Perturbation in Macroeconomics -...

Perturbation in Macroeconomics

A Short Course for the 2011 CRC 649 Annual Conference

Hong LanHumboldt-Universität zu Berlin

Alexander Meyer-GohdeHumboldt-Universität zu Berlin

Motivation 2 | 87

Economic Risk

Risk and uncertainty [...] influence microeconomic decisionswhich ultimately sum up to macroeconomic outcomes.

—Wolfgang Härdle & Michael C. Burda, “About the CRC 649”


Third-Order Approximation

Response of consumption to a persistent increase in volatility

0 20 40 60 80−4

−3

−2

−1

0

1

2x 10

−7D

evia

tions

Periods since Shock Realization

Linear Approximation

Response of consumption to a persistent increase in volatility

0 20 40 60 80−4

−3

−2

−1

0

1

2x 10

−7D

evia

tions


Motivation | Consequences 5 | 87

Limitations of Linear Methods

I Linearization eliminates many phenomena of interest

. Precautionary behavior, risk sensitivity, asymmetries, large shocks

I Nonlinear methods needed to capture these components of themacroeconomic consequences of economic risk.

This short course will review one such method for the study of DSGE

models.


Outline | 6 | 87

Outline

I Part I: An introduction to perturbation methods

I Part II: A State-space approach

I Part III: A Nonlinear MA approach

I Part IV: Applications

I Part V: Frontiers of Current Research

I Appendix: Detailed derivations


Intro | 7 | 87

Part I: An introduction to perturbation methods

I Perturbation: The basic idea

I Linear methods

I Perturbation methods


Intro | 8 | 87

The basic idea of perturbation


Intro | Perturbation: Basic idea 9 | 87

Background

I Macroeconomists are frequently faced with a functional equation

of the form

f (y) = 0

for an unknown function y(x).I Many solution methods have been developed

. Global methods : projection, numerical dynamic programming...

. Local methods : perturbation, linear methods...

I Focus of this presentation: Perturbation. Pioneered by Fleming (1970) for continuous-time control problems

. Applied by Judd and Guu (1997) and Judd (1998) to a specific

DSGE model.


Intro | Perturbation: Basic idea 10 | 87

Basic idea of perturbation methods

I Perturbation solves the functional problem by specifying

y [n](x) =n∑

i=0

θi(x − x0)i

I Implicit function theorem pins down θi ’s.

I A local approximation, but perform far better than purely local(i.e., linear) methods.

I Perturbation is a generalization of traditional linear methods.


Intro | Linear Methods 11 | 87

A review of linear methods



Model Setup

I Throughout the presentation, we analyze a system of dynamic,discrete-time rational expectations equations

0 = Et [f (yt−1, yt , yt+1, εt)]

where εs ∼ iid Ψ(z), s > t

I f is an (ny × 1) function, sufficiently smooth in all its arguments;

I yt : (ny × 1) endogenous variables;

I εt : (ne × 1) exogenous stochastic process;

I The time-invariant solution takes the form

yt = g(yt−1, εt), and

yt+1 = g(yt , εt+1) = g(g(yt−1, εt), εt+1)



Linear Methods

1. Solve for the non-stochastic steady state

y = g(y , 0), and y satisfies 0 = f (y , y , y , 0)

2. Then linearize the model by around the non-stochastic steadystate and rearrange

A

[Et yt+1

yt

]= B

[yt

yt−1

]+ Cεt where yt = yt − y

3. The solution (e.g., Blanchard and Kahn (1980), Uhlig (1999),

Klein(2000)) takes the form

yt = αyt−1 + βεt



Limitation of the Linear Methods

I Spurious welfare reversal: Tesar (1995) and Kim & Kim (2003)

I The effects of volatility shocks: Fernandez-Villaverde et. al (RES,2007; AER, forthcoming)

I Neglects precautionary behavior: Schmitt-Grohé & Uribe (JEDC2004), Fernandez-Villaverde et. al (JEDC, 2005)

I Notably poor for analysis of asset pricing: Rudebusch &Swanson (JME 2008)


Intro | Perturbation 15 | 87

Perturbation Methods



Perturbation Parameter σ

I σ ∈ [0, 1] denotes an auxiliary, ”scaling” parameter for the

distribution of stochastic shocks

I Scales uncertainty, i.e., stochastic shocks in period t +1 and later

I Effectively, consider a continuum of auxiliary models

parameterized by σ

0 = Et [f (yt−1, yt , yt+1, εt)], εs ∼ iid Ψ(z/σ), s > t

I with a family of solutions indexed by σ

yt = g(yt−1, εt , σ)

yt+1 = g(yt , σεt+1, σ) = g(g(yt−1, εt , σ), σεt+1, σ)

σ = 1—original, stochastic model; σ = 0—deterministic model



State Space Solution

I The policy function


defines the state vector to be (yt−1, εt) and σ;

I Often, this policy function is referred to as the state spacesolution of the model;

I Collard and Juillard (2001), Schmitt-Grohé and Uribe (2004) andothers construct a second-order Taylor expansion of the state

space solution

I Anderson et al. (2006), Dynare++ and others construct a

higher-order Taylor expansion of the state space solution



Nonlinear Moving Average Solution

I The policy function

yt = y(σ; εt , εt−1, . . .)

defines the state vector to be σ and the infinite sequence of pastshock realizations;

I Extension of linear MA methods of Muth (1961), Whiteman(1983), Taylor (1984) and others

I We are developing methods based on this form of the policyfunction

I Allows, e.g., natural extension of the IRF property of linear MA tononlinear spaces


State Space | 19 | 87

Part II : State Space Solution

I Intuition via the stochastic growth model

I Numerical expansion

I Limitation of the state space solution


State Space | Intuition 20 | 87

The Stochastic Growth Model




I An economy is populated by infinitely-lived agents;

I A representative agent maximizes the discounted sum of herexpected utility

I given a resource constraint;

I The state of the economy evolves according to a Markov

process.



The Stochastic Growth Model-Cont.

I The consumption Euler equation characterizes the agent’s

utility-maximization behavior

c−γ

t = βEt [c−γ

t+1(αezt+1 kα−1t + 1 − δ)]

I under following constraints

yt = ct + it

yt = ezt kα

t−1

it = kt − (1 − δ)kt−1

I The state of the economy evolves according to

zt = ρzt−1 + εt




I From the state-space perspective, we seek the policy function

kt = k(kt−1, εt)

I The model has a closed-form solution when γ = δ = 1 (The

Brock-Mirman model)

ln(kt ) = (1 − α) ln(k ) + αln(kt−1) + ρzt−1 + εt

I Otherwise, we resort to perturbation methods: kt = k(kt−1, εt , σ)


State Space | Expansion 24 | 87

Numerical expansion



Setting Up the Expansion

I Insert the solution function into the model


yt+1 = g(y(yt−1, εt , σ), εt+1, σ);where εt+1 ≡ σεt+1

→0 = Et [f (yt−1, g(yt−1, εt , σ), g(g(yt−1, εt , σ), εt+1, σ), εt )]

I Successive differentiation wrt yt−1, εt , σ

I evaluated atyt−1 = y , εt = σ = 0 identifies coefficients in approx.

The zeroth order expansion identifies the non-stochastic steady state

f (y , y , y , 0) = 0,where y = y(y , 0, 0)



First Order Expansion

I We aim to construct

yt = y + gy (yt − y) + gεεt + gσσ

I Using the implicit function theorem, we can pin down all thecoefficients in the Taylor expansion;

I In particular

gσ = 0

I The first order expansion is a certainty equivalent solution;

I First order perturbation cannot capture the effect of uncertainty

in the model!

I gy and gε as in traditional linearizations



Second Order Expansion

I We aim to construct (where yt abbreviates yt − y )

yt =gy yt−1 + gεεt + gσσ +12

gy2(yt−1 ⊗ yt−1) + gεy (yt−1 ⊗ εt)

+ gσy yt−1σ +12

gε2(εt ⊗ εt) + gσεσεt +12

gσσσ2

I In particular

gσy = 0 gσε = 0, and gσ = 0

I Thus, up to second order, uncertainty only affects the constantterm σ2 in the expansion of the policy function;

I This reflects that the second order expansion depends on the

size of the variance of the exogenous shocks!



Third Order Expansion

I Analogous to before

I In particular

gσy2 = gσε2 = gσyε = 0

but, in general gσ2y 6= gσ2ε 6= 0

I Third order captures time-varying correction for uncertainty (e.g.,

risk-sensitive dynamics);

I This reflects that the second order expansion depends on the

size of the variance of the exogenous shocks!

I gσ3 the shape (skewness) of the distribution matters.


Policy Function: Stoch. Growth

-4 -2 2 4k[t-1]

-2

-1

1

2

3

4

Consumption


State-Space Policy Function

I Provides an approximation accurate up to the order of approx.

I For the mapping yt−1, εt , σ, 7→ yt

I Often interested in a different mapping

I IRF’s and simulations: ..., εt−1, εt , σ, 7→ yt


State Space | Limitations 31 | 87

Limitations of the state space solution



Iterating a State-Space Perturbation

From Kim et al. (2008): Consider a 2nd-order solution

yt = ρyt−1 + αy2t−1 + εt , |ρ| < 1 and α > 0

|ρ| < 1 follows from stability of linear solution.

I Iterating forward generates spurious higher order terms;

yt+1 = α3y4t−1 + 2α2ρy3

t−1 + 2α2yt−1εt + (αρ2 + αρ)y2t−1 + ρ2yt−1 + . . .

I Iterating again yields sixth and fifth order terms

I ...

Obviously not a 2nd-order simulation or impulse response.



What Goes Wrong

Moreover: Examine an impulse response with

I yt−1 = 0

I and a single shock of size εt = (1 − ρ)/α

yt =1 − ρ

α

yt+1 = ρ1 − ρ

α+ α

(1 − ρ

α

)2

=1 − ρ

α

...

I yt+s is constant: εt = (1 − ρ)/α is a threshold

I if εt > (1 − ρ)/α: yt+s → ∞!

Potentially explosive paths despite stability of linear solution.


Source of explosive paths

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1

45 line

true Þxed point

truth

additional undesirable

Þxed point

second-order

approximation

Notes: This �gure plots the function f(x�1) described in Section 1 and its second-order

Taylor-series approximation.


Pruning?

Kim et al.(2008) suggest “pruning” the solution

I Let the first-order solution be yFt : yF

t = ρyFt−1 + εt

I The pruned second-order solution is ySt = ρyS

t−1 + α(yFt−1)

2 + εt

I Intuition: Replace ySt−1 with yF

t−1 for the quadratic terms;

I Repeat throughout simulation/impulse responses to maintain

desired order of approximation;

I Discard all spurious higher order terms.

I Pruning is an ad-hoc procedure, Lombardo (2010), and

I does not represent a valid perturbation approximation, Den Haan

& De Wind (2010).


Nonlinear MA | 36 | 87

Part III : Nonlinear MA Solution

I Overview: Nonlinear MA solution

I An equivalent solution in Brock-Mirman model

I Numerical expansion


Nonlinear MA | Overview: Nonlinear MA 37 | 87

Overview: Nonlinear MA solution


Nonlinear MA | Overview: Nonlinear MA 38 | 87

Nonlinear MA Solution

I The Nonlinear MA policy function takes the form

yt = y(σ; εt , εt−1, . . .)

I Nonlinear MA policy function is a direct mapping:

. . . , εt−1, εt , σ, 7→ yt ,

I explicitly taking the history of the exogenous shocks intoconsideration;

I Stability of first-order approx. carries over to higher-order approx.

I It avoids any ” pruning” in generating simulation and impulse

responses, and

I demonstrate the specific contribution of the different moments ofthe exogenous shocks to the impulse responses.


Nonlinear MA | Special Case: Brock-Mirman 39 | 87

An equivalent solution in Brock-Mirman model


Nonlinear MA | Special Case: Brock-Mirman 40 | 87

Nonlinear MA solution of Brock-Mirman model

I Recall the closed-form state space solution of Brock-Mirman

model

ln(kt ) = (1 − α) ln(k ) + αln(kt−1) + ρzt−1 + εt

I This is is equivalent to the following nonlinear MA solution

ln (kt)− ln(k)=

∞∑

i=0

kiεt−i

I with k = (αβ)1

1−α , ki = αki−1 + ρi , with k−1 = 0

I Generally, though, numerical approximation unavoidable.

I The one-to-one mapping btw. state space and MA breaks down

under approx.


Nonlinear MA | Expansion 41 | 87

Solution Form

The time-invariant solution takes the form

yt = y(σ; εt , εt−1, . . .)

yt−1 = y−(σ; εt−1, εt−2, . . .)

yt+1 = y+(σ; εt+1, εt , εt−1, . . .) where εt+1 ≡ σεt+1

Inserting into the model

0 = Et [f (y−(σ; εt−1, εt−2, . . .), y(σ; εt , εt−1, . . .), y+(σ; εt+1, εt , εt−1, . . .), εt)]

Differentiate with respect to σ, εt , εt−1, ... to identify Taylor series

coefficients.Zeroth-order solution the same as before: non stochastic steady

state.




I The first order expansion of the policy function takes the form

yt = y + yσσ +

∞∑

i=0

yiεt−i , i = 0, 1, 2, . . .

I Using the implicit function theorem, we pin down the coefficientsvia a recursion

yi = αyi−1 + β1ui , y−1 = 0

I Additionally, like the state space solution

yσ = 0

I The first-order expansion is still a certainty equivalent solution

I Equivalent to state-space solution (recursion in coefficients notvariables)




I Uncertainty again only affects the constant term up to second

order

yiσ = 0

I thus

yt = y +12

yσ2σ2 +

∞∑

i=0

yiεt−i +12

∞∑

j=0

∞∑

i=0

yj,i(εt−j ⊗ εt−i)




I The uncertainty affects the policy function not only through theconstant term, but also through a time-variant term!

yjiσ = 0, but, in general yiσ2 6= 0

I Third order expansion can capture time-varying precautionarybehavior.

I Third order expansion writes

yt =y +12

yσ2σ2 +

∞∑

i=0

(yi +

12

yσ2,iσ2)εt−i +

12

∞∑

j=0

∞∑

i=0


+16

∞∑

k=0

∞∑

j=0

∞∑

i=0

yk ,j,i(εt−k ⊗ εt−j ⊗ εt−i)



Discussion of Nonlinear MA

I Work in progress (look for an SFB Discussion Paper soon!)

I Alternate state basis (infinite history of shocks) for policy function

I Straightforward approach to impulse responses and simulations

I With stability properties inherited from first-order

I Numerical methods: Don’t use a knife to turn a screw...


Examples | 46 | 87

Part IV : Nonlinear MA Solution

I RBC with stochastic volatility

I Int’l RBC with real interest rate risk

I Nominal Asset Pricing with risk-sensitive preferences


Examples | RBC: Stochastic Volatility 47 | 87

Basic RBC with Stochastic Volatilty



Basic RBC with Stochastic Volatility

I Add disutility from working

I yt = ezt l1−αkα

t−1

I to stochastic growth model above

I Standard RBC (e.g., Hansen (198))

I Add time varying volatility to tech shock

I ln(εt ) is now AR(1)

I mean as as in constant volatility case

I persistence and stand. dev.: post-war US (Fernandez-Villaverde(RES 2007))


IRF: Technology Shock

0 10 20 30 40

0

2

4

6

8

10

x 10−3


Dev

iatio

ns

Y L Y/L C K

IRF: Volatility Shock

0 10 20 30 40−4

−2

0

2

4

x 10−7

Dev

iatio

ns


KL Y Y/L C


Consequences of Stochastic Volatility

I Weakens the positive correlation of consumption and labor

productivity with output

I But note the scale: tech. shocks overwhelm vol. shocks

I Additional fundamental nonlinearities required for vol. shocks tocontribute significantly


Examples | Int’l RBC: Interest Rate Volatility 52 | 87

Int’l RBC with real interest rate riskFernández-Villaverde et. al (AER forthcoming)



Fernández-Villaverde et. al (AER forthcoming)

I Small open economy populated by infinitely-lived agents;

I A representative household maximizes the discounted sum ofher expected utility w.r.t some constraint;

I The household can invest in the stock of physical capital and aninternationally-traded bond;

I The real interest rate in international markets follows

rt = r + εtb,t + εr ,t

where εr ,t = ρrεr ,t−1 + eσr,t ur ,t

and σr ,t = (1 − ρσr ) + ρσrσr ,t−1 + ηr uσr ,t

I Consider a positive shock to uσr ,t .


0 20 40−0.1

−0.05

0

0.05

0.1Consumption

0 20 40−0.4

−0.3

−0.2

−0.1

0Investment

0 20 40−0.03

−0.02

−0.01

0

0.01Output

0 20 40−2

0

2

4x 10

−4 Hours

0 20 40−1

−0.5

0

0.5

1Real Interest Rate

0 20 40−4

−3

−2

−1

0Debt


Fernández-Villaverde et. al (AER forthcoming)

I Increase in riskiness of financing debt in int’l capital mkt’s

I causes a quantitatively significant and protracted contraction

I induced by a precautionary winding down of exposure to int’lfinancing

Compared to simple RBC above

I quantitatively significant impact of volatility shocks

I linear methods miss this “important force behind the

I distinctive size and pattern of business cycle fluctuations inemerging economies”


Examples | Nominal Asset Pricing with risk-sensitive preferences 56 | 87

Nominal Asset Pricing with risk-sensitive preferencesRudebusch & Swanson (AEJ forthcoming)



Rudebusch & Swanson (AEJ forthcoming)

I Representative agent model with Calvo sticky prices

I Epstein-Zin preferences (separate RRA from IES)

I SDF mt,t+1 =

(Vt+1

(Et V1−α

t+1 )1

1−α

)α

βUc,t+1

Uc,t

1πt+1

I α = 0 standard exp. utility framework

I Avg. term premium 10 year zero coupon bond (US postwar):

100 bp

I Linear Methods: 0 bp, Exp. util: 4 bp, Epstein-Zin: 10´0 bp

I DSGE can reproduce the upward sloping yield curve.




I Backus-Gregory-Zin (1989), Den Haan (1995)

I Low interest rates in recession imply increasing bond prices

I Thus, bonds pay out high when consumption is low

I Implies a negative premium and downward sloping yield curve


Monetary Policy Shock

0 5 10 15 20−0.1

−0.05

0

0.05

0.1C

Dev

iatio

ns

0 5 10 15

0

0.1

0.2

0.3

termprem

Dev

iatio

ns


0 5 10 15 20−0.3

−0.2

−0.1

0

0.1pi

Dev

iatio

ns

0 5 10 15 20−0.8

−0.6

−0.4

−0.2

0

0.2bond price

Dev

iatio

ns




I Structural interpretation

I Tech. shocks main contributor of variance

I Negative tech. shock recession produces increase in inflation

I Causing nominal bond prices to fall

I Positive premium and upward sloping yield curve!


Tech. Shock

0 5 10 15 20−5

−4

−3

−2

−1

0termprem

Dev

iatio

ns


0 5 10 15 200

1

2

3

4bond price

Dev

iatio

ns


3rd Order 2nd Order 1st Order0 5 10 15 20

−1.5

−1

−0.5

0pi

Dev

iatio

ns

0 5 10 15 200

0.05

0.1

0.15

0.2C

Dev

iatio

ns

Frontiers | 62 | 87

Part V : Frontiers

I Measuring the quality of approximation

I Estimation

I Generalized transfer functions


Frontiers | Measuring the quality of approximation 63 | 87

Measuring the quality of approximation



Euler Equation Error

I Currently, only one accepted method to measure quality ofapproximation

I insert approximation back into functional

I measure residuals over range in state space of particular interest

I In simple stoch. growth model: the functional is the Euler

equation

I Interpretation: one-period optimization error given current state


Euler equation errors


Euler Equation Error

For the stochastic growth model

I Error on the magnitude of 1E-6

I implies a $1 mistake

I for every $1,000,000 of expenses

I Judd & Guu (1997): “Few economists would seriously argue that

real-world agents do better than this.”

Nonlinear MA has an infinite dimensional state space...what to put on

the x-axis?Alternative measures? How to interpret Euler equation errors in

multi-state DSGE model?


Frontiers | Estimation 67 | 87

Estimation


Frontiers | Estimation 68 | 87

Estimation

I Can apply Kalman filter for ML to estimate Gaussian model

I With nonlinear models, observables no longer inherit Gaussian

distribution from shocksI Hence, no comp. advantage to Gaussianity in the first place

So

I How do we estimate fully parameterized, nonlinear time seriesmodels

I where the mapping from parameters to reduced form time series

model is highly nonlinear and available only numerically?I Current cutting edge: Particle filter (extends Kalman filter idea of

tracking conditional distributions to evaluate likelihood function)I Alternative approaches?


Frontiers | Generalized transfer functions 69 | 87

Generalized transfer functions



Generalized transfer functions

I Breakdown of superposition

I History of shocks impacts current response nonlinearly

I What is an impulse response?

I Initial point: stoch. steady state, non stoch. steady state, ergodicmean?


0 50 100 150 200 250 300 350 400−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Second Order Contributions to Response of k to 2 100 Std. Dev. Shocks in e

Periods

Dev

iatio

ns

Sum of individual second−order contributionsTotal second−order contributions

0 200 4000

0.1

0.2

0.3

0.42nd−Ord Contr. of 1st Shock

Dev

iatio

ns

Periods0 200 400

0

0.1

0.2

0.3

0.4

Indiv. 2nd−Ord Contr. of 2nd Shock

Dev

iatio

ns

Periods0 200 400

−0.1

−0.05

0

Cross Correction to 2nd−Ord Contr. of 2st Shock

Dev

iatio

ns

Periods


Thank you very much for your attention!


Detailed Derivations | 74 | 87

Part V : The detailed derivations

I State space solution

I Nonlinear MA solution



State Space Solution




I To determine gy , Et

[DyT

t−1f∣∣∣y

]= 0

fy− + fy gy + fy+gy gy = 0

I This is a version of Blanchard and Kahn (1980), Anderson andMoore (1985), Uhlig (1999), Klein (2000) saddle-point problem,

and can be solved using, i.e., the QZ algorithm proposed byKlein (2000).



First Order Expansion-Cont.

I To determine gε, Et

[D

εTtf∣∣∣y

]= 0

fε + fy gε + fy+gy gε = 0

I Therefore

gε = −(fy + fy+gy )−1fε

I To determine gσ, Et

[Dσf

∣∣∣y

]= 0

fy gσ + fy+gy gσ = 0

I Therefore

gσ = 0




I To determine gy2 , Et

[DyT

t−1yTt−1

f∣∣∣y

]= 0

(fy+ + fy )gy2 + fy+gy2(gy ⊗ gy ) = B

I This is a specific Sylvester equation studied in, and solution

methods proposed by Kamenik (2005).



Second Order Expansion-Cont.

I To determine gεy , gσy , gσε and gε2

Et

[D

εTt yT

t−1f∣∣∣y

]= 0 : fy+(gy2(gy ⊗ gε) + gy gεy ) + fy gεy = B

⇒ gεy = (fy+gy + fy )−1(B − fy+gy2(gy ⊗ gε))

Et

[D

σyTt−1

f∣∣∣y

]= 0 : fy+gy gσy + fy gσy = 0 ⇒ gσy = 0

Et

[D

σεTtf∣∣∣y

]= 0 : fy+gy gσε + fy gσε = 0 ⇒ gσε = 0

Et

[D

εTt ε

Ttf∣∣∣y

]= 0 : fy+(gy2(gε ⊗ gε) + gy gεε) + fy gεε = B

⇒ gεε = (fy+gy + fy )−1(B − fy+gy2(gε ⊗ gε))




I To determine gσσ, Et

[Dσσf

∣∣∣y

]= 0

fy+(gσσ + gy gσσ) + fy gσσ + (fy+2(gε ⊗ gε) + fy+gεε)Et (σ2εt ⊗ εt) = 0

I Therefore

gσσ = −(fy+(I + gy ) + fy )−1(fy+2(gε ⊗ gε) + fy+gεε)Et (σ2εt ⊗ εt)



Nonlinear MA Solution




I To determine yi , evaluate Et

[D

εTt−i

f∣∣∣y

]= 0

fy−yi−1 + fy yi + fy+yi+1 + fuui = 0

I This is an inhomogeneous version of Blanchard and Kahn

(1980), Anderson and Moore (1985), Uhlig (1999), Klein (2000)saddle-point problem, solved in detail by Anderson (2010).

I Anderson (2010) method can be applied under our assumption,and it delivers a convergent inhomogeneous solution in the form

yi = αyi−1 + β1ui



First Order Expansion-Cont.

I To determine yσ, Et

[Dσf

∣∣∣y

]= 0

(fy− + fy + fy+)yσ = 0

I From our no-unit-roots assumption, it follows that

det(fy− + fy + fy+) 6= 0

and hence yσ = 0.

The first order expansion of the policy function therefore takes theform

yt = y +

∞∑

i=0

yiεt−i , i = 0, 1, 2, . . .



I To determine yj,i , evaluate Et

[D2

εTt−jε

Tt−i

f∣∣∣y

]= 0

fy−yj−1,i−1 + fy yj,i + fy+yj+1,i+1 + fx2(xj ⊗ xi) = 0

I The inhomogeneous part xj ⊗ xi , by construction, is

xj ⊗ xi = (γ1 ⊗ γ1)(Sj ⊗ Si)

where Si =

[yi−1

ui

]

and Si has the 1st order Markov representation: Si+1 = δ1Si .

I Therefore, yj,i will take the form

yj,i = αyj−1,i−1 + β2(Sj ⊗ Si )

I β2 solves the following

(fy + fy+α)β2 + fy+β2(δ1 ⊗ δ1) = −fx2(γ1 ⊗ γ1)

I The foregoing is a specific Sylvester equation studied in and the

solution method developed by Kamenik (2005).


I To determine yiσ, evaluate Et

[D2

σεTt−i

f∣∣∣y

]= 0

(fy− + fy + fy+)yiσ = 0

With no-unit-roots, the forgoing delivers: yiσ = 0.

I To determine yσ2 , evaluate Et

[D2

σ2 f∣∣∣y

]= 0

[fy+y02 + fy+2(y0 ⊗ y0)]Et (εt+1 ⊗ εt+1)− (fy− + fy + fy+)yσ2 = 0

hence yσ2 can be recovered as follows

yσ2 = (fy− + fy + fy+)−1[fy+y02 + fy+2(y0 ⊗ y0)]Et (εt+1 ⊗ εt+1)

Therefore the 2nd order expansion of the policy function writes

yt = y +12

yσ2σ2 +∞∑

i=0

yiεt−i +12

∞∑

j=0

∞∑

i=0




I To determine yk ,j,i , evaluate Et

[D3

εTt−kε

Tt−jε

Tt−i

f∣∣∣y

]= 0

fy−yk−1,j−1,i−1 + fy yk ,j,i + fy+yk+1,j+1,i+1

+ fx3(xk ⊗ xj ⊗ xi) + fx2(xk ⊗ xj,i) + fx2(xk ,j ⊗ xi)

+ fx2(xj ⊗ xk ,i)Kne,ne2 (Ine ⊗ Kne,ne) = 0

I The solution will take the form

yk ,j,i = αyk−1,j−1,i−1 + β3Sk ,j,i

I β3 solves the following Sylvester equation

(fy + fy+α)β3 + fy+β3δ3 = −[fx3 fx2 fx2 fx2

]γ3


Third Order Expansion-Cont.

I The state space Sk ,j,i contains not only the triple Kroneckerproduct of the first order state spaces Sk ⊗ Sj ⊗ Si , but also the

combinations of both first and second order state spaces, i.e.,

Sk ⊗ Sj,i , Sk ,j ⊗ Si . . .

I Solving Et

[D3

σεTt−jε

Tt−i

f∣∣∣y

]= 0 leads to yσ,j,i = 0, whereas

Et

[D3

σ2εTt−i

f∣∣∣y

]= 0 leads to

yσ2,i = αyσ2,i−1 + β3Sσ2,i

I yσ3 = 0 if the exogenous variables are normally distributed.

The 3rd order expansion of the policy function writes

yt =y +12

yσ2σ2 +∞∑

i=0

(yi +

12

yσ2,iσ2)εt−i +

12

∞∑

j=0

∞∑

i=0


+16

∞∑

k=0

∞∑

j=0

∞∑

i=0

yk ,j,i(εt−k ⊗ εt−j ⊗ εt−i)

Perturbation in Macroeconomics -...

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