of the growth rate of realfederal spending as implied by the SPF Importantly for our purpose Ramey (2011) shows that while a defensenews variable based on military spending is not very informative in a sample that excludes the WWII or the KoreanWar like our own a news variable based on professional forecasters is a powerful instrument for government spendingshocks in such a sample We also follow Perotti (2011) and we use forecast revisions rather than forecast errors SeeSection 54 in Perotti (2011) for an in-depth discussion More specifically let ft be the log of federal governmentspending and denote with fet|tminus1 the SPF expectation of federal spending We further define ∆fet|t = fet|t minus fetminus1|tThe revision of expectation of ∆ft = ft minus ftminus1 is given by ∆fet|t minus ∆fet|tminus1 Our instrument is the residual of aregression of spending revision onto the output gap and federal surplus (see Auerbach 2003) Such a constructionof the instrument is essential to address the ldquoanticipationrdquo or ldquonon-fundamentalnessrdquo problem (see eg Lippi andReichlin 1994)

Our instrument for capital tax rates is given by the narrative account of legislated federal corporate incometax liability changes in the United States developed by Mertens and Ravn (2013) To comply with the exogeneitycondition which requires that the instruments are orthogonal to all nontax structural shocks Mertens and Ravn(2013) follow the Romer and Romer (2009) approach and retain only those changes in tax liabilities that are unrelatedto the current state of the economy The final narrative measure contains 16 observations for corporate income taxliability changes Importantly the average corporate income tax rate used in the VAR by Mertens and Ravn (2013)has a high correlation (over the common sample) of about 92 with our capital tax rate series described in SectionA

Lastly to recover the uncertainty shocks we use a Cholesky decomposition with the following ordering four fiscalpolicy variables (gt σgt τ

k and στkt) output inflation the one-quarter yield and the 5-year yields Changing

the ordering of the fiscal instruments ie using (τk and στkt followed by gt σgt) does not affect the results Bothorderings are motivated by our view that the fiscal uncertainty shocks are exogenous This identification approachhas been used in the literature on uncertainty see eg Baker et al (2016) Basu and Bundick (2017) and Fernandez-Villaverde et al (2015)

E Additional Results

Table E1 reports a series of robustness checks for the main results of Table Each regression in TableE1 includes G G vol and MWDGDP and controls for variables that proxy for the state of the economy Morespecifically we include non-farm payroll output gap and GDP growth We also control for the CP (Cochraneand Piazzesi 2005) factor since Koijen et al (2017) show that it forecasts future economic activity at businesscycle horizons Finally to address the concern that each of these series can capture different aspects of economicgrowth we also include as a control variable a measure of ldquoReal activityrdquo which is obtained from more than 130macroeconomic and financial variables (Ludvigson and Ng 2009)4 For each specification where we control for thestate of the macroeconomy we also run a companion regression which - besides the macroeconomic state - controls forthe information from the term structure that is contained in first three principal components of the yield curve Thesole exception is the CP factor since this variable is already constructed from the yield curve The Table conveys anunequivocal message At two years maturity G and G vol are significant predictors of bond excess returns across allspecifications At long maturity G is again significant across all specifications and G vol is always significant except

4Ludvigson and Ng (2009) call the first principal component ldquoreal activityrdquo because it is highly correlated withstandard measures of real activity For example its correlation with log differenced industrial production exceeds08

10

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

when the slope (or a variable highly correlated with the slope like CP) is included among the control variables Thisis fully consistent with our analysis in the main text ldquo[] the correlation between the slope and the governmentspending uncertainty series makes it hard for OLS to discern between the two predictorsrdquo Interestingly even theinclusion of output gap (a very robust macro predictor of bond returns see Cooper and Priestley (2009)) does notoverturn the statistical significance of G vol (see specification (7) of Panels A and B) In fact using output gaptogether with government spending variables delivers an impressive R2 of 30 for long maturities (relative to a 17when only information from the term structure is included in the forecasting regression ndash see specification (6) in PanelB of Table )

Table E2 reports the pricing errors Each row of the table reports the error for a specific portfolio (the first sixrows refer to bond portfolios the seventh is the market the next 25 rows are the Fama-French book-to-market andsize portfolios) Each column reports a different model The first column contains the risk-neutral SDF and thereforereports the average pricing errors to be explained The model in the second column has the market return as theonly factor (MKT) The last three columns refer to our fiscal models the first includes only government spendinglevel the second includes exclusively government spending uncertainty and the last one includes both governmentspending level and uncertainty There are two important takeaways from this Table First with regard to the modelwhich includes only government spending level (specification 3) the portfolio error improves in 20 instances (out of32) when compared to the CAPM Second the model with both level and uncertainty (specification 5) is the bestmodel in 17 instances among all five candidate models Hence the improvement of the fiscal model over the CAPMand the improvement of the fiscal model with level and uncertainty relative to a model with just level or uncertaintyare not due to few outliers but rather due to an improvement across asset classes (bonds and stocks) and withinstocks across size and book-to-market quintiles

Moreover Table E3 adds industry portfolios to the cross-section of test assets used in Table This helpsbreaking the factor structure in book-to-market and size sorted portfolios Adding industry portfolios reduces the fitof our fiscal models only by 5 (the R2 in Panels B C and D of Table are 67 72 74 compare to 62 67 69in Table E3) without affecting the statistical significance of our fiscal factors Importantly the sampling variabilityof our cross-sectional R2 remains low across all specifications

Further Table E4 quantifies the contribution of each shock to the variability of macroeconomic and financialvariables by shutting down one shock at the time and examine the volatility of the endogenous variables PanelA shows that transitory productivity level shocks are an important driver of consumption and output volatilitieswhereas uncertainty shocks to transitory productivity contribute to inflation volatility Moreover government spend-ing and capital tax (level and uncertainty) shocks also generate sizeable effects on investment hours and inflation Inparticular government spending level and uncertainty shocks are significant drivers of the variability in hours Taxrate level and uncertainty shocks in turn have strong influence on the variability of investment Panel B of Table E4shows that uncertainty in government spending is a key driver of the variation in the slope of the term structure Allshocks are important drivers of nominal yields movements except for permanent productivity and monetary shocksTo summarize we find that stochastic volatility in government spending generates sizeable variation in the slope ofthe term structure without distorting the ability of the model to match key macroeconomic moments

Finally Table E5 reports the unconditional means of nominal and real yields when the model is simulated withall but one shocks active at the time Both transitory productivity and government spending uncertainty contributepositively to the slope of the nominal and real term structures in the model

Figure E1 reports the autocorrelation functions in the data and in the model Figure E2 reports the impulseresponse functions for structural shocks other than fiscal shocks in the model The four Panels show responses ofoutput price level nominal one quarter and nominal five year rates to one standard deviation shocks to transitoryproductivity level and uncertainty permanent productivity and monetary policy

Finally Figure E3 plots yield shock decompositions for the baseline model and two alternative models one withlow persistence in fiscal variables and another one without stochastic volatility in fiscal variables

11

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Table E1 Forecasting Excess Returns to Treasury Bonds 1970Q1 to 2016Q4 This table reportscoefficient estimates corresponding reverse regression p-values and R2s for regressions of annual excess returns ofTreasury bonds (for 2- and 5-year maturities) on fiscal variables an indicator variable for the zero lower bound andother predictors measured in quarter t The column F -test reports the p-value for the hypothesis that the fiscalvariables have jointly no incremental explanatory power beyond the other control variables Reverse regression p-values (in parentheses) are calculated using the delta method of Wei and Wright (2013) Control variables includethe maturity-weighted debt-to-GDP ratio MWDGDP (see Greenwood and Vayanos 2014) the first three PCs ofthe Treasury yield curve the first PC of many macroeconomic time series (LN) constructed by Ludvigson and Ng(2009) the CP (Cochrane and Piazzesi 2005) factor three measures of the state of the economy namely Non-FarmPayroll Output Gap and Output Growth Bold values indicate significance at least at the 10 level

Predictors

G G vol MWDGDP PC1 PC2 PC3 LN CP Payroll Output Gap Output Growth R2 F -test

Panel A Excess Returns on 2-year Treasury Bond

(1) 048 029 051 090 023(003) (005) (001) (015) (001)

(2) 064 025 054 059 011 -040 099 031(001) (005) (000) (016) (073) (011) (004) (000)

(3) 052 029 044 009 020(002) (004) (002) (047) (001)

(4) 044 030 053 -060 022(008) (005) (001) (041) (002)

(5) 060 026 057 064 008 -038 -069 030(002) (004) (000) (014) (086) (014) (027) (000)

(6) 066 035 054 020 020(003) (002) (002) (084) (002)

(7) 101 038 063 078 013 -030 100 029(000) (001) (000) (006) (066) (024) (031) (000)

(8) 054 032 053 -039 020(001) (004) (001) (066) (001)

(9) 076 031 060 063 002 -037 -031 028(000) (002) (000) (014) (095) (015) (077) (000)

Panel B Excess Returns on 5-year Treasury Bond

(1) 182 075 161 110 018(001) (005) (001) (054) (001)

(2) 159 038 124 110 133 -105 202 025(005) (021) (002) (037) (015) (022) (021) (003)

(3) 122 046 102 073 022(006) (018) (007) (007) (005)

(4) 179 075 164 -068 017(003) (005) (001) (083) (001)

(5) 154 040 131 120 127 -100 -135 025(006) (019) (001) (035) (018) (024) (053) (003)

(6) 249 100 180 220 018(001) (001) (001) (048) (000)

(7) 307 092 156 202 193 -061 612 030(000) (002) (001) (010) (003) (052) (006) (000)

(8) 207 080 162 042 017(000) (004) (001) (057) (000)

(9) 205 052 137 124 108 -097 023 024(001) (013) (001) (032) (026) (026) (057) (001)

12

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Table E2 Model for Stocks and Bonds Pricing Errors This table reports pricing errors for the 25 book-to-market and size sorted stock portfolios the market portfolio and six bond portfolios of maturities 1-2 2-3 3-44-5 5-10 and more than 10 years They are expressed in percent per year (quarterly numbers multiplied by 400)Each column corresponds to a different stochastic discount factor (SDF) model MAPE stands for the mean absolutepricing error Specification (1) column contains the risk-neutral SDF and therefore reports the average pricing errorsto be explained The SDF model of specification (2) has the market return as the only factor (MKT) Specification(3) presents the model including government spending level and the market Specification (4) presents the results forthe model with government spending uncertainty and the market Finally the last specification refers to the modelincluding government spending level and uncertainty and the market The sample is from 1970Q1 to 2016Q4

(1) (2) (3) (4) (5)RN SDF MKT MKT + G level MKT + G vola MKT + G level + G vola

1-2 yr 072 -168 -100 -142 -1412-3 yr 118 -146 -124 -149 -1253-4 yr 158 -122 -123 -122 -1214-5 yr 170 -123 -141 -125 -1195-10 yr 215 -117 -153 -105 -056gt 10 yr 332 -085 -138 -028 083

Market 649 -029 109 151 170

SG 274 -854 -680 -578 -505S12 1005 028 -039 043 131S13 1008 106 -102 -077 -073S14 1287 445 201 267 363SV 1425 525 239 190 1852G 603 -501 -500 -456 -41522 966 024 -191 -171 -12123 1075 194 129 066 01524 1183 319 102 024 -0202V 1236 340 021 -026 -0263G 644 -413 -315 -317 -34232 1001 073 117 087 05233 954 120 089 -021 -12134 1118 271 171 067 -0173V 1323 458 121 086 1014G 771 -256 077 114 10142 811 -087 -001 021 02843 907 037 133 147 14744 1042 185 130 092 0844V 1094 192 218 166 114BG 599 -293 106 101 039B2 759 -076 036 097 138B3 749 -004 294 286 238B4 660 -141 -071 -084 -106BV 882 101 384 397 369

MAPE 213 167 150 146

13

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Table E3 Pricing Model for Stocks and Bonds Robustness We estimate cross-sectional regressionswith and without a constant In particular the table reports results from running the cross-sectional regressionRei = (γ) +βiλ+αi where Rei is the mean excess return of portfolio i and βi is the vector of factor betas of portfolioi estimated in the first-pass regression We use the following test assets 25 equity portfolios sorted on size and book-to-market five industry portfolios the market portfolio (consisting of a value-weighted stock index and a long-termgovernment bond index) and six maturity-sorted Fama bond portfolios obtained from the CRSP The table reports

the estimates of the factor risk premia λ on the factors and the constant term Fama and MacBeth (1973) p-values(in parentheses) and the GMM-VARHAC p-values which account for sampling error in the betas (in braces) Thepenultimate column reports asymptotic p-values of chi-squared tests of the null hypothesis that all pricing errors arejointly zero (Pr err = 0) To compute the test statistic we use the OLS covariance matrix of α The last columnreports the R2 of the cross-sectional regression and for the model with the constant its standard error In additionwe also report the root mean square alpha (RMSE) and the mean absolute pricing error (MAPE) across all testassets These are expressed as percentages per year Return data is quarterly from 1970Q1 to 2016Q4 Bold valuesare significant at least at the 10 level

14

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Table E3 Pricing Model for Stocks and Bonds Robustness (continued)

Panel A Rei = (γ) + βiMKTλMKT + αi

Constant λMKT RMSE MAPE H0 Pr error = 0 p-value R2

0060 2786 1973 0024 037(0003)0005

0005 0046 2684 2019 0016 041(0097) (0067) (027)0099 0079

Panel B Rei = (γ) + βigλg + βiMKTλMKT + αi

Constant λg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-0967 0055 2193 1636 0024 060(0003) (0004)0041 0018

0002 -0920 0049 2170 1692 0015 062(0506) (0008) (0052) (027)0689 0079 0151

Panel C Rei = (γ) + βiσgλσg + βiMKTλMKT + αi

Constant λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

1164 0050 2058 1399 0000 066(0002) (0009)0064 0024

0003 1119 0041 2002 1495 0000 067(0319) (0004) (0089) (024)0549 0081 0202

Panel D Rei = (γ) + βigλg + βiσgλσg + βiMKTλMKT + αi

Constant λg λσg λMKT RMSE MAPE H0 Pr error = 0 p-value R2

-1010 1255 0048 2043 1351 0000 066(0002) (0002) (0077)0091 0075 0100

0005 -0962 1288 0033 1945 1378 0000 069(0097) (0006) (0002) (0189) (022)0406 0099 0079 0399

15

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Table E4 Quantitative Importance of Structural Shocks This table reports the quantitative importance ofthe structural shocks in the model A and Z denote permanent and transitory productivity respectively G denotesgovernment spending Panel A (Panel B) reports the standard deviations of macro variables (asset prices) with allbut one structural shocks active at the time

Panel A Macro Variables

Output Consumption Investment Wages Hours Inflation

All Shocks 173 148 587 131 152 063All except A 164 143 576 123 149 063All except Monetary 168 144 578 121 139 061All except Z Level 112 093 531 070 149 059All except Z Uncertainty 166 141 571 123 150 036All except G Level 145 136 554 129 081 060All except G Uncertainty 153 138 563 128 101 057All except Tax Level 170 147 410 130 143 063All except Tax Uncertainty 171 146 376 130 143 063

Panel B Asset Prices

Nominal Yields

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 370 317 291 268 236 177All except A 369 313 288 265 235 175All except Monetary 363 314 288 266 235 162All except Z Level 347 292 267 245 216 171All except Z Uncertainty 200 131 110 095 080 149All except G Level 352 307 283 262 231 155All except G Uncertainty 335 305 284 263 233 135All except Tax Level 369 315 289 266 235 166All except Tax Uncertainty 369 313 287 264 233 169

16

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Table E5 Nominal and Real Term Structure The Effect of Structural Shocks This table reports themean of the nominal and real term structure under different simulations In particular it shows the nominal andreal yields across different maturities resulting from simulations with all but one structural shock active at the timeA and Z denote permanent and transitory productivity respectively G denotes government spending All reportedyields are expressed in annualized percentages

Nominal Term Structure

1Q 3Y 5Y 7Y 10Y Slope

All Shocks 562 585 609 638 685 123All except A 561 583 608 637 685 124All except Monetary 567 585 610 639 686 119All except Z Level 564 586 611 640 687 123All except Z Uncertainty 641 659 679 703 743 101All except G Level 575 593 617 645 692 117All except G Uncertainty 578 593 616 644 691 113All except Tax Level 569 588 613 641 688 119All except Tax Uncertainty 565 587 612 641 688 123

Real Term Structure

2Y 3Y 5Y 7Y 10Y Slope

All Shocks 388 391 398 408 423 037All except A 387 390 398 407 423 038All except Monetary 389 391 399 408 424 036All except Z Level 388 391 399 408 424 037All except Z Uncertainty 415 418 424 431 444 031All except G Level 392 394 401 410 426 035All except G Uncertainty 393 395 401 410 426 033All except Tax Level 390 393 400 409 424 036All except Tax Uncertainty 389 392 399 409 424 037

17

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Figure E1 Autocorrelation Functions

0 10 20

-05

0

05

1

acf

Output

datamodel

0 10 20

-05

0

05

1 Consumption

0 10 20

-05

0

05

1 Investment

0 10 20

-05

0

05

1

acf

Wages

0 10 20

-05

0

05

1 Hours

0 10 20

04

06

08

1 Price Level

0 10 20quarters

02

04

06

08

1

acf

Nominal Rate 1Q

0 10 20quarters

04

06

08

1Nominal Rate 10Y

0 10 20quarters

02

04

06

08

1 Slope

In this figure we plot autocorrelation functions of the observable variables in the model and the data The dashed

line corresponds to the data The solid line is the model-implied median and the shaded areas correspond to 95

confidence bands when considering parameter uncertainty The sample period for the data is from 1970Q1 to 2016Q4

18

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Figure E2 Impulse Responses for Structural Shocks

5 10 15 20quarters

0

02

04

06

08

perc

ent

Output

5 10 15 20quarters

-002

-0015

-001

-0005

0

perc

ent

Price Level

5 10 15 20quarters

-01

-005

0

perc

ent

Nominal 1Q

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Nominal 5Y

(a) Transitory Productivity Level Shock

5 10 15 20quarters

0

002

004

006

perc

ent

Output

5 10 15 20quarters

-006

-004

-002

0

perc

ent

Price Level

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 1Q

5 10 15 20quarters

-035

-03

-025

-02

-015

-01

-005

perc

ent

Nominal 5Y

(b) Transitory Productivity Uncertainty Shock

5 10 15 20quarters

-01

-008

-006

-004

-002

0

perc

ent

Output

5 10 15 20quarters

0

0005

001

0015

002

0025

003

perc

ent

Price Level

5 10 15 20quarters

0

002

004

006

008

01

perc

ent

Nominal 1Q

5 10 15 20quarters

0

001

002

003

004pe

rcen

t

Nominal 5Y

(c) Permanent Productivity Level Shock

5 10 15 20quarters

-04

-03

-02

-01

0

perc

ent

Output

5 10 15 20quarters

-015

-01

-005

0

perc

ent

Price Level

5 10 15 20quarters

0

01

02

03

04

05

06

perc

ent

Nominal 1Q

5 10 15 20quarters

0

002

004

006

008

perc

ent

Nominal 5Y

(d) Monetary Shock

In this figure we plot the impulse responses of output inflation the nominal short- and long-term bond yields toa positive one standard deviation shock to transitory productivity level and uncertainty to permanent productivityand to monetary policy The blue shaded areas correspond to 95 confidence bands when considering parameteruncertainty

19

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Figure E3 Yield Shock Decomposition

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(a) Baseline Model - Theoretical

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(b) Low Persistence in Fiscal Variables

0 2 4 6 8 10years

0

20

40

60

80

100

basi

s po

ints

std yieldsstd news expected inflation

(c) No SV in Fiscal Variables

In this figure we plot in Panel a the theoretical model-implied unconditional standard deviations of quarterly shocks

Unconditional model-implied standard deviations of yield shocks (circles) and news about expected inflation (Xs)

are determined from our baseline model Panels b and c show corresponding results for model variants with low

persistence in fiscal variables and no stochastic volatility in fiscal variables respectively

20

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

References

Andreasen M M Fernandez-Villaverde J Rubio-Ramırez J F 2017 The Pruned State-SpaceSystem for Non-Linear DSGE Models Theory and Empirical Applications The Review of Eco-nomics Studies 28 755ndash775

Auerbach A J 2003 Fiscal Policy Past and Present Brookings Papers on Economic Activity 3475ndash138

Baker S Bloom N Davis S 2016 Measuring economic policy uncertainty The QuarterlyJournal of Economics 131 1593ndash1636

Basu S Bundick B 2017 Uncertainty Shocks in a Model of Effective Demand Econometrica85 937ndash958

Calvo G 1983 Staggered Prices in a Utility Maximizing Framework Journal of Monetary Eco-nomics 12 383ndash398

Campbell J Y Shiller R J Viceira L M 2009 Understanding Inflation-Indexed Bond MarketsBrookings Papers on Economic Activity 40 79ndash138

Chernov M Mueller P 2012 The term structure of inflation expectations Journal of FinancialEconomics 106 367ndash394

Cochrane J H Piazzesi M 2005 Bond Risk Premia American Economic Review 95 138ndash160

Cooper I Priestley R 2009 Time-varying risk premiums and the output gap Review of FinancialStudies 22 2601ndash2633

Fama E F French K R 1992 The cross-section of expected stock returns The Journal ofFinance 47 427ndash465

Fernandez-Villaverde J Guerron-Quintana P Kuester K Rubio-Ramırez J 2015 FiscalVolatility Shocks and Economic Activity American Economic Review 105 3352ndash84

Gertler M Karadi P 2015 Monetary Policy Surprises Credit Costs and Economic ActivityAmerican Economic Journal Macroeconomics 7 44ndash76

Greenwood R Vayanos D 2014 Bond Supply and Excess Bond Returns Review of FinancialStudies 27 663ndash713

Gurkaynak R S Sack B Wright J H 2007 The US Treasury yield curve 1961 to the presentJournal of Monetary Economics 54 2291ndash2304

Gurkaynak R S Sack B Wright J H 2010 The TIPS Yield Curve and Inflation CompensationAmerican Economic Journal Macroeconomics 2 70ndash92

21

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Jones J B 2002 Has fiscal policy helped stabilize the postwar US economy Journal of MonetaryEconomics 49 709ndash746

Koijen R S Lustig H Van Nieuwerburgh S 2017 The cross-section and time series of stockand bond returns Journal of Monetary Economics 88 50ndash69

Leeper E M Plante M Traum N 2010 Dynamics of fiscal financing in the United StatesJournal of Econometrics 156 304ndash321

Levintal O 2017 Fifth-Order Perturbation Solution to DSGE Models Journal of Economic Dy-namics and Control pp ndash

Lippi M Reichlin L 1994 VAR analysis nonfundamental representations blaschke matricesJournal of Econometrics 63 307ndash325

Ludvigson S C Ng S 2009 Macro Factors in Bond Risk Premia Review of Financial Studies22 5027ndash5067

Mertens K Ravn M O 2013 The dynamic effects of personal and corporate income tax changesin the united states American Economic Review 103 1212ndash47

Montiel Olea J L Stock J Watson M W 2016 Inference in svars with an external instrumentharvard University

Perotti R 2011 Expectations and Fiscal Policy An Empirical Investigation Working Papers429 IGIER (Innocenzo Gasparini Institute for Economic Research) Bocconi University

Ramey V A 2011 Identifying Government Spending Shocks Itrsquos all in the Timing The QuarterlyJournal of Economics 126 1ndash50

Ramey V A 2016 Macroeconomic Shocks and Their Propagation NBER Working Papers 21978National Bureau of Economic Research Inc

Ramey V A Zubairy S 2018 Government Spending Multipliers in Good Times and in BadEvidence from US Historical Data Journal of Political Economy 126 850ndash901

Romer C D Romer D H 2009 A narrative analysis of postwar tax changes Unpublished notesUniversity of Chicago

Sack B P Elsasser R 2004 Treasury inflation-indexed debt a review of the US experienceEconomic Policy Review pp 47ndash63

Sims C A Kim J Kim S Schaumburg E 2008 Calculating and Using Second Order AccurateSolution of Discrete Time Dynamic Equilibrium Models Journal of Economic Dynamics andControl 32 3397ndash3414

Stock J 2008 Whatrsquos New in Econometrics-Time Series Lecture 7 Structural VARs Minicourse2008 NBER Summer Institute Cambridge Mass National Institute for Economic Research

22

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results

Stock J H Watson M W 2012 Disentangling the Channels of the 2007-09 Recession BrookingsPapers on Economic Activity 44 81ndash156

Stock J H Watson M W 2017 Identification and estimation of dynamic causal effects inmacroeconomics Tech rep Sargan Lecture

Wei M Wright J H 2013 Reverse regressions and long-horizon forecasting Journal of AppliedEconometrics 28 353ndash371

23

• Data
• Solution and Estimation
• Solving the Benchmark Model
• • Households with Epstein-Zin Preference
  • Aggregation
  • Loglinearized Phillips Curve
  • The System of Equations for the Model with Growth
  • • VAR Analysis
    • Additional Results