Optimal Monetary Policy under Uncertainty in DSGE...
Transcript of Optimal Monetary Policy under Uncertainty in DSGE...
Optimal Monetary Policy under Uncertaintyin DSGE Models:
A Markov Jump-Linear-Quadratic Approach
Lars E.O. Svensson1 Noah Williams2
1Sveriges Riksbank
2University of Wisconsin - Madison
November 2009
Svensson and Williams Optimal Model Policy Under Uncertainty
Introduction
I have long been interested in the analysis of monetary policyunder uncertainty. The problems arise from what we do notknow; we must deal with the uncertainty from the base of whatwe do know. [...]
The Fed faces many uncertainties, and must adjust its onepolicy instrument to navigate as best it can this sea ofuncertainty. Our fundamental principle is that we must use thatone policy instrument to achieve long-run price stability. [...]
My bottom line is that market participants should concentrateon the fundamentals. If the bond traders can get it right,they’ll do most of the stabilization work for us, and we at theFed can sit back and enjoy life.
William Poole, (1998) “A Policymaker Confronts Uncertainty”
Svensson and Williams Optimal Model Policy Under Uncertainty
Overview
Develop methods for policy analysis under uncertainty.Methods have broad potential applications.Consider optimal policy when policymakers don’t observetrue economic structure, must learn from observations.Classic problem of learning and control: actions haveinformational component. Motive to alter actions tomitigate future uncertainty (“experimentation”).Unlike most previous literature we consider forward-lookingmodels. Particular focus on DSGE models.Issues:
How does uncertainty affect policy?How does learning affect losses?How does experimentation motive affect policy and losses?
Svensson and Williams Optimal Model Policy Under Uncertainty
Overview
Develop methods for policy analysis under uncertainty.Methods have broad potential applications.Consider optimal policy when policymakers don’t observetrue economic structure, must learn from observations.Classic problem of learning and control: actions haveinformational component. Motive to alter actions tomitigate future uncertainty (“experimentation”).Unlike most previous literature we consider forward-lookingmodels. Particular focus on DSGE models.Issues:
How does uncertainty affect policy?How does learning affect losses?How does experimentation motive affect policy and losses?
Svensson and Williams Optimal Model Policy Under Uncertainty
Some Related Literature
This paper: application of our work inSvensson-Williams (2007- . . .)Aoki (1967), Chow (1973): Multiplicative uncertainty inLQ model (only backward-looking/control case)Control theory: Costa-Fragoso-Marques (2005), othersRecursive saddlepoint method: Marcet-Marimon (1998)Blake-Zampolli (2005), Zampolli (2005): similar observablemodes case, less generalWieland (2000, 2006), Beck and Wieland (2002): Optimalexperimentation with backward looking modelsCogley, Colacito, Sargent (2007): Adaptive policy asapproximation to Bayesian, expectational variablesTesfaselassie, Schaling, Eijffinger (2006), Ellison (2006):similar, less general
Svensson and Williams Optimal Model Policy Under Uncertainty
Some Related Literature
This paper: application of our work inSvensson-Williams (2007- . . .)Aoki (1967), Chow (1973): Multiplicative uncertainty inLQ model (only backward-looking/control case)Control theory: Costa-Fragoso-Marques (2005), othersRecursive saddlepoint method: Marcet-Marimon (1998)Blake-Zampolli (2005), Zampolli (2005): similar observablemodes case, less generalWieland (2000, 2006), Beck and Wieland (2002): Optimalexperimentation with backward looking modelsCogley, Colacito, Sargent (2007): Adaptive policy asapproximation to Bayesian, expectational variablesTesfaselassie, Schaling, Eijffinger (2006), Ellison (2006):similar, less general
Svensson and Williams Optimal Model Policy Under Uncertainty
The Model
Standard linear rational expectations framework:
Xt+1 = A11Xt + A12xt + B1it + C1εt+1
EtHxt+1 = A21Xt + A22xt + B2it + C2εt
Xt predetermined, xt forward-looking, it CB instruments(controls)εt i.i.d. shocks, N (0, I )
Matrices can take Nj different values in period t,corresponding to n modes jt = 1, 2, ..., Nj .Modes jt follow Markov chain w/transition matrixP = [Pjk ].
Svensson and Williams Optimal Model Policy Under Uncertainty
The Model
Markov jump-linear-quadratic framework:
Xt+1 = A11,t+1Xt + A12,t+1xt + B1,t+1it + C1,t+1εt+1
EtHt+1xt+1 = A21,tXt + A22,txt + B2,tit + C2,tεt
Xt predetermined, xt forward-looking, it CB instruments(controls)εt i.i.d. shocks, N (0, I )
Matrices can take Nj different values in period t,corresponding to n modes jt = 1, 2, ..., Nj .Modes jt follow Markov chain w/transition matrixP = [Pjk ].
Svensson and Williams Optimal Model Policy Under Uncertainty
Beliefs and Loss
Central bank (and aggregate private sector) observe Xt , it ,do not (in general) observe jt or εt .pt|t : perceived probabilities of modes in period tPrediction equation: pt+1|t = P ′pt|t
CB intertemporal loss function:
Et
∞∑τ=0
δτLt+τ (1)
Period loss:
Lt ≡12
Xtxtit
′
Wt
Xtxtit
(2)
Svensson and Williams Optimal Model Policy Under Uncertainty
Beliefs and Loss
Central bank (and aggregate private sector) observe Xt , it ,do not (in general) observe jt or εt .pt|t : perceived probabilities of modes in period tPrediction equation: pt+1|t = P ′pt|t
CB intertemporal loss function:
Et
∞∑τ=0
δτLt+τ (1)
Period loss:
Lt ≡12
Xtxtit
′
Wt
Xtxtit
(2)
Svensson and Williams Optimal Model Policy Under Uncertainty
General and Tractable Way to Model Uncertainty
Large variety of uncertainty configurations. Approximate most(all?) relevant kinds of model uncertainty
Regime switching modelsi.i.d. and serially correlated random model coefficients(generalized Brainard-type uncertainty)Different structural models
Different variables, different number of leads and lagsBackward- or forward-looking models
Particular variablePrivate-sector expectations
Ambiguity aversion, robust control (P ∈ P)Different forms of CB judgment (for instance, perceiveduncertainty)And many more . . .
Svensson and Williams Optimal Model Policy Under Uncertainty
General and Tractable Way to Model Uncertainty
Large variety of uncertainty configurations. Approximate most(all?) relevant kinds of model uncertainty
Regime switching modelsi.i.d. and serially correlated random model coefficients(generalized Brainard-type uncertainty)Different structural models
Different variables, different number of leads and lagsBackward- or forward-looking models
Particular variablePrivate-sector expectations
Ambiguity aversion, robust control (P ∈ P)Different forms of CB judgment (for instance, perceiveduncertainty)And many more . . .
Svensson and Williams Optimal Model Policy Under Uncertainty
General and Tractable Way to Model Uncertainty
Large variety of uncertainty configurations. Approximate most(all?) relevant kinds of model uncertainty
Regime switching modelsi.i.d. and serially correlated random model coefficients(generalized Brainard-type uncertainty)Different structural models
Different variables, different number of leads and lagsBackward- or forward-looking models
Particular variablePrivate-sector expectations
Ambiguity aversion, robust control (P ∈ P)Different forms of CB judgment (for instance, perceiveduncertainty)And many more . . .
Svensson and Williams Optimal Model Policy Under Uncertainty
General and Tractable Way to Model Uncertainty
Large variety of uncertainty configurations. Approximate most(all?) relevant kinds of model uncertainty
Regime switching modelsi.i.d. and serially correlated random model coefficients(generalized Brainard-type uncertainty)Different structural models
Different variables, different number of leads and lagsBackward- or forward-looking models
Particular variablePrivate-sector expectations
Ambiguity aversion, robust control (P ∈ P)Different forms of CB judgment (for instance, perceiveduncertainty)And many more . . .
Svensson and Williams Optimal Model Policy Under Uncertainty
Approximate MJLQ Models
MJLQ models provide convenient approximations fornonlinear DSGE models.Underlying function of interest: f (X , θ), where X iscontinuous, θ ∈ {θ1, . . . , θnj}.Taylor approximation around (X , θ):
f (X , θj) ≈ f (X , θ) + fX (X , θ)(X − X) + fθ(X , θ)(θj − θ).
Valid as X → X and θ → θ: small shocks to X and θ.MJLQ approximation around (Xj , θj):
f (X , θj) ≈ f (Xj , θj) + fX (Xj , θj)(X − Xj).
Valid as X → Xj : small shocks to X , slow variation in θ(P → I ).
Svensson and Williams Optimal Model Policy Under Uncertainty
Approximate MJLQ Models
MJLQ models provide convenient approximations fornonlinear DSGE models.Underlying function of interest: f (X , θ), where X iscontinuous, θ ∈ {θ1, . . . , θnj}.Taylor approximation around (X , θ):
f (X , θj) ≈ f (X , θ) + fX (X , θ)(X − X) + fθ(X , θ)(θj − θ).
Valid as X → X and θ → θ: small shocks to X and θ.MJLQ approximation around (Xj , θj):
f (X , θj) ≈ f (Xj , θj) + fX (Xj , θj)(X − Xj).
Valid as X → Xj : small shocks to X , slow variation in θ(P → I ).
Svensson and Williams Optimal Model Policy Under Uncertainty
Four Different CasesIn each we assume commitment under timeless perspective.Follow Marcet-Marimon (1999). Convert tosaddlepoint/min-max problem. Extended state vector includeslagged Lagrange multipliers, controls include currentmultipliers.
1 Observable modes (OBS)Current mode known, uncertainty about future modes
2 Optimal policy with no learning (NL)Naive updating equation: pt+1|t+1 = P ′pt|t
3 Adaptive optimal policy (AOP)Policy as in NL, Bayesian updating of pt+1|t+1 each periodNo experimentation
4 Bayesian optimal policy (BOP)Optimal policy taking Bayesian updating into accountOptimal experimentation
Svensson and Williams Optimal Model Policy Under Uncertainty
Four Different CasesIn each we assume commitment under timeless perspective.Follow Marcet-Marimon (1999). Convert tosaddlepoint/min-max problem. Extended state vector includeslagged Lagrange multipliers, controls include currentmultipliers.
1 Observable modes (OBS)Current mode known, uncertainty about future modes
2 Optimal policy with no learning (NL)Naive updating equation: pt+1|t+1 = P ′pt|t
3 Adaptive optimal policy (AOP)Policy as in NL, Bayesian updating of pt+1|t+1 each periodNo experimentation
4 Bayesian optimal policy (BOP)Optimal policy taking Bayesian updating into accountOptimal experimentation
Svensson and Williams Optimal Model Policy Under Uncertainty
Four Different CasesIn each we assume commitment under timeless perspective.Follow Marcet-Marimon (1999). Convert tosaddlepoint/min-max problem. Extended state vector includeslagged Lagrange multipliers, controls include currentmultipliers.
1 Observable modes (OBS)Current mode known, uncertainty about future modes
2 Optimal policy with no learning (NL)Naive updating equation: pt+1|t+1 = P ′pt|t
3 Adaptive optimal policy (AOP)Policy as in NL, Bayesian updating of pt+1|t+1 each periodNo experimentation
4 Bayesian optimal policy (BOP)Optimal policy taking Bayesian updating into accountOptimal experimentation
Svensson and Williams Optimal Model Policy Under Uncertainty
Four Different CasesIn each we assume commitment under timeless perspective.Follow Marcet-Marimon (1999). Convert tosaddlepoint/min-max problem. Extended state vector includeslagged Lagrange multipliers, controls include currentmultipliers.
1 Observable modes (OBS)Current mode known, uncertainty about future modes
2 Optimal policy with no learning (NL)Naive updating equation: pt+1|t+1 = P ′pt|t
3 Adaptive optimal policy (AOP)Policy as in NL, Bayesian updating of pt+1|t+1 each periodNo experimentation
4 Bayesian optimal policy (BOP)Optimal policy taking Bayesian updating into accountOptimal experimentation
Svensson and Williams Optimal Model Policy Under Uncertainty
1. Observable modes (OBS)
Policymakers (and public) observe jt , know jt+1 drawnaccording to P.Analogue of regime-switching models in econometrics.Law of motion for Xt linear, preferences quadratic in Xtconditional on modes.Solution linear in Xt for given j
it = Fjt Xt ,
Value function quadratic in Xt for given j
V (Xt , jt) ≡12X ′
tVXX ,jtXt + wjt .
Svensson and Williams Optimal Model Policy Under Uncertainty
1. Observable modes (OBS)
Policymakers (and public) observe jt , know jt+1 drawnaccording to P.Analogue of regime-switching models in econometrics.Law of motion for Xt linear, preferences quadratic in Xtconditional on modes.Solution linear in Xt for given j
it = Fjt Xt ,
Value function quadratic in Xt for given j
V (Xt , jt) ≡12X ′
tVXX ,jtXt + wjt .
Svensson and Williams Optimal Model Policy Under Uncertainty
2. Optimal policy with no learning (NL)
Interpretation: Policymakers forget past Xt−1, . . . in periodt when choosing it .Allows for persistence of modes. But means beliefs don’tsatisfy law of iterated expectations. Requires slightly morecomplicated Bellman equation.Law of motion linear in Xt , dual preferences quadratic inXt . pt|t exogenous.Solution linear in Xt for given pt|t
it = Fi(pt|t)Xt ,
Value function quadratic in Xt for given pt|t
V (Xt , pt|t) ≡12X ′
tVXX (pt|t)Xt + w(pt|t).
Svensson and Williams Optimal Model Policy Under Uncertainty
2. Optimal policy with no learning (NL)
Interpretation: Policymakers forget past Xt−1, . . . in periodt when choosing it .Allows for persistence of modes. But means beliefs don’tsatisfy law of iterated expectations. Requires slightly morecomplicated Bellman equation.Law of motion linear in Xt , dual preferences quadratic inXt . pt|t exogenous.Solution linear in Xt for given pt|t
it = Fi(pt|t)Xt ,
Value function quadratic in Xt for given pt|t
V (Xt , pt|t) ≡12X ′
tVXX (pt|t)Xt + w(pt|t).
Svensson and Williams Optimal Model Policy Under Uncertainty
3. Adaptive optimal policy (AOP)
Similar to adaptive learning, anticipated utility, passivelearning.Policy as under NL (disregarding Bayesian updating),it = i(Xt , pt|t), xt = z(Xt , pt|t)
Transition equation for pt+1|t+1 from Bayes rule:
pt+1|t+1 = Q(Xt , pt|t , xt , it , jt , εt , jt+1, εt+1).
Nonlinear, interacts with Xt . True AOP value function notquadratic in Xt .Evaluation of loss more complex numerically, but recursiveimplementation simple.
Svensson and Williams Optimal Model Policy Under Uncertainty
3. Adaptive optimal policy (AOP)
Similar to adaptive learning, anticipated utility, passivelearning.Policy as under NL (disregarding Bayesian updating),it = i(Xt , pt|t), xt = z(Xt , pt|t)
Transition equation for pt+1|t+1 from Bayes rule:
pt+1|t+1 = Q(Xt , pt|t , xt , it , jt , εt , jt+1, εt+1).
Nonlinear, interacts with Xt . True AOP value function notquadratic in Xt .Evaluation of loss more complex numerically, but recursiveimplementation simple.
Svensson and Williams Optimal Model Policy Under Uncertainty
Bayesian Updating Makes Beliefs Random
Ex post,
pt+1|t+1 = Q(Xt , pt|t , xt , it , jt , εt , jt+1, εt+1)
is random variable, depends on jt+1 and εt+1.Note that
Etpt+1|t+1 = pt+1|t = P ′pt|t .
Bayesian updating gives a mean-preserving spread ofpt+1|t+1.If V (Xt , pt|t) concave in pt|t , lower loss under AOP, and it’sbeneficial to learn.Note that we assume symmetric beliefs. Learning bypublic changes the nature of policy problem, may makestabilization more difficult.
Svensson and Williams Optimal Model Policy Under Uncertainty
Bayesian Updating Makes Beliefs Random
Ex post,
pt+1|t+1 = Q(Xt , pt|t , xt , it , jt , εt , jt+1, εt+1)
is random variable, depends on jt+1 and εt+1.Note that
Etpt+1|t+1 = pt+1|t = P ′pt|t .
Bayesian updating gives a mean-preserving spread ofpt+1|t+1.If V (Xt , pt|t) concave in pt|t , lower loss under AOP, and it’sbeneficial to learn.Note that we assume symmetric beliefs. Learning bypublic changes the nature of policy problem, may makestabilization more difficult.
Svensson and Williams Optimal Model Policy Under Uncertainty
Bayesian Updating Makes Beliefs Random
Ex post,
pt+1|t+1 = Q(Xt , pt|t , xt , it , jt , εt , jt+1, εt+1)
is random variable, depends on jt+1 and εt+1.Note that
Etpt+1|t+1 = pt+1|t = P ′pt|t .
Bayesian updating gives a mean-preserving spread ofpt+1|t+1.If V (Xt , pt|t) concave in pt|t , lower loss under AOP, and it’sbeneficial to learn.Note that we assume symmetric beliefs. Learning bypublic changes the nature of policy problem, may makestabilization more difficult.
Svensson and Williams Optimal Model Policy Under Uncertainty
4. Bayesian optimal policy (BOP)
Optimal “experimentation” incorporated: may alter actionsto mitigate future uncertainty. More complex numerically.Dual Bellman equation as in AOP, but now with beliefupdating equation incorporated in optimization.Because of the nonlinearity of Bayesian updating, solutionno longer linear in Xt for given pt|t .Dual value function V (Xt , pt|t), primal value functionV (Xt , pt|t), no longer quadratic in Xt for given pt|tAlways weakly better than AOP in backward-lookingmodels. Not necessarily true in forward-looking:experimentation by public changes policymaker constraints.
Backward: V = mini∈I
Et [L + δV ]
Forward: V = maxγ∈Γ
mini∈I
Et
[L + δV
]Svensson and Williams Optimal Model Policy Under Uncertainty
4. Bayesian optimal policy (BOP)
Optimal “experimentation” incorporated: may alter actionsto mitigate future uncertainty. More complex numerically.Dual Bellman equation as in AOP, but now with beliefupdating equation incorporated in optimization.Because of the nonlinearity of Bayesian updating, solutionno longer linear in Xt for given pt|t .Dual value function V (Xt , pt|t), primal value functionV (Xt , pt|t), no longer quadratic in Xt for given pt|tAlways weakly better than AOP in backward-lookingmodels. Not necessarily true in forward-looking:experimentation by public changes policymaker constraints.
Backward: V = mini∈I
Et [L + δV ]
Forward: V = maxγ∈Γ
mini∈I
Et
[L + δV
]Svensson and Williams Optimal Model Policy Under Uncertainty
4. Bayesian optimal policy (BOP)
Optimal “experimentation” incorporated: may alter actionsto mitigate future uncertainty. More complex numerically.Dual Bellman equation as in AOP, but now with beliefupdating equation incorporated in optimization.Because of the nonlinearity of Bayesian updating, solutionno longer linear in Xt for given pt|t .Dual value function V (Xt , pt|t), primal value functionV (Xt , pt|t), no longer quadratic in Xt for given pt|tAlways weakly better than AOP in backward-lookingmodels. Not necessarily true in forward-looking:experimentation by public changes policymaker constraints.
Backward: V = mini∈I
Et [L + δV ]
Forward: V = maxγ∈Γ
mini∈I
Et
[L + δV
]Svensson and Williams Optimal Model Policy Under Uncertainty
Numerical Methods & Summary of Results
Suite of programs available on my website for OBS, NLcases. Very fast, efficient, and adaptable.For AOP and BOP, use Miranda-Fackler collocationmethods, CompEcon toolbox.Under NL, V (Xt , pt|t) is not always concave in pt|t .AOP significantly different from NL, but not necessarilylower. Learning typically beneficial in backward-lookingmodels, not always in forward-looking.May be easier to control expectations when agents don’tlearn: the bond traders may get it (more) right, but thatdoesn’t always improve welfare.BOP modestly lower loss than AOPEthical and other issues with BOP relative to AOP:Perhaps not much of a practical problem?
Svensson and Williams Optimal Model Policy Under Uncertainty
Numerical Methods & Summary of Results
Suite of programs available on my website for OBS, NLcases. Very fast, efficient, and adaptable.For AOP and BOP, use Miranda-Fackler collocationmethods, CompEcon toolbox.Under NL, V (Xt , pt|t) is not always concave in pt|t .AOP significantly different from NL, but not necessarilylower. Learning typically beneficial in backward-lookingmodels, not always in forward-looking.May be easier to control expectations when agents don’tlearn: the bond traders may get it (more) right, but thatdoesn’t always improve welfare.BOP modestly lower loss than AOPEthical and other issues with BOP relative to AOP:Perhaps not much of a practical problem?
Svensson and Williams Optimal Model Policy Under Uncertainty
Numerical Methods & Summary of Results
Suite of programs available on my website for OBS, NLcases. Very fast, efficient, and adaptable.For AOP and BOP, use Miranda-Fackler collocationmethods, CompEcon toolbox.Under NL, V (Xt , pt|t) is not always concave in pt|t .AOP significantly different from NL, but not necessarilylower. Learning typically beneficial in backward-lookingmodels, not always in forward-looking.May be easier to control expectations when agents don’tlearn: the bond traders may get it (more) right, but thatdoesn’t always improve welfare.BOP modestly lower loss than AOPEthical and other issues with BOP relative to AOP:Perhaps not much of a practical problem?
Svensson and Williams Optimal Model Policy Under Uncertainty
New Keynesian Phillips Curve Examples
πt = (1− ωjt )πt−1 + ωjt Etπt+1 + γjt yt + cjtεt
Assume policymakers directly control output gap yt .Period loss function
Lt = π2t + 0.1y2
t , δ = 0.98
Example 1: How forward-looking is inflation? Assumeω1 = 0.2, ω2 = 0.8. E(ωj) = 0.5. Fix other parameters:γ = 0.1, c = 0.5.Example 2: What is the slope of the Phillips curve?Assume γ1 = 0.05, γ2 = 0.25. E(γj) = 0.15. Fix otherparameters: ω = 0.5, c = 0.5.In both cases, highly persistent modes:
P =
[0.98 0.020.02 0.98
]Svensson and Williams Optimal Model Policy Under Uncertainty
New Keynesian Phillips Curve Examples
πt = (1− ωjt )πt−1 + ωjt Etπt+1 + γjt yt + cjtεt
Assume policymakers directly control output gap yt .Period loss function
Lt = π2t + 0.1y2
t , δ = 0.98
Example 1: How forward-looking is inflation? Assumeω1 = 0.2, ω2 = 0.8. E(ωj) = 0.5. Fix other parameters:γ = 0.1, c = 0.5.Example 2: What is the slope of the Phillips curve?Assume γ1 = 0.05, γ2 = 0.25. E(γj) = 0.15. Fix otherparameters: ω = 0.5, c = 0.5.In both cases, highly persistent modes:
P =
[0.98 0.020.02 0.98
]Svensson and Williams Optimal Model Policy Under Uncertainty
Example 1: Effect of UncertaintyConstant coefficients vs. OBS
−5 0 5
−8
−6
−4
−2
0
2
4
6
8
Policy: OBS and Constant Modes
πt
yt
−5 0 50
5
10
15
20
25
30
35
Loss: OBS and Constant Modes
πt
Lo
ss
OBS 1OBS 2E(OBS)Constant
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 1: Value functions
0.2 0.4 0.6 0.830
40
50
60
70
80
Loss: NL
p1t
Lo
ss
0.2 0.4 0.6 0.8
40
50
60
70
80
Loss: BOP
p1t
Lo
ss
0.2 0.4 0.6 0.8
40
50
60
70
80
Loss: AOP
p1t
Lo
ss
π
t=0
πt=−5
πt=3.33
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 1: Loss Differences
0.2 0.4 0.6 0.80
0.5
1
1.5
Loss difference: BOP−NL
p1t
Lo
ss
0.2 0.4 0.6 0.8
−6.5
−6
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
x 10−9 Loss differences: BOP−AOP
p1t
Lo
ss
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 1: Optimal Policies
−5 0 5
−8
−6
−4
−2
0
2
4
6
8
Policy: AOP
πt
yt
−5 0 5
−8
−6
−4
−2
0
2
4
6
8
Policy: BOP
πt
yt
p
1t=0.89
p1t
=0.5
p1t
=0.11
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 1: Policy Differences: BOP-AOP
−5 0 5−3
−2
−1
0
1
2
3x 10
−9 Policy difference: BOP−AOP
πt
yt
−5
0
5
0.20.4
0.60.8
−10
−5
0
5
x 10−9
πt
Policy difference: BOP−AOP
p1t
yt
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 2: Effect of UncertaintyConstant coefficients vs. OBS
−5 0 5
−4
−3
−2
−1
0
1
2
3
4
Policy: OBS and Constant Modes
πt
yt
OBS 1OBS 2E(OBS)Constant
−5 0 50
2
4
6
8
10
12
14
16
18
Loss: OBS and Constant Modes
πt
Lo
ss
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 2: Value functions
0.2 0.4 0.6 0.8
20
22
24
26
28
Loss: NL
p1t
Lo
ss
0.2 0.4 0.6 0.8
20
22
24
26
28
Loss: BOP
p1t
Lo
ss
0.2 0.4 0.6 0.8
20
22
24
26
28
Loss: AOP
p1t
Lo
ss
π
t=0
πt=−2
πt=3
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 2: Loss Differences
0.2 0.4 0.6 0.80.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Loss difference: BOP−NL
p1t
Lo
ss
0.2 0.4 0.6 0.8−0.026
−0.024
−0.022
−0.02
−0.018
−0.016
−0.014
−0.012
Loss differences: BOP−AOP
p1t
Lo
ss
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 2: Optimal Policies
−5 0 5
−4
−3
−2
−1
0
1
2
3
4
Policy: AOP
πt
yt
−5 0 5
−4
−3
−2
−1
0
1
2
3
4
Policy: BOP
πt
yt
p
1t=0.92
p1t
=0.5
p1t
=0.08
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 2: Policy Differences: BOP-AOP
−5 0 5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Policy difference: BOP−AOP
πt
yt
−5
0
5
0.20.4
0.60.8
−1
−0.5
0
0.5
πt
Policy difference: BOP−AOP
p1t
yt
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 3: Estimated New Keynesian Model
πt = ωfjEtπt+1 + (1− ωfj)πt−1 + γjyt + cπjεπt ,
yt = βfjEtyt+1 + (1− βfj) [βyjyt−1 + (1− βyj)yt−2]
−βrj (it − Etπt+1) + cyjεyt .
Estimated hybrid model constrained to have one modebackward-looking, one partially forward-looking.
Parameter Mean Mode 1 Mode 2ωf 0.0938 0.3272 0γ 0.0474 0.0580 0.0432βf 0.1375 0.4801 0βr 0.0304 0.0114 0.0380βy 1.3331 1.5308 1.2538cπ 0.8966 1.0621 0.8301cy 0.5572 0.5080 0.5769
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 3: Estimated New Keynesian Model
πt = ωfjEtπt+1 + (1− ωfj)πt−1 + γjyt + cπjεπt ,
yt = βfjEtyt+1 + (1− βfj) [βyjyt−1 + (1− βyj)yt−2]
−βrj (it − Etπt+1) + cyjεyt .
Estimated hybrid model constrained to have one modebackward-looking, one partially forward-looking.
Parameter Mean Mode 1 Mode 2ωf 0.0938 0.3272 0γ 0.0474 0.0580 0.0432βf 0.1375 0.4801 0βr 0.0304 0.0114 0.0380βy 1.3331 1.5308 1.2538cπ 0.8966 1.0621 0.8301cy 0.5572 0.5080 0.5769
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 3: More Detail
Estimated transition probabilities
P =
[0.9579 0.04210.0169 0.9831
]Loss function:
Lt = π2t + y2
t + 0.2(it − it−1)2, δ = 1
Only feasible to consider NL and AOP. Evaluate them via1000 simulations of 1000 periods each.
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 3: Simulated Impulse Responses
0 10 20 30 40 50
0.20.4
0.60.8
1Response of π to π shock
0 10 20 30 40 50
−1
−0.5
0Response of y to π shock
0 10 20 30 40 50
0
1
2
3
Response of i to π shock
0 10 20 30 40 50
0.05
0.1
0.15
0.2
Response of π to y shock
0 10 20 30 40 50−0.2
0
0.2
0.4
0.6
Response of y to y shock
0 10 20 30 40 500
1
2
3Response of i to y shock
ConstantAOP MedianNL Median
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 3: Simulated Distributions
0 20 40 60 80 1000
0.05
0.1
0.15
Distribution of Eπ2
t
AOPNL
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
Distribution of Ey2
t
50 100 150 2000
0.005
0.01
0.015
0.02
0.025
0.03
Distribution of Ei2t
0 50 100 150 2000
0.01
0.02
0.03
0.04
0.05
0.06
Distribution of ELt
Svensson and Williams Optimal Model Policy Under Uncertainty
Example 3: Representative Simulation
0 100 200 300 400 500 600 700 800 900 1000−20
−10
0
10
Inflation
0 100 200 300 400 500 600 700 800 900 1000−20
0
20
Output Gap
0 100 200 300 400 500 600 700 800 900 1000−50
0
50Interest Rate
0 100 200 300 400 500 600 700 800 900 10000
0.5
1Probability in Mode 1
AOPNL
Svensson and Williams Optimal Model Policy Under Uncertainty
Conclusion
MJLQ framework flexible, powerful, yet tractable way ofhandling model uncertainty and non-certainty equivalence.Large variety of uncertainty configurations, also able toincorporate a large variety of CB judgment.Extension to forward-looking variables via recursivesaddlepoint method.Straightforward to incorporate unobservable modes w/olearning.Adaptive policy as easy to implement, harder to evaluate.Bayesian optimal policy more complex, particularly inforward-looking cases.Learning has sizeable effects, may or may not be beneficial.Experimentation seems to have relatively little effect.
Svensson and Williams Optimal Model Policy Under Uncertainty
Conclusion
MJLQ framework flexible, powerful, yet tractable way ofhandling model uncertainty and non-certainty equivalence.Large variety of uncertainty configurations, also able toincorporate a large variety of CB judgment.Extension to forward-looking variables via recursivesaddlepoint method.Straightforward to incorporate unobservable modes w/olearning.Adaptive policy as easy to implement, harder to evaluate.Bayesian optimal policy more complex, particularly inforward-looking cases.Learning has sizeable effects, may or may not be beneficial.Experimentation seems to have relatively little effect.
Svensson and Williams Optimal Model Policy Under Uncertainty
Conclusion
MJLQ framework flexible, powerful, yet tractable way ofhandling model uncertainty and non-certainty equivalence.Large variety of uncertainty configurations, also able toincorporate a large variety of CB judgment.Extension to forward-looking variables via recursivesaddlepoint method.Straightforward to incorporate unobservable modes w/olearning.Adaptive policy as easy to implement, harder to evaluate.Bayesian optimal policy more complex, particularly inforward-looking cases.Learning has sizeable effects, may or may not be beneficial.Experimentation seems to have relatively little effect.
Svensson and Williams Optimal Model Policy Under Uncertainty