Optimal Monetary Policy 3. Model Uncertainty Ulf S¨oderstr¨om Bocconi University and Sveriges Riksbank [email protected] ulf.c.soderstrom.googlepages.com

Uppsala University, August 20, 2008

Optimal Monetary Policy

3. Model Uncertainty

Introduction • General model uncertainty. – Specification uncertainty within same model – Uncertainty across competing models – Parameter uncertainty special case • How deal with general 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

1

Optimal Monetary Policy

3. Model Uncertainty

Agenda 1. Bayesian approach v. worst-case approach: • Finite set of competing models • Known probabilities 2. Robust control: • Worst-case approach • Continuum of models in a neighborhood of a reference model • Knightian uncertainty: unknown probabilities 3. Application from Cateau (2006) 4. Conclusions 5. Matlab application

2

Optimal Monetary Policy

3. Model Uncertainty

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} .

(1.1)

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

(1.2)

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

3

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

4

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)} .

(1.3)

• 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, . . . )

5

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 6

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

(1.4)

• 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

7

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 the 8 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))} ,

(1.5)

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 – 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

9

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 (1.6) 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 10

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

11

Optimal Monetary Policy

3. Model Uncertainty

Determining the probability distribution across models • State and observation equations for model k: xkt+1 = axkt + ck εkt+1,

(1.7)

yt+1 = hk xkt+1.

(1.8)

• Prior probability on model k: pk,0 = P (mk ). • CB observes y t ≡ (y0, y1, . . . , yt), updates using Bayes’s law 

t



pk,t = P mk |y P (y t|mk ) P (mk ) = P t h P (y |mh ) P (mh )   t ∝ P y |mk P (mk ) ≡ αk,t.

(1.9)

• Posterior probability of model k is proportional to marginal likelihood of the data given model k, weighted by the prior probability of model k.

12

Optimal Monetary Policy

3. Model Uncertainty

• Recursion for αk,t: 

t+1



αk,t+1 P y |mk = . αk,t P (y t|mk )

(1.10)

• The Kalman filter implies 

P y

t+1



|mk =

t Y τ =0

P (yτ +1|y τ , mk ) .

(1.11)

• Thus,   αk,t+1 t = P yt+1|y , mk . αk,t

(1.12)

• Posterior probability for model k: pk,t =

αk,t . P α h h,t

(1.13)

• Caveat: Posterior probabilities tend to approach 0 or 1 (Lindley’s paradox). See Sims (2002, 2006).

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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

pk φ (vk ) .

(1.14)

k=1

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

pk eη(vk −v

∗ −δ)

= 1.

(1.15)

k=1

Caveats: Multiple solutions possible, η may be increasing or decreasing in δ • Local approach: How does η behave with small across-model risk? v) φ (v ∗ + δ) = Eφ (˜

(1.16)

First-order approximation of LHS, second-order approximation of RHS around v ∗ gives v) 2 (δ + v ∗ − E˜ η= v − v ∗ )2 E (˜

(1.17)

η increasing in δ

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Optimal Monetary Policy

3. Model Uncertainty

2 Robust control and Knightian uncertainty • References: Walsh (2003), Cateau (2006), Giordani and S¨oderlind (2004). • For details: Hansen and Sargent (2007). • 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) = CC 0. • Central bank loss function: E0

∞ X t=0

β t [x0tQxt + 2x0tU ut + u0tRut] ,

where xt ≡

"

x01t

x02t

(2.2)

#0

.

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

16

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.

17

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 + 2x0tU ut + 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 + 2x0tU ut + 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

18

(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

β

t



t=0

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.

19

(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.

20

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

     t     t+1    

u v

 

= N 



x1t ρ2t

   

.

(2.10)

ρ1t

• 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, . . .

21

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 = M x1t + Cεt+1,

(2.12)

x2t = N x1t,     

(2.13)



ut vt+1

   

= F x1t.

(2.14)

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

22

Optimal Monetary Policy

3. Model Uncertainty

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



ut vt+1

   

= F xt .

(2.15)

• Solution: x1t+1 = M x1t + Cεt+1,

(2.16)

x2t = N x1t.

(2.17)

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

23

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 (2007): 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

24

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.

25

Optimal Monetary Policy

3. Model Uncertainty

Robust control: An analytical example Reference: Leitemo and S¨oderstr¨om (2008). 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

β

t



πt2

t=0

+

λx2t

Distorted model:

Budget for evil agent: E0

∞ X t=0

β

t

(vtπ )2



+

(vtx)2



≤ η.

(2.26)

26

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 = − πt , 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).

27

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)

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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 (2007): 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 (2007): Larger estimated open-economy model. Central bank fears in particular shocks to exchange rate and inflation. Output shocks not very costly.

29

Optimal Monetary Policy

3. Model Uncertainty

Robust control: An alternative approach • Dennis, Leitemo, and S¨oderstr¨om (2006): 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.

30

(2.37)

Optimal Monetary Policy

3. Model Uncertainty

3 Model uncertainty: An application • Reference: Cateau (2006) • Two competing models • Simplified version of Fuhrer and Moore (1995): – 3-period staggered wage contracts ⇒ hybrid Phillips curve – Backward-looking IS equation – Long-term interest rate – Short rate ⇒ long rate ⇒ output ⇒ inflation – Estimated with maximum likelihood on Canadian data, 1962:Q1–2005:Q1 • Simplified version of Christiano, Eichenbaum, and Evans (2005): – Staggered prices (Calvo, 1983) with indexation to past inflation ⇒ hybrid Phillips curve – Habits in consumption ⇒ hybrid IS equation – Short rate ⇒ output ⇒ inflation – Calibrated

31

Optimal Monetary Policy

3. Model Uncertainty

Monetary policy • Central bank uses simple rule it = ρπ πt + ρxxt + ρiit−1

(3.1)

• Loss function E0

∞  X t=0

πt2

+

ωx2t

+

η∆i2t



(3.2)

• Two approaches 1. Generalized Bayesian 2. Robust control

32

Optimal Monetary Policy

value αi,0 at time 0, we can use (45) to update αi,t every period. We can then calculate the posterior probability for each model by normalizing αi,t as follows: πi,t

αi,t =P . αj,t

(46)

j

Bayesian approach:Figure Model probability weights, 11 uses quarterly data on inflation and the output2000:Q1– gap from 2000Q1 to update 1

0.8

probability

CEE FM 0.6

0.4

0.2

0

2

4

6

8 Quarters

10

12

14

Figure 11: Updating probability weights of model weights to the CEE and FM model. Initially, the models are assigned equal weights, but as

data arrived, weights6toquarters. each model are recalculated using equations (45) and (46). • FM weight 1, CEEnewweight 0 theafter The weight to the FM model initially declines, but we see that after 6 quarters of data have been accumulated, the statistical evidence in favour of the FM model causes its weight to converge to 1.

• FM estimated, CEE calibrated. 10

• Lindley’s paradox?

Appendix C explains how we can use the Kalman filter to construct yt+1 |y t and ∆t+1 .

28

33

3. Model Uncertainty

Optimal Monetary Policy

3. Model Uncertainty

Bayesian approach: Optimized rules with different degrees of across-model risk aversion Table 3: Simple rule coefficients, the degree of across-model risk aversion and model losses η 0 0.5 1 1.5 2 3 5 7 10 15 20 25 30

ρi 0.785 0.794 0.803 0.811 0.818 0.831 0.847 0.855 0.861 0.863 0.864 0.864 0.864

ρπ 1.337 1.292 1.250 1.210 1.173 1.112 1.031 0.989 0.961 0.950 0.948 0.947 0.947

ρx 0.496 0.502 0.507 0.511 0.514 0.517 0.520 0.520 0.520 0.520 0.520 0.520 0.520

loss CEE 0.1138 0.1144 0.1152 0.1159 0.1167 0.1181 0.1203 0.1217 0.1226 0.1230 0.1231 0.1231 0.1231

loss FM 0.4932 0.4926 0.4920 0.4916 0.4912 0.4906 0.4901 0.4899 0.4899 0.4899 0.4899 0.4899 0.4899

• Probabilityrule weight 0.5 on each model. is more restrictive for the FM model than for the CEE model, increasing the degree of aversion makes the policy-maker more worried about improving the performance of the

• FM has larger loss, η agives weightη,on ⇒ FM chooses loss falls, rule in the increasing FM model. As result,larger as we increase theFM policy-maker rulesCEE that loss increases. • More

exhibit more inertia, a slightly higher contemporaneous response to the output gap but a robustness ⇒ smaller response inflation coefficient, output and interest rate lower contemporaneous to inflation. Theselarger rules perform better in the FM model but worse in the CEE model. It is in this sense that increasing η makes the policy-maker more conservative and gradually convinces the policy-maker to act on a worst-case scenario. Now that we know the effect of different degrees of across-model risk aversion on the optimal policy rule, it is important to interpret their economic significance. We do that by computing the implied premium that each 34 degree of aversion leads to. We use (54) and

coefficient.

Figure 4: Detection-error probabilities for the FM model 3. Model Uncertainty

Optimal Monetary Policy

model declines.and Hansen and Sargent (2004) recommend choosing values of θ Robust control: The preferenceworst-case for robustness detection-error probabilities

−1

0.5

0.5

0.45

0.45

0.4

0.4

detection probability

detection probability

that correspond to detection-error probabilities from 10 to 25 per cent, on a case-by-case basis. For the FM model, this range suggest values of θ−1 from 0.66 to 0.55.

0.35

0.3

0.25

0.2

0.15

0.35

0.3

0.25

0.2

0.15

0.1

0.1

0.05

0.05

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.7

θ −1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

θ −1

Figure 4: Detection-error probabilities for the FM model Figure 5: Detection-error probabilities for the CEE model

worst-case model declines. Hansen and Sargent (2004) recommend choosing values of θ−1 similarly the detection-error probabilities for the CEE model. The values of that correspond to detection-error probabilities from 10 to Figure 25 per 5cent, on a plots case-by-case −1 θ 0.66 corresponding basis. For the FM model, this range suggest values of θ−1 from to 0.55. to detection-error probabilities of 10 and 25 per cent are 0.2 and 0.11, respectively. 0.5

16

0.45

detection probability

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

35

Optimal Monetary Policy

3. Model Uncertainty

Robust control: Optimized rulesforforrobustness different on preference robustness 3.3 Effects of concerns monetaryfor policy Table 1: FM Model: how coefficients of the simple rule vary with θ−1 θ−1 0 0.1000 0.2000 0.3000 0.4000 0.5000

ρi 0.8629 0.8524 0.8390 0.8222 0.7993 0.7666

ρπ 0.9594 1.0238 1.1059 1.2114 1.3605 1.5883

ρx 0.5316 0.4962 0.4581 0.4161 0.3696 0.3162

Table 2: How coefficients of the simple rule vary with θ−1 −1 Table 1 shows how the coefficients change θ−1 of theρrule ρπ as weρxincrease θ from 0 to 0.5 when i 0.6607 the reference model is the FM 0model.0.9502 Recall 1.8859 that when θ−1 = 0, there is no concern for 0.22 0.9501 1.8862 0.6604 robustness. The optimal rule in that case requires the policy-maker to put a weight of 0.86 1.00 0.9498 1.8870 0.6596 on interest rate inertia and weights of 0.96 and 0.53 on contemporaneous inflation and the 5.00 −1 0.9482 1.8912 0.6552 output gap, respectively. But10.00 as θ increases, optimal 0.6497 policy changes in three ways: first, 0.9461 1.8966 the policy-maker gives less and less importance to interest rate inertia; second, the policy30.00 0.9367 1.9191 0.6262 0.9255 1.9432 0.6008 maker responds with a smaller50.00 weight to contemporaneous output gap; and third, the policy70.00 to 0.9122 1.9691 0.5729 maker responds more aggressively contemporaneous inflation. Why does that happen? 100.00 0.8878 2.0115 0.5250 Figure 6 shows how the dynamic responses of inflation, the output gap, and interest rate vary with respect to the two shocks in the FM model: ²w,t , the real contract price shock, and

²x,t , the output shock. We see that in contrastsmaller to the approximating therate worst• More robustness ⇒gap larger inflation coefficient, output and model, interest coefficient. −1

caseapproximating when θmodel. = 0.5 makes the effect of a real contract price shock and output gap the Opposite tomodel Bayesian approach.

shock on inflation and output more persistent. Indeed, in the worst-case model, a contract price inflation one morestand importantly approximating model,to and its Theshock aboveincreases result illustrates on whichthan the the robust control approach dealing effectmodel takes uncertainty 25 quarters has to die outcriticized. (relative to 17 quarters the approximating model). On with been Robust controlinassumes that the policy-maker’s 36

the other hand,model a realiscontract shock yields The a more pronounced output approximating a good price reference model. reference modeldecline posits in thethe structural

Optimal Monetary Policy

3. Model Uncertainty

Robust control: Impulse responses in FM model

inflation

real contract price shock

output gap shock

0.04

0.004

0.02

0.002

0

0

−0.02

0

10

20

30

output gap

0.02

−0.002

0

10

20

30

0

10

20

30

0

10

20

30

0.01 0.005

0

0 −0.02 −0.04

−0.005 0

10

20

30

interest rate

0.2

−0.01

0.04

0.1

0.02

0 0

−0.1 −0.2

0

10

20

30 approximating

−0.02

worst-case

Figure 6: FM model: Approximating model vs worst-case model (θ = 0.5)

Table 2 displays the coefficients of the policy rule various choices of θ−1 • Distortion to inflation equation more important. (Cf.for simple model.)

for the Dennis model. When there is no concern for robustness, the optimal policy rule requires a high degree of inertia in the interest rate (0.95), a relatively high weight to contemporaneous

inflation (1.89), and a moderate response to the output gap (0.66). When we increase θ−1 , we first notice that for θ−1 = 0.22 (the size of θ−1 ) corresponding to a detection-error probability of 25 per cent, the optimal policy rule remains virtually unchanged. Even for very high values 37 of θ−1 , which according to the detection-error probability criterion is too high, we end up

(Binette, Murchison, Perrier, and Rennison 2004) - a new model that will be used for both economic projections and policy analysis. Since these models will be non-nested models, it is 3. Model Uncertainty Optimal Monetary Policy probably not far-fetched to imagine that they will have different predictions along different economic dimensions.7 So, how should the Bank deal with model uncertainty when it does not have one but two or more reference models that it considers relevant for policy making?

Robust control: Impulse responses in CEE model

In the next section, we present various approaches for making decisions when the policymaker considers more than one model for policy-making.

inflation

output gap shock 0.005

0.05

0

0

output gap

−0.005 0

5

10

15

20

0.005

0.05

0

0

−0.005

0

5

10

15

20

0.02 interest rate

−0.05

−0.05

inflation shock

0

5

10

15

20

0

5

10

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20

0

5

10

15

20

0.05

0.01 0

0

−0.01 −0.02

0

5

10

15

20

−0.05

approximating

worst-case

Figure 7: CEE model: Approximating model vs worst-case model (θ = 10)

• Small effects4.ofMultiple-Model robustness. Approaches Suppose that the true data-generating model is G, but that the policy-maker does not know it. Suppose also that, being faced with competing theories suggesting different models, the 7

The models cited in this paragraph are not an exhaustive list of the models that can be used for policy analysis at the Bank of Canada. Ortega and Rebei (2005), for instance, can analyze optimal policy in the 38 context of a multi-sector small open-economy model estimated for the Canadian economy.

Optimal Monetary Policy

3. Model Uncertainty

Application: Conclusions • Robust control policy robust in a neighborhood to reference model. More attractive if reference model fairly good? • Bayesian approach more attractive if competing models are very different? • Generalized Bayesian approach: Robust policy responds less aggressively to inflation, more to output, and is more inertial. • Robust control: Robust policy responds more aggressively to inflation, less to output, and is less inertial. • Policy implications very different.

39

Optimal Monetary Policy

3. Model Uncertainty

4 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

3. Model Uncertainty

• 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???

41

Optimal Monetary Policy

3. Model Uncertainty

5 Matlab application • Again, consider the hybrid New Keynesian model πt = απt−1 + (1 − α)βEtπt+1 + κ [yt − y¯t] + vtπ , 1 yt = δyt−1 + (1 − δ)Etyt+1 − [it − Etπt+1] + vty , σ π π π vt = ρπ vt−1 + εt ,

(5.2)

y vty = ρy vt−1 + εyt ,

(5.4)

y¯t = ρy¯y¯t−1 + εyt¯,

(5.5)

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

42

(5.1)

(5.3)

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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Optimal Monetary Policy

3. Model Uncertainty

References Calvo, Guillermo A. (1983), “Staggered prices in a utility-maximizing framework,” Journal of Monetary Economics, 12 (3), 383–398. 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. Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans (2005), “Nominal rigidities and the dynamic effects of a shock to monetary policy,” Journal of Political Economy, 113 (1), 1–45. Dennis, Richard (2007), “Optimal policy rules in rational-expectations models: New solution algorithms,” Macroeconomic Dynamics, 11 (1), 31–55. Dennis, Richard, Kai Leitemo, and Ulf S¨oderstr¨om (2006), “Methods for robust control,” Discussion Paper No. 5638, Centre for Economic Policy Research. ——— (2007), “Monetary policy in a small open economy with a preference for robustness,” Discussion Paper No. 6067, Centre for Economic Policy Research. Fuhrer, Jeffrey C. and George Moore (1995), “Monetary policy trade-offs and the correlation between nominal interest rates and real output,” American Economic Review , 85 (1), 219–239. 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 (2007), Robustness, Princeton University Press. Knight, Frank H. (1921), Risk, Uncertainty and Profit, Boston, MA: Houghton Mifflin Company. 44

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Leitemo, Kai and Ulf S¨oderstr¨om (2007), “Robust monetary policy in a small open economy,” Manuscript, IGIER, Bocconi University. Forthcoming, Journal of Economic Dynamics and Control. ——— (2008), “Robust monetary policy in the New-Keynesian framework,” Macroeconomic Dynamics, 12 (S1), 126–135. Sims, Christopher A. (2002), “The role of models and probabilities in the monetary policy process,” Brookings Papers on Economic Activity, 2, 1–62. ——— (2006), “Improving monetary policy models,” Manuscript, Princeton University. Forthcoming, Journal of Economic Dynamics and Control. S¨oderlind, Paul (1999), “Solution and estimation of RE macromodels with optimal policy,” European Economic Review Papers and Proceedings, 43 (4–6), 813–823. 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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