Robustness and Monetary Policy Experimentation

Robustness and Monetary PolicyExperimentation

Tim Cogley (UC Davis)Ric Colacito (UNC Chapel Hill)

Lars Hansen (University of Chicago)Tom Sargent (NYU)

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Introduction Motivation

Motivation

When a policy maker has multiple submodels, Bayes’ lawand a Bellman equation tell him to experiment.Nevertheless, Blinder, Lucas, and others have told policymakers not to experiment (i.e., to ignore the Bellmanequation).

In Cogley, Colacito, and Sargent (2007), we studied thebenefits from listening to Bellman (and not Lucas andBlinder).We now study experimentation when the policy makerdoubts both models and his prior over them.

Policy maker wants robust decision rules.

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Introduction Motivation

Motivation

When a policy maker has multiple submodels, Bayes’ lawand a Bellman equation tell him to experiment.Nevertheless, Blinder, Lucas, and others have told policymakers not to experiment (i.e., to ignore the Bellmanequation).

In Cogley, Colacito, and Sargent (2007), we studied thebenefits from listening to Bellman (and not Lucas andBlinder).We now study experimentation when the policy makerdoubts both models and his prior over them.Policy maker wants robust decision rules.

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Introduction Roadmap

Plan of the talk

Two Bellman equations

Policy and value functions

Different θ1’s and θ2’s: risk sensitivity parameters.

Different λ ’s: relative importance of unemployment andinflation.

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Plan of the talk


1. Bayesian problem




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Plan of the talk


1. Bayesian problem

2. Robust problem




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Plan of the talk


1. Bayesian problem

2. Robust problem




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Plan of the talk


1. Bayesian problem

2. Robust problem




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The Economy Setup

The economy

Central Bank chooses vt to

min E0

∞

∑t=0

.995t(U2t +λv2

t ),s.t.

Model z=1 a Samuelson-Solow model that specifies apermanently exploitable inflation-unemployment tradeoff

Model z=2 a Lucas-Phelps model with no exploitabletrade-off

Bayesian updating:α∗ = πα(α,U∗)

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The Economy Setup

The economy


min E0

∞

∑t=0

.995t(U2t +λv2

t ),s.t.

Model z=1

Ut+1 = 0.0023+0.7971Ut−0.2761πt+1 +0.0054η1,t+1

πt+1 = vt +0.0055η3,t

Model z=2

Ut+1 = 0.0007+0.8468Ut−0.2489(πt+1− vt)+0.0055η2,t+1

πt+1 = vt +0.0055η3,t+1


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The Economy Setup

The economy


min E0

∞

∑t=0

.995t(U2t +λv2

t ),s.t.

Model z=1

U∗1 = U1 +A1U +B1v+C1ε∗1

Model z=2

U∗2 = U2 +A2U +C2ε∗2


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The Economy Setup

The economy


min E0

∞

∑t=0

βtr(Ut +λvt),s.t.

Model z=1 (α)

U∗1 = U1 +A1U +B1v+C1ε∗1

Model z=2 (1−α)

U∗2 = U2 +A2U +C2ε∗2


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The Economy Setup

Evolution of αt

Using Bayes’ law:

logαt

1−αt= log

αt−1

1−αt−1+ log

p1(Ut|Ut−1,vt−1)p2(Ut|Ut−1,vt−1)

Timing protocol

vt-1 t , Ut t vt…

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The Economy Setup

Evolution of αt

Using Bayes’ law:

logαt

1−αt= log

αt−1

1−αt−1+ log

p1(Ut|Ut−1,vt−1)p2(Ut|Ut−1,vt−1)

Timing protocol

vt-1 t , Ut t vt…

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The Economy Bayesian problem

Bayesian Problem

V (U,α) = maxv

{r(U,α)+β

[α

∫V (U∗1 ,α∗)dF(ε∗1 )+(1−α)

∫V (U∗2 ,α∗)dF(ε∗2 )

]}

subject to:

U∗1 = U1 +A1U +B1v+C1ε∗1

U∗2 = U2 +A2U +C2ε∗2

α∗ = πα(α,U∗z ), z = {1,2}

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Bayesian Problem

V (U,α) = maxv

{r(U,α)+β

[α

∫V (U∗1 ,α∗)dF(ε∗1 )+(1−α)

∫V (U∗2 ,α∗)dF(ε∗2 )

]}

subject to:

U∗1 = U1 +A1U +B1v+C1ε∗1

U∗2 = U2 +A2U +C2ε∗2

α∗ = πα(α,U∗z ), z = {1,2}

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Bayesian Problem

V (U,α) = maxv

{r(U,α)+β

[α

∫V (U∗1 ,α∗)dF(ε∗1 )+(1−α)

∫V (U∗2 ,α∗)dF(ε∗2 )

]}

subject to:

U∗1 = U1 +A1U +B1v+C1ε∗1

U∗2 = U2 +A2U +C2ε∗2

α∗ = πα(α,U∗z ), z = {1,2}

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Bayesian Problem

V(s,α) = maxv

{r(U,v)+Ez

[EU∗,α∗(βV(U∗,α∗)|U,v,α,z)|U,v,α

]}

subject to:

U∗ = πU(U,v,z,ε∗)α∗ = πα(α,πU(U,v,z,ε∗))z = {1,2}

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Bayesian Problem

V(s,α) = maxv

{r(U,v)+Ez


]}

subject to:


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Bayesian Problem

V(s,α) = maxv

{r(U,v)+Ez


]}

subject to:


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The Economy T1 operator

T1 operator: misspecification of a submodel

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T1(V(U∗,α∗))(U,α,v,z;θ1) =−θ1 logEU∗,α∗

[exp(−V(U∗,α∗)

θ1

)∣∣∣(U,α,v,z)]

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T1(V(U∗,α∗))(U,α,v,z;θ1) =−θ1 logEU∗,α∗


θ1

)∣∣∣(U,α,v,z)]

This is the indirect utility function for a penalized utility min-imization problem that yields a worst-case case distortion tothe distribution over (U∗,α∗) conditional on z that is propor-tional to

exp(−V(U∗,α∗)

θ1

)7 / 22


T2 operator: prior misspecification

T2(V(U∗,α∗))(U,α,v;θ2) =−θ2 logEz


θ2

)∣∣∣(U,α,v)]

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T2 operator: prior misspecification

T2(V(U∗,α∗))(U,α,v;θ2) =−θ2 logEz


θ2

)∣∣∣(U,α,v)]

The associated distortion to the worst-case prior over z is pro-portional to

exp(−V(U,α,v,z)

θ2

)

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The Economy Robust Bellman Equation

Robust Bellman equation

V(U,α) = maxv

{r(U,v)+T2[T1 [(βV(U∗,α∗)(U,v,α,z;θ1))

](U,v,α;θ2)

]}

θ1 measures concern about misspecification of asubmodel.θ2 measures concern about misspecification of the priorα.Idea: replace EU∗,α∗ with T1 and Ez with T2.

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V(U,α) = maxv

{r(U,v)+T2[T1 [(βV(U∗,α∗)(U,v,α,z;θ1))

](U,v,α;θ2)

]}

θ1 measures concern about misspecification of asubmodel.θ2 measures concern about misspecification of the priorα.

Idea: replace EU∗,α∗ with T1 and Ez with T2.

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V(U,α) = maxv

{r(U,v)+T2[T1 [(βV(U∗,α∗)(U,v,α,z;θ1))

](U,v,α;θ2)

]}

θ1 measures concern about misspecification of asubmodel.θ2 measures concern about misspecification of the priorα.Idea: replace EU∗,α∗ with T1 and Ez with T2.

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Quantitative findings Roadmap

Quantitative findings

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Quantitative findings Roadmap

Quantitative findings

Roadmap

1. Risk sensitivity operator T2 only.

2. Risk sensitivity operator T1 only.

3. Both risk sensitivity operators are turned on.

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Quantitative findings T2 only

T2 only: messages

Slants α toward worst case model

When λ is small, Lucas model is worst case model.

When λ is big, the SS is the worst-case model.

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T2 only: messages




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T2 only: messages



→ Therefore, robust policy is less countercyclical.


→ Therefore, robust policy is more countercyclical.

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Non-Robust value function, λ = 0.1

00.2

0.40.6

0.81

−0.02

−0.01

0

0.01

0.02

0.03−0.03

−0.029

−0.028

−0.027

−0.026

−0.025

−0.024

−0.023

−0.022

Prior on Samuelson and SolowUnemployement

Val

ue fu

nctio

n

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T2 only, λ = 0.1 and θ2 = .1

0

0.5

1

−0.01

0

0.01

0.02

0.03

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

αt

Ut

d(U

t,αt)

α ≈ 0

0

0.02

0.04

α = 0.2

Infla

tion

α = 0.4

0.00

0.02

0.04

α = 0.6

Infla

tion

−.015 0 .015 .03

α = 0.8

Unemployment−.015 0 .015 .03

0.00

0.02

0.04

α ≈ 1

Unemployment

Infla

tion

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Value function (no robustness) for λ = 16

00.2

0.40.6

0.81

−0.02

−0.01

0

0.01

0.02

0.03−0.044

−0.042

−0.04

−0.038

−0.036

−0.034

−0.032

−0.03

−0.028

−0.026

Prior on Samuelson and Solow

Unemployement

Val

ue fu

nctio

n

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T2 only, λ = 16 and θ2 = 0.001

α ≈ 0

0

5

10

x 10−4α = 0.2

Infla

tion

α = 0.4

0

5

10

x 10−4α = 0.6

Infla

tion

−.015 0 .015 .03

α = 0.8


0

5

10

x 10−4α ≈ 1

Unemployment

Infla

tion

0

0.5

1

−0.01

0

0.01

0.02

0.03

0.95

1

1.05

1.1

1.15

1.2

1.25

αt

Ut

d(U

t,αt)

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T1 only: messages

Worst case slants shock distribution toward higherprobabilities of deviation-amplifying shock when U is large.

Affects both conditional means and variances.

The robust policy maker adopts a more aggressivecountercyclical stance.

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T1 only: messages




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T1 only: messages




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Distorted shocks to SS model (θ1 = .1)

00.5

1

−0.010

0.010.02

0.03−0.04

−0.02

0

0.02

0.04

α

Distorted Et [η

1,t+1]

U 00.5

1

−0.010

0.010.02

0.03

1

1.01

1.02

α

Distorted Vart [η

1,t+1]

U

00.5

1

−0.010

0.010.02

0.03

−0.01

0

0.01

α

Distorted Et [η

3,t+1]

U 00.5

1

−0.010

0.010.02

0.03

1

1.001

1.002

α

Distorted Vart [η

3,t+1]

U

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Distorted shocks to Lucas model (θ1 = .1)

00.5

1

−0.010

0.010.02

0.03−0.04

−0.02

0

0.02

0.04

0.06

α

Distorted Et [η

2,t+1]

U 00.5

1

−0.010

0.010.02

0.03

0.98

1

1.02

α

Distorted Vart [η

2,t+1]

U

00.5

1

−0.010

0.010.02

0.03

−0.01

0

0.01

α

Distorted Et [η

4,t+1]

U 00.5

1

−0.010

0.010.02

0.03

0.999

1

1.001

α

Distorted Vart [η

4,t+1]

U

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T1 only robust policy: θ1 = .1

α ≈ 0

−0.02

0

0.02

0.04

α = 0.2

Infla

tion

α = 0.4

−0.02

0

0.02

0.04

α = 0.6

Infla

tion

−0.01 0 0.01 0.02 0.03

α = 0.8

Unemployment−0.01 0 0.01 0.02 0.03

−0.02

0

0.02

0.04

α ≈ 1

Unemployment

Infla

tion

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Quantitative findings Both T j ’s

T1 and T2: prediction

T1 only: policy is more countercyclical.

T2 only: policy is less countercyclical.

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T1 and T2: prediction

T1 only: policy is more countercyclical.

T2 only: policy is less countercyclical.

Optimal Bayesian decision rule with experimentation is robustto a mixture of concerns about the two types of

misspecification.

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Both Tj’s, θ1 = .1 and θ2 = .1

α ≈ 0

0

0.02

0.04

α = 0.2

Infla

tion

α = 0.4

0

0.02

0.04

α = 0.6

Infla

tion

−.015 0 .015 .03

α = 0.8


0

0.02

0.04

α ≈ 1

Unemployment

Infla

tion

0

0.5

1

−0.01

0

0.01

0.02

0.03

0.95

0.96

0.97

0.98

0.99

1

αt

Ut

d(U

t,αt)

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Concluding Remarks

Conclusions

Robustness attained by calculating bounds on valuefunctions.This automatically leads to a worst case analysis.The T1 operator checks robustness of a submodel.⇒ Calls for more countercyclical policy.The T2 operator checks robustness w.r.t. prior oversubmodels.⇒ Calls for less countercyclical policy.Bayesian policy with experimentation is robust to bothfears of misspecification.

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Documents

Robustness and Monetary Policy Experimentation