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”

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?

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?

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

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

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 ].

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 ].

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:

∞∑τ=0

δτLt+τ (1)

Period loss:

Lt ≡12

Xtxtit

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:

∞∑τ=0

δτLt+τ (1)

Period loss:

Lt ≡12

Xtxtit

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 . . .

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 ).

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 ).

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

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 .

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 .

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).

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).

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.

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.

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.

Ex post,

Etpt+1|t+1 = pt+1|t = P ′pt|t .

Ex post,

Etpt+1|t+1 = pt+1|t = P ′pt|t .

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

[L + δV

]Svensson and Williams Optimal Model Policy Under Uncertainty

Et [L + δV ]

mini∈I

[L + δV

Et [L + δV ]

mini∈I

[L + δV

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?

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:

[0.98 0.020.02 0.98

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:

[0.98 0.020.02 0.98

Example 1: Effect of UncertaintyConstant coefficients vs. OBS

−5 0 5

Policy: OBS and Constant Modes

−5 0 50

Loss: OBS and Constant Modes

OBS 1OBS 2E(OBS)Constant

Example 1: Value functions

0.2 0.4 0.6 0.830

Loss: NL

0.2 0.4 0.6 0.8

Loss: BOP

0.2 0.4 0.6 0.8

Loss: AOP

πt=−5

πt=3.33

Example 1: Loss Differences

0.2 0.4 0.6 0.80

Loss difference: BOP−NL

0.2 0.4 0.6 0.8

−6.5

−5.5

−4.5

−3.5

−2.5

x 10−9 Loss differences: BOP−AOP

Example 1: Optimal Policies

−5 0 5

Policy: AOP

−5 0 5

Policy: BOP

1t=0.89

Example 1: Policy Differences: BOP-AOP

−5 0 5−3

−9 Policy difference: BOP−AOP

0.20.4

0.60.8

x 10−9

Policy difference: BOP−AOP

Example 2: Effect of UncertaintyConstant coefficients vs. OBS

−5 0 5

Policy: OBS and Constant Modes

OBS 1OBS 2E(OBS)Constant

−5 0 50

Loss: OBS and Constant Modes

Example 2: Value functions

0.2 0.4 0.6 0.8

Loss: NL

0.2 0.4 0.6 0.8

Loss: BOP

0.2 0.4 0.6 0.8

Loss: AOP

πt=−2

Example 2: Loss Differences

0.2 0.4 0.6 0.80.25

Loss difference: BOP−NL

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

Example 2: Optimal Policies

−5 0 5

Policy: AOP

−5 0 5

Policy: BOP

1t=0.92

Example 2: Policy Differences: BOP-AOP

−5 0 5

−0.4

−0.3

−0.2

−0.1

0.20.4

0.60.8

−0.5

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

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

Example 3: More Detail

Estimated transition probabilities

[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.

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

−0.5

0Response of y to π shock

0 10 20 30 40 50

Response of i to π shock

0 10 20 30 40 50

Response of π to y shock

0 10 20 30 40 50−0.2

Response of y to y shock

0 10 20 30 40 500

3Response of i to y shock

ConstantAOP MedianNL Median

Example 3: Simulated Distributions

0 20 40 60 80 1000

Distribution of Eπ2

0 20 40 60 80 1000

Distribution of Ey2

50 100 150 2000

Distribution of Ei2t

0 50 100 150 2000

Distribution of ELt

Example 3: Representative Simulation

0 100 200 300 400 500 600 700 800 900 1000−20

Inflation

0 100 200 300 400 500 600 700 800 900 1000−20

Output Gap

0 100 200 300 400 500 600 700 800 900 1000−50

50Interest Rate

0 100 200 300 400 500 600 700 800 900 10000

1Probability in Mode 1

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.

Conclusion

Optimal Monetary Policy under Uncertainty in DSGE...

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