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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Introduction Roadmap
Plan of the talk
Two Bellman equations
1. Bayesian problem
Policy and value functions
Different θ1’s and θ2’s: risk sensitivity parameters.
Different λ ’s: relative importance of unemployment andinflation.
3 / 22
Introduction Roadmap
Plan of the talk
Two Bellman equations
1. Bayesian problem
2. Robust problem
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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Introduction Roadmap
Plan of the talk
Two Bellman equations
1. Bayesian problem
2. Robust problem
Policy and value functions
Different θ1’s and θ2’s: risk sensitivity parameters.
Different λ ’s: relative importance of unemployment andinflation.
3 / 22
Introduction Roadmap
Plan of the talk
Two Bellman equations
1. Bayesian problem
2. Robust problem
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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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∗)
4 / 22
The Economy Setup
The economy
Central Bank chooses vt to
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
Bayesian updating:α∗ = πα(α,U∗)
4 / 22
The Economy Setup
The economy
Central Bank chooses vt to
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
Bayesian updating:α∗ = πα(α,U∗)
4 / 22
The Economy Setup
The economy
Central Bank chooses vt to
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
Bayesian updating:α∗ = πα(α,U∗)
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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…
5 / 22
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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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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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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The Economy Bayesian problem
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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The Economy Bayesian problem
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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The Economy Bayesian problem
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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The Economy T1 operator
T1 operator: misspecification of a submodel
7 / 22
The Economy T1 operator
T1 operator: misspecification of a submodel
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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The Economy T1 operator
T1 operator: misspecification of a submodel
T1(V(U∗,α∗))(U,α,v,z;θ1) =−θ1 logEU∗,α∗
[exp(−V(U∗,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
The Economy T2 operator
T2 operator: prior misspecification
T2(V(U∗,α∗))(U,α,v;θ2) =−θ2 logEz
[exp(−V(U∗,α∗)
θ2
)∣∣∣(U,α,v)]
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The Economy T2 operator
T2 operator: prior misspecification
T2(V(U∗,α∗))(U,α,v;θ2) =−θ2 logEz
[exp(−V(U∗,α∗)
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.
9 / 22
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.
9 / 22
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.
9 / 22
Quantitative findings Roadmap
Quantitative findings
10 / 22
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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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.
11 / 22
Quantitative findings T2 only
T2 only: messages
Slants α toward worst case model
When λ is small, Lucas model is worst case model.
→ Therefore, robust policy is less countercyclical.
When λ is big, the SS is the worst-case model.
→ Therefore, robust policy is more countercyclical.
11 / 22
Quantitative findings T2 only
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
12 / 22
Quantitative findings T2 only
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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Quantitative findings T2 only
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
14 / 22
Quantitative findings T2 only
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
Unemployment−.015 0 .015 .03
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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Quantitative findings T1 only
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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Quantitative findings T1 only
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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Quantitative findings T1 only
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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Quantitative findings T1 only
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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Quantitative findings T1 only
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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Quantitative findings T1 only
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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Quantitative findings Both T j ’s
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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Quantitative findings Both T j ’s
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
Unemployment−.015 0 .015 .03
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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