Optimal Monetary Policy 2. Model Uncertainty Ulf S¨oderstr¨om Sveriges Riksbank [email protected] www.riksbank.se/research/soderstrom

Uppsala University August 2009

Optimal Monetary Policy

3. Model Uncertainty

Introduction • Optimal policy depends on model specification and parameters. But these are very uncertain. • Sources of parameter uncertainty – Truly time-varying parameters – Estimation uncertainty • General model uncertainty – Specification uncertainty within same model – Uncertainty across competing models – Parameter uncertainty special case

1

Optimal Monetary Policy

3. Model Uncertainty

• What is optimal policy when parameters or model specification are uncertain? • Certainty equivalence does not hold – Policy can reduce effects of uncertainty • How deal with model uncertainty? – Bayesian approach: minimize average loss across models – Worst-case approach: minimize loss in worst-case model • Knight (1921): Risk v. Uncertainty – Risk: randomness with known probabilities – Uncertainty: randomness with unknown probabilities • Trade-off average performance against robustness

2

Optimal Monetary Policy

3. Model Uncertainty

Agenda 1. Parameter uncertainty • Uncertainty about the effects of policy • Uncertainty about dynamics 2. General model uncertainty • Bayesian approach v. worst-case approach: – Finite set of competing models – Known probabilities • Robust control: – Worst-case approach – Continuum of models in a neighborhood of a reference model – Knightian uncertainty: unknown probabilities 3. Conclusions 4. Matlab application

3

Optimal Monetary Policy

3. Model Uncertainty

1 Parameter uncertainty • Suppose we’re confident in the model specification, but not in the exact parameter values – Truly time-varying parameters – Estimation uncertainty • What is optimal policy? • Minimize expected loss given probability distribution of parameters: Bayesian approach

4

Optimal Monetary Policy

3. Model Uncertainty

1.1 Uncertainty about the effects of policy • References: Brainard (1967), Walsh (2003) • Benchmark New Keynesian model xt = Etxt+1 − st (it − Etπt+1 − rtn) ,

(1.1)

πt = βEtπt+1 + κtxt + et.

(1.2)

¯ and variance σs2, σκ2 . st, κt independent random variables with mean s¯, κ • Objectives   ∞ 1 X 2 2 2 j min Et β πt+j + λxxt+j + λiit+j . {it } 2 j=0

• Note: Microfoundations ⇒ κt, st, λx, λi correlated. Not taken into account for now. • At t, central bank knows rtn, et, β before setting policy, but not st, κt. • Disregard learning: CB faces same distibution of st, κt in every period.

5

(1.3)

Optimal Monetary Policy

3. Model Uncertainty

• Optimal policy with discretion: – Take expectations as given – Series of one-period problems – Assume et, rtn white noise ⇒ Etxt+1 = Etπt+1 = 0 • Lagrangian:  1  2 2 2 n Lt = Et πt + λxxt + λiit − 2φx [xt + st (it − rt )] − 2φπ [πt − κtxt − et] . 2

(1.4)

• First-order conditions for πt, xt, it: Et {πt − φπ } = 0,

(1.5)

Et {λxxt − φx + φπ κt} = 0,

(1.6)

Et {λiit − φxst} = 0.

(1.7)

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Optimal Monetary Policy

3. Model Uncertainty

• Combining the FOCs and (1.1)–(1.2) gives Et {λiit − λxstxt − κtstπt} = 0,

combining the FOCs,

Et {λiit − λxstxt − κtst [κtxt + et]} = 0, 







Et λiit − λx +

κ2t



st [−st (it −



rtn)]

using (1.2), 

− κtstet = 0,

using (1.1),



Et λiit + λx + κ2t s2t [it − rtn] − κtstet = 0.

(1.8) (1.9) (1.10) (1.11)

• rtn, et are known at t and κt, st are independent, so 

2

¯ + λiit + λx + κ

σκ2



2

s¯ +

σs2



[it − rtn] − κ ¯ s¯et = 0.

(1.12)

• So the optimal policy rule is ¯ 2 + σκ2 s¯2 + σs2 ¯ s¯ κ λx + κ n it = r + et. ¯ 2 + σκ2 ) (¯ s2 + σs2) t λi + (λx + κ ¯ 2 + σκ2 ) (¯ s2 + σs2) λi + (λx + κ 





• Look at special cases.

7

(1.13)

Optimal Monetary Policy

3. Model Uncertainty

Case 1: No parameter uncertainty, λi = 0 • Optimal rule it = rtn +

κ ¯ et . (λx + κ ¯ 2) s¯

(1.14)

• Completely offset movements in rtn. • Lean against movements in et: Raise it if et > 0. More aggressively if λx small or s small.

• Certainty equivalence: Optimal policy independent of σe, σrn Optimal policy same as in non-stochastic economy

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Optimal Monetary Policy

3. Model Uncertainty

Case 2: No parameter uncertainty, λi > 0 • Optimal rule κ ¯ s¯ ¯ 2 s¯2 λx + κ n + it = r et. ¯ 2) s¯2 t λi + (λx + κ ¯ 2) s¯2 λi + (λx + κ 



• Interest rate volatility costly: – Do not offset movements in rtn completely. – Respond less to et movements. – λi → ∞ ⇒ i = 0 always. • Certainty equivalence

9

(1.15)

Optimal Monetary Policy

3. Model Uncertainty

Case 3: Parameter uncertainty, λi = 0 • Optimal rule it = rtn +

κ ¯ s¯ et . (λx + κ ¯ 2 + σκ2 ) (¯ s2 + σs2)

(1.16)

• Completely offset movements in rtn. • More cautious response to movements in et.

σs2 or σκ2 → ∞ ⇒ Do not respond to et at all.

• No certainty equivalence: optimal policy different from non-stochastic economy • Brainard (1967): Uncertainty about the effects of policy implies that optimal policy should respond more cautiously to shocks.

• Not necessarily true if parameters are correlated with shocks. • Intuition: – Uncertainty about st, κt increases the effect of it volatility on xt, πt volatility. – Optimal policy therefore reduces volatility in it.

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Optimal Monetary Policy

3. Model Uncertainty

Case 4: Parameter uncertainty, λi > 0 • Optimal rule κ ¯ s¯ ¯ 2 + σκ2 s¯2 + σs2 λx + κ n + it = r et. ¯ 2 + σκ2 ) (¯ ¯ 2 + σκ2 ) (¯ s2 + σs2) t λi + (λx + κ s2 + σs2) λi + (λx + κ 





• Do not offset movements in rtn completely. • More cautious response to movements in et. • More aggressive response to movements in rtn. • Counterexample to the Brainard (1967) result. • Intuition: – Uncertainty about st, κt increases effect of rtn volatility on xt, πt volatility. – Optimal policy therefore offsets rtn movements more. – σs2 or σκ2 → ∞ ⇒

∗ Offset rtn movements completely ∗ Do not respond to et at all.

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(1.17)

Optimal Monetary Policy

3. Model Uncertainty

Uncertainty about the effects of policy • Uncertainty about the effects of policy can make policy more aggressive. • But: with the output equation xt = Etxt+1 − st (it − Etπt+1 − rtn)

(1.18)

extreme uncertainty makes it = rtn always, neutral policy. • So in a sense the Brainard intuition holds: uncertainty about the effects of policy implies that optimal policy should be closer to “neutral,” that is, more cautious.

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Optimal Monetary Policy

3. Model Uncertainty

1.2 Uncertainty about dynamics • Add persistence: xt = αf Etxt+1 + αbxt−1 − σ˜ −1 (it − Etπt+1 − rtn) ,

(1.1)

˜ xt . πt = γf Etπt+1 + γbπt−1 + κ

(1.2)

• No analytical solution. • Backward-looking models: – Craine (1979): one-equation model, persistence uncertain ⇒ policy more aggressive. – S¨oderstr¨om (2002): two-equation model, inflation persistence uncertain ⇒ policy more aggressive, output persistence uncertain ⇒ makes policy more cautious.

• Forward-looking models (two equations): – Kimura and Kurozumi (2007): commitment, inflation or output persistence uncertain ⇒ policy more aggressive.

– Moessner (2005): discretion, inflation persistence uncertain ⇒ policy more aggressive.

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Optimal Monetary Policy

3. Model Uncertainty

1.3 Parameter uncertainty: Conclusions • Brainard (1967): Respond less to shocks if uncertain about the effects of policy. • Several counterexamples: – Correlation between parameters and shocks ⇒ more aggressive response to shocks. – Uncertainty about interest rate effect on x and π ⇒ more aggressive response to rtn. – Uncertainty about inflation (or output) persistence ⇒ more aggressive response to shocks. • But policymakers often cite Brainard, e.g., Blinder (1998, p. 12): My intuition is that [Brainard’s] finding is more general—or at least more wise—in the real world than the mathematics will support. • How important in practice? Often small effects on the performance and design of monetary

policy in estimated models (Levin, Onatski, Williams, and Williams, 2005; Edge, Laubach, and Williams, 2007; Sala, S¨oderstr¨om, and Trigari, 2008).

• Model uncertainty more important? Next topic.

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Optimal Monetary Policy

3. Model Uncertainty

2 Model uncertainty • What if there’s uncertainty about which model is the true one? • How deal with such uncertainty? • Two approaches: – Average across models: Bayesian approach – Focus on worst case model: worst-case approach (robust control)

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Optimal Monetary Policy

3. Model Uncertainty

2.1 Bayesian approach v. worst-case approach • References: Cateau (2007, 2006). • True data-generating process: G • Models: Gk (θk ) for k = 1, . . . , n with parameters θk ∈ Θk • Probability of model k: pk • Prior on Θk : Pk • Policy rules with coefficients γ: {K(γ), γ ∈ Γ} .

(2.1)

• Loss function for model k: vk (θk , γ) = V (Gk (θk ) , K(γ)) .

(2.2)

• CB seeks one rule which minimizes loss, but is unable to decide between models.

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Optimal Monetary Policy

3. Model Uncertainty

• Uncertainty facing the CB: – Within models: θk ∈ Θk , prior Pk over Θk . – Across models: k ∈ {1, . . . , n}, with probability pk . • Two common approaches: – Bayesian – Worst-case • General approach

17

Optimal Monetary Policy

3. Model Uncertainty

Bayesian approach • Choose γ to minimize average loss: av(γ) =

n X k=1

pk EPk {vk (θk , γ)} .

(2.3)

• Minimize expected value across models of expected value within each model. • Pro: – Weight models by its probability: Most likely model gets largest weight, least likely model gets smallest weight • Con: – CB not concerned with very bad outcomes (banking crisis, exchange rate crisis, . . . )

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Optimal Monetary Policy

3. Model Uncertainty

Example of Bayesian approach (p1 = 0.9, p2 = 0.1) 5

Model 2

variance of inflation

4

3

Bayesian 2

1

0

Model 1

0

0.5

1

1.5

2

2.5

3

change in interest rate with respect to change in inflation

Figure 8: The Bayesian approach

interest rate, with respect to changes in inflation. The policy-maker has two competing 19

Optimal Monetary Policy

3. Model Uncertainty

A worst-case approach • Choose γ to minimize loss in worst-case outcome:  

 

wc(γ) = max  max v1 (θ1, γ) , max v2 (θ2, γ) , . . . , max vn (θn, γ) . θ1 ∈Θ1

θ2 ∈Θ2

θn ∈Θn

(2.4)

• Pros: – No need to specify CB beliefs (probability distribution), can handle Knightian uncertainty – Avoids very bad outcomes • Cons: – Does not allow for different degrees of robustness, only considers extreme view – Disregards beliefs: worst-case outcome often very unlikely

20

worst-case approach. In the worst-case approach, the policy-maker chooses policy according Optimal Monetary Policy to the following criterion:

3. Model Uncertainty

wc(γ) = max {v1 (γ), v2 (γ), ..., vn (γ)} . Example of worst-case approach (p1 = 0.9, p2 = 0.1)

(38)

What the criterion above entails can again be illustrated by way of an example. Consider 5

Model 2

variance of inflation

4

3

2

Model 1

1

0

0

0.5

1

1.5

2

2.5

3

change in interest rate with respect to change in inflation

Figure 9: The worst-case approach again the policy-maker with Model 1 and Model 2 as competing reference models for the economy. With the worst-case approach, the policy-maker’s objective is to ensure that the policy decision rule works reasonably well no matter which of the two models is true. To do that, the policy-maker contemplates 21 the policy choices and determines which model

Optimal Monetary Policy

3. Model Uncertainty

A general approach • Choose γ to minimize expected value of transformation of loss: h(γ) =

n X

k=1

pk EPk {φ (vk (θk , γ))} ,

(2.5)

where φ(·) transforms the value of the loss function in model k. • Transformation function φ(·) characterizes the central bank’s attitude to across-model risk. • Degree of aversion φ00(x)/φ0(x) – φ(x) convex: φ00(x)/φ0(x) > 0, CB averse to across-model risk (weigh bad outcomes more than good) – φ(x) linear: φ00(x)/φ0(x) = 0, CB neutral to across-model risk, Bayesian approach – φ00(x)/φ0(x) → +∞: Worst-case approach • φ(x) distinguishes between within-model risk and across-model risk, determines trade-off average performance/robustness.

• General approach nests Bayesian and worst-case approaches • Note: Need probability distribution, cannot handle Knightian uncertainty 22

balance between average performance and robustness. Indeed, Cateau (2005) shows that his Optimal Monetary Policy 3. Model Uncertainty framework nests both the Bayesian approach and the worst-case approach as special cases: the Bayesian approach is the special case where the decision-maker is neutral to the acrossmodel risk (the decision-maker’s degree of aversion towards the across-model risk is 0), and the worst-case approach is the special case where the decision-maker’s degree of aversion ηx e towards − 1 the φ00across-model (x) φ(x) = , = η. risk is infinite. Therefore, the degree of aversion towards the (2.6) 0 risk which reflects the attitude of the decision-maker towards model uncertainty ηacross-model φ (x) determines the extent to which the decision-maker wants to trade-off average performance for robustness.

Example of general approach

3.5

η=0 η = 0.1 η = 0.2

variance of inflation

Model 2 3 2.5 2 1.5

Model 1 1 0.5

0

0.5

1

1.5

2

2.5

3

variance of inflation

3.5 η=0 η = 0.1 η = 0.2

3 2.5 2 1.5 1 0.5

0

0.5

1

1.5

2

2.5

3

change in interest rate with respect to inflation

Figure 10: Aversion to across-model risk Figure 10 shows how accounting for the across-model risk helps the policy-maker to balance 23

Optimal Monetary Policy

3. Model Uncertainty

Determining the probability distribution within a model • Classical econometrics: Parameter fixed, estimators random • Bayesian econometrics: – Parameters random variables – Estimate posterior distribution of parameters ⇒ probabilities within model – Estimate posterior odds of competing models ⇒ probabilities across models

24

Optimal Monetary Policy

3. Model Uncertainty

Determining the degree of aversion to across-model risk • Acceptable loss: v ∗ • Define premium δ such that the CB is indifferent between achieving v ∗ for sure and facing model uncertainty: ∗

φ (v + δ) =

N X k=1

pk φ (vk ) .

(2.7)

• Global approach: With φ(x) = (eηx − 1) /η, η satisfies N X k=1

pk eη(vk −v

∗ −δ)

= 1.

(2.8)

• Local approach: How does η behave with small across-model risk? φ (v ∗ + δ) = Eφ (˜ v)

(2.9)

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Optimal Monetary Policy

3. Model Uncertainty

2.2 Knightian uncertainty and Robust control • References: Walsh (2003), Cateau (2006), Giordani and S¨oderlind (2004). • For details: Hansen and Sargent (2008). • Central bank has a reference model, but wants to be robust against model misspecification. • No probability distribution over possible models, Knightian uncertainty. • Minimize loss in worst-case model in a neighborhood of the reference model.

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Optimal Monetary Policy

3. Model Uncertainty

Model setup • Reference: Giordani and S¨oderlind (2004). • Reference model:     

x1t+1 Etx2t+1





   

= A 



x1t x2t

    

+ But + Cεt+1.

(2.1)

• Σε = I, so Var (Cεt+1) = CC0. • Central bank loss function: E0

∞ X t=0

β t [x0tQxt + 2x0tUut + u0tRut] , "

(2.2)

#0

where xt ≡ x01t x02t . • Central bank sets ut to minimize loss, but fears that the model is misspecified and wants to be robust against such misspecification.

27

Optimal Monetary Policy

3. Model Uncertainty

• Distorted model:     

x1t+1 Etx2t+1





   

   

=A

x1t x2t

    

+ But + C (εt+1 + vt+1) .

(2.3)

• One distortion for each shock: nv = nε. • Distortions vt+1 disguised by shocks εt+1, so multiplied by C. • Worst-case distortions vt+1 chosen by “evil agent” to maximize loss subject to “budget constraint” E0

∞ X t=0

0 vt+1 ≤ η. β tvt+1

(2.4)

• η measures central bank’s preference for robustness:

Larger η ⇒ larger distortions, so central bank more robust.

• Robust central bank surrounds reference model with a ball of models with radius η. Wants to be robust to worst-case model within this set of models.

• General model misspecification: distortions feed back from current state variables. • Note dating convention: vt+1 chosen at t, affects economy at t + 1. 28

Optimal Monetary Policy

3. Model Uncertainty

Robust control • Central bank minimizes loss, evil agent maximizes loss: min max E0 {ut } {vt+1 }

∞ X t=0

β t [x0tQxt + 2x0tUut + u0tRut] ,

(2.5)

subject to distorted model (2.3) and budget for evil agent (2.4). • The solution is a Nash equilibrium between the central bank and the evil agent. • Equivalent multiplier problem: min max E0 {ut } {vt+1 }

∞ X t=0

0 β t [x0tQxt + 2x0tUut + u0tRut − θvt+1 vt+1]

subject to distorted model (2.3). • Multiplier θ inversely related to budget η: – No robustness: η → 0 ⇒ θ → ∞ – Infinite robustness (H∞): η → η¯ from below ⇒ θ → θ from above – η¯ upper bound on η: above η¯, the evil agent destabilizes the model

29

(2.6)

Optimal Monetary Policy

3. Model Uncertainty

Solving the robust control problem • Rewrite robust control problem as min max E0 {ut } {vt+1 }

∞ X t=0

β

t

x0tQxt



+

2x0tU∗u∗t

+

u∗t 0R∗u∗t



,

(2.7)

subject to     

x1t+1 Etx2t+1





   

   

=A

x1t x2t

    

+ B∗u∗t + Cεt+1,

where 

R∗ ≡ ∗

B ≡

    "

R

0

0 −θI #

B C ,

    



,

u∗t ≡  

∗

"



ut vt+1

   

,

#

U ≡ U 0 .

• Standard control problem, solve with standard methods.

First-order condition for minimization and maximization problem are the same.

30

(2.8)

Optimal Monetary Policy

3. Model Uncertainty

Equilibrium concepts • Worst-case equilibrium: Central bank uses robust policy, private expectations consistent, distortions optimal.

• Approximating equilibrium: Central bank uses robust policy, private expectations consistent, no distortions.

• The worst-case model is the outcome the central bank fears most and wants to guard against. But not very likely.

• The approximating equilibrium is a more likely outcome (the most likely outcome?). • Central bank loss is larger in approximating equilibrium than in reference model with optimal non-robust policy. Difference = Cost of insuring against worst-case outcome.

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Optimal Monetary Policy

3. Model Uncertainty

Solution with commitment • Symmetry: both central bank and evil agent able to commit. • Solution:     

x1t+1 ρ2t+1

             



   

   

= M

x1t ρ2t

    

+ Cεt+1,

(2.9)



x2t ut vt+1 ρ1t



            

 

= N 

x1t ρ2t

    

.

(2.10)

• Optimal policy and distortions:     

ut vt+1





   

   

=F

x1t ρ2t

    

.

(2.11)

• Optimality, Time-inconsistency, History dependence. • Distortions feed back from state variables (x1t, ρ2t). Captures general misspecification: parameters, covariances, . . .

32

Optimal Monetary Policy

3. Model Uncertainty

Solution with discretion • Symmetry: both central bank and evil agent unable to commit, reoptimize every period. • Solution: x1t+1 = Mx1t + Cεt+1,

(2.12)

x2t = Nx1t,     

ut vt+1

(2.13)

    

= Fx1t.

(2.14)

• Suboptimality, Time-consistency, No history dependence. • Distortions feed back from state variables (x1t).

33

Optimal Monetary Policy

3. Model Uncertainty

Solution with simple rules • Suppose the central bank and the evil agent use simple rule:     

ut vt+1

    

= Fxt.

(2.15)

• Solution: x1t+1 = Mx1t + Cεt+1,

(2.16)

x2t = Nx1t.

(2.17)

• Find optimal simple rules using a non-linear optimization routine.

34

Optimal Monetary Policy

3. Model Uncertainty

Choosing the preference for robustness • Preference for robustness (η or θ) determines the size of set of surrounding models. • In LQ framework the evil agent’s budget constraint always binds: worst-case model is on the boundary of the set.

• Larger budget ⇒ larger distortions ⇒ easier detect model misspecification. • Hansen and Sargent (2008): Choose θ to achieve desired “detection-error probability” 1. Take agnostic position on whether DGP is the approximating model (A) or the worst-case model (B). 2. Compute detection-error probability: probability of making the wrong choice between the two models given in-sample fit in finite sample. 3. Choose θ to achieve desired level of detection-error probability. • No longer Knightian uncertainty.

Pure Knightian: use H∞ control, setting θ = θ

35

Optimal Monetary Policy

3. Model Uncertainty

Calculating the detection-error probability • Lij = likelihood of model j if model i generated the data. • Log likelihood ratio when model i generated the data: ri ≡ log

Lii , Lij

j 6= i,

i = A, B.

(2.18)

• Detection error if ri ≤ 0: mistakenly conclude model j generated data. • Probability of making detection error if model i generated the data: pi = freq (ri ≤ 0)

(2.19)

over many simulations of a finite sample. • Detection-error probability: p(θ) =

1 (pA + pB ) . 2

(2.20)

• θ → +∞ ⇒ no distortions (A = B), so p(θ) = 0.5. • θ → 0 ⇒ p(θ) → 0. • Choose θ to obtain desired p(θ), for instance, 0.2, 0.1, 0.05. 36

Optimal Monetary Policy

3. Model Uncertainty

Robust control: An analytical example • Reference: Leitemo and S¨oderstr¨om (2008b) • New Keynesian reference model: πt = βEtπt+1 + κxt + Σπ επt ,

(2.21)

xt = Etxt+1 − σ −1 [it − Etπt+1] + Σxεxt.

(2.22)

• Loss function: ∞ X



(2.23)

πt = βEtπt+1 + κxt + Σπ [επt + vtπ ] ,

(2.24)

xt = Etxt+1 − σ −1 [it − Etπt+1] + Σx [εxt + vtx] .

(2.25)

E0

t=0

β

t



πt2

+

λx2t

• Distorted model:

• Budget for evil agent: E0

∞ X

t=0

β

t

(vtπ )2



+

(vtx)2



≤ η.

(2.26)

37

Optimal Monetary Policy

3. Model Uncertainty

Optimal policy and worst-case distortions with discretion • Lagrangian E0

∞ X t=0

β

t



πt2

+

λx2t

−θ

(vtπ )2



+

(vtx)2



−µπt [πt − βEtπt+1 − κxt − Σπ vtπ − Σπ επt ]

−µxt[xt

− Etxt+1 + σ

−1

(it − Etπt+1) −

Σxvtx

(2.27) −

Σxεxt]



,

• First-order conditions imply κ xt = − π t , λ Σπ vtπ = πt , θ vtx = 0.

(2.28) (2.29) (2.30)

– Optimal trade-off (2.28) independent of preference for robustness. – No distortions to output equation: can be offset (without cost) by central bank. – Worst-case distortions to inflation equation increasing in πt and Σπ , and decreasing in θ (increasing in the preference for robustness).

38

Optimal Monetary Policy

3. Model Uncertainty

Equilibrium • Optimal interest rate rule: it = cN επt + σΣxεxt, cN ≡

(2.31)

σκ > 0. λ(1 − Σ2π /θ) + κ2

(2.32)

• Approximating equilbrium for πt and xt: πt = aN Σπ επt ,

(2.33)

xt = bN Σπ επt ,

(2.34)

where aN bN

κ2 ≡ 1− > 0, λ(1 − Σ2π /θ) + κ2 κ ≡ − < 0. λ(1 − Σ2π /θ) + κ2

(2.35) (2.36)

39

Optimal Monetary Policy

3. Model Uncertainty

The effects of robustness • An increased preference for robustness (smaller θ) implies – Worst-case distortions larger – Policy more aggressive (cN larger) – Output more volatile (bN larger) – Inflation less volatile (aN smaller) • Central bank fears that cost-push shocks (επt ) have larger effect on inflation. Therefore, the optimal policy responds more aggressively to these shocks. As a consequence, when there are no distortions, output is more volatile and inflation is less volatile. • Very simple model, only επt creates trade-off. • Leitemo and S¨oderstr¨om (2008a): Open-economy model, more trade-offs, preference for

robustness differs across equations. Optimal policy more or less aggressive depending on type of shock and source of misspecification.

• Dennis, Leitemo, and S¨oderstr¨om (2009b): Larger estimated open-economy model. Central

bank fears in particular shocks to exchange rate and inflation. Output shocks not very costly.

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Optimal Monetary Policy

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Robust control: An alternative approach • Dennis, Leitemo, and S¨oderstr¨om (2009a): Adapt the algorithms of Dennis (2007) to robust control.

• Distorted model given by A0xt = A1xt−1 + A2Etxt+1 + A3ut + A4Etut+1 + A5 (εt + vt) . • No robustness: Same solution as standard algorithms. • Robustness: Not necessarily same solution. – If written as in (2.37), distortions chosen at t affect economy at t. Evil agent distorts conditional mean and variance. – In standard setup (2.3), distortions chosen at t affect economy at t1. Evil agent distorts only conditional mean.

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(2.37)

Optimal Monetary Policy

3. Model Uncertainty

2.3 Model uncertainty: Conclusions • Bayesian approach: – Weight models with probabilities – Requires probabilities – No particular weight on bad outcomes • Worst-case approach: – All weight on worst outcome, even if very unlikely – No need for probabilities • Generalized approach (Cateau): – Nests Bayesian and worst-case approaches – Requires probabilities – Works well also if very different models

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Optimal Monetary Policy

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• Robust control: – All weight on worst outcome, even if very unlikely – Does not require probabilities – But set of models restricted using detection-error probabilities – Works better if good reference model(?) • Does uncertainty imply caution???

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Optimal Monetary Policy

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3 Conclusion: How does uncertainty affect monetary policy? • Parameter uncertainty: – Uncertainty about effects of policy ⇒ optimal policy typically more cautious – Uncertainty about dynamics ⇒ optimal policy typically more aggressive – But exceptions in both cases • Model uncertainty: – Bayesian approach: No general conclusions – Avoid worst-case scenario ⇒ optimal policy often more aggressive • Data uncertainty (not covered here): – Certainty equivalence to efficient estimate – Respond less to noisy indicator – But how do we know if estimate is efficient? • In general: Optimal policy is more or less aggressive depending on type and source of uncertainty

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Optimal Monetary Policy

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Empirical importance • Parameter uncertainty – Does not seem to have large effects: Rudebusch (2001); Edge, Laubach, Williams (2007); Levin, Onatski, Williams, Williams (2005), Sala, S¨oderstr¨om, Trigari (2008) – Large effects if large model with many imprecisely estimated parameters: Sack (2000), S¨oderstr¨om (2000) • Model uncertainty – Not much work – Caveat: Bayesian methods tend to put much weight on one model, posterior odds depend on prior • Data uncertainty – Seems very important ∗ Data revisions: Croushore and Stark (2000)

∗ Real time measures of natural rates: Orphanides and van Norden (2002), Sala et al. (2009).

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Optimal Monetary Policy

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Central bank learning • Central bank learning about private agents’ behavior • Private agents learning about the economy and monetary policy • Learning ⇒ more gradual behavior (cf. Kalman filter)

More efficient if discretionary policy? (Dennis and Ravenna, 2008)

• Experimentation: “Active learning” – Sacrifice stability today for more efficiency in future – Can imply more aggressive policy – Reasonable? Blinder (1998): “You don’t conduct experiments on a real economy solely to sharpen your econometric estimates” • Svensson and Williams (2008) – Optimal active and passive learning in small forward-looking model – Active learning computationally intensive – Gains from experimentation small: Active learning not very different from passive

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Optimal Monetary Policy

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4 Matlab application • Again, consider the hybrid New Keynesian model πt = (1 − ψπ )βEtπt+1 + ψπ πt−1 + κxt + ut, 1 xt = (1 − ψx)Etxt+1 + ψxxt−1 − [it − Etπt+1 − rte] , σ u ut = ρuut−1 + εt ,

(4.2)

e rte = ρr rt−1 + εrt .

(4.4)

• Now assume three different specifications: 1. Forward-looking: ψπ = ψx = 0.1 2. Backward-looking: ψπ = ψx = 0.9 3. Hybrid: ψπ = ψx = 0.5 • Other parameters constant at same values as in other application

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(4.1)

(4.3)

Optimal Monetary Policy

3. Model Uncertainty

Analysis • Use Matlab code described in S¨oderlind (1999) and Giordani and S¨oderlind (2004). This code is available at Paul S¨oderlind’s webpage.

• The script ModelUncertainty.m proceeds in five steps: 1. Solves the three models with a benchmark simple rule 2. Optimizes simple rules in each model 3. Evaluates the simple rules across models and selects rule that performs best in worst case 4. Optimizes a simple rule across models, minimizing average loss (Bayesian approach) 5. Calculates robust policy under commitment and discretion (Robust control)

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References Blinder, Alan S. (1998), Central Banking in Theory and Practice, The MIT Press. Brainard, William (1967), “Uncertainty and the effectiveness of policy,” American Economic Review , 57 (2), 411–425. Cateau, Gino (2006), “Guarding against large policy errors under model uncertainty,” Working Paper No. 2006-13, Bank of Canada. ——— (2007), “Monetary policy under model and data-parameter uncertainty,” Journal of Monetary Economics, 54 (7), 2083– 2101. Craine, Roger (1979), “Optimal monetary policy with uncertainty,” Journal of Economic Dynamics and Control , 1 (1), 59–83. Croushore, Dean and Tom Stark (2000), “A funny thing happened on the way to the data bank: A real-time data set for macroeconomists,” Federal Reserve Bank of Philadelphia Business Review , 15–27. Dennis, Richard (2007), “Optimal policy rules in rational-expectations models: New solution algorithms,” Macroeconomic Dynamics, 11 (1), 31–55. Dennis, Richard and Federico Ravenna (2008), “Learning and optimal monetary policy,” Journal of Economic Dynamics and Control , 32 (6), 1964–1994. Dennis, Richard, Kai Leitemo, and Ulf S¨oderstr¨om (2009a), “Methods for robust control,” Journal of Economic Dynamics and Control , 33 (8), 1604–1616. ——— (2009b), “Monetary policy in a small open economy with a preference for robustness,” Manuscript, Sveriges Riksbank. Edge, Rochelle M., Thomas Laubach, and John C. Williams (2007), “Welfare-maximizing monetary policy under parameter uncertainty,” Working Paper No. 2007-11, Federal Reserve Bank of San Francisco. Forthcoming, Journal of Applied Econometrics.

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Giordani, Paolo and Paul S¨oderlind (2004), “Solution of macromodels with tbpansen-Sargent robust policies: Some extensions,” Journal of Economic Dynamics and Control , 28 (12), 2367–2397. Hansen, Lars Peter and Thomas J. Sargent (2008), Robustness, Princeton University Press. Kimura, Takeshi and Takushi Kurozumi (2007), “Optimal monetary policy in a micro-founded model with parameter uncertainty,” Journal of Economic Dynamics and Control , 31 (2), 399–431. Knight, Frank H. (1921), Risk, Uncertainty and Profit, Boston, MA: Houghton Mifflin Company. Leitemo, Kai and Ulf S¨oderstr¨om (2008a), “Robust monetary policy in a small open economy,” Journal of Economic Dynamics and Control , 32 (10), 3218–3252. ——— (2008b), “Robust monetary policy in the New-Keynesian framework,” Macroeconomic Dynamics, 12 (S1), 126–135. Levin, Andrew T., Alexei Onatski, John C. Williams, and Noah Williams (2005), “Monetary policy under uncertainty in microfounded macroeconometric models,” in Mark Gertler and Kenneth Rogoff (eds.), NBER Macroeconomics Annual , The MIT Press. Moessner, Richhild (2005), “Optimal discretionary policy and uncertainty about inflation persistence,” Working Paper No. 540, European Central Bank. Orphanides, Athanasios and Simon van Norden (2002), “The unreliability of output-gap estimates in real time,” Review of Economics and Statistics, 84 (4), 569–583. Rudebusch, Glenn D. (2001), “Is the Fed too timid? Monetary policy in an uncertain world,” Review of Economics and Statistics, 83 (2), 203–217. Sack, Brian (2000), “Does the Fed act gradually? A VAR analysis,” Journal of Monetary Economics, 46 (1), 229–256. Sala, Luca, Ulf S¨oderstr¨om, and Antonella Trigari (2008), “Monetary policy under uncertainty in an estimated model with labor market frictions,” Journal of Monetary Economics, 55 (5), 983–1006.

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——— (2009), “Estimating potential output in a modern business cycle model,” Manuscript, Sveriges Riksbank. S¨oderlind, Paul (1999), “Solution and estimation of RE macromodels with optimal policy,” European Economic Review Papers and Proceedings, 43 (4–6), 813–823. S¨oderstr¨om, Ulf (2000), “Should central banks be more aggressive?” Manuscript, Sveriges Riksbank. ——— (2002), “Monetary policy with uncertain parameters,” Scandinavian Journal of Economics, 104 (1), 125–145. Svensson, Lars E. O. and Noah Williams (2008), “Optimal monetary policy under uncertainty in DSGE models: A Markov jump-linear-quadratic approach,” Working Paper No. 13892, National Bureau of Economic Research. Walsh, Carl E. (2003), “Implications of a changing economic structure for the strategy of monetary policy,” in Monetary Policy and Uncertainty: Adapting to a Changing Economy, Federal Reserve Bank of Kansas City.

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