Does Model Uncertainty Justify Conservatism? Robustness and the Delegation of Monetary Policy Peter Tillmann1 Swiss National Bank June 5, 2009

Abstract: This paper analyzes the rationale for delegating monetary policy to an inflation-averse central banker when there is a preference for robustness of optimal policy with respect to misspecifications of the underlying model of the economy. We use a simple New Keynesian model to show how the optimal output gap weight in the central bank’s objective function depends on the degree of model uncertainty. In particular, we show numerically that the rationale for appointing a conservative central bank prevails if the central bank is concerned about the robustness of policy with respect to model misspecifications. Moreover, we find that the central bank should put more relative weight on inflation stabilization if the degree of uncertainty increases. Interestingly, if the degree of uncertainty is large, monetary policy should be delegated to a conservative central banker even in the absence of shock persistence. Keywords: optimal monetary policy, delegation, robust control, New Keynesian model, stabilization bias JEL classification: E32, E52, E58

1

Contact: Swiss National Bank, Börsenstr. 15, CH-8022 Zurich, Email: [email protected] I am grateful to the editor, John Leahy, and two anonymous referees for insightful comments and suggestions. I also thank Carl Walsh, Federico Ravenna, and Paul Söderlind for important comments and suggestions. This paper was written while I was visiting the department of economics at the University of California Santa Cruz. I am grateful for the department’s hospitality. Moreover, I thank seminar participants at the fourth meeting of the DFG network "Quantitative Macroeconomics", the 2006 annual conference of the European Economic Association (Vienna), the University of Copenhagen, and the ZEI summer school 2007 for insightful comments. Financial support from Deutsche Forschungsgemeinschaft is gratefully acknowledged. The views expressed in this paper do not necessarily reflect those of the Swiss National Bank.

"Indeed, intuition suggests that stronger action by the central bank may be warranted to prevent particularly costly outcomes". Fed Chairman Ben Bernanke (2007)

1

Introduction

Uncertainty is now recognized to be of central importance for the design of monetary policy.2 In this paper, we analyze optimal monetary policy in an economy that is plagued by uncertainty about the basic structure of the underlying economic model. In particular, this paper evaluates monetary delegation under model uncertainty, that is, the question of delegating monetary policy to a central bank, whose preferences with respect to the relative weights assigned to output and inflation stabilization differ from that of the social planner. Since Rogoff’s (1985) seminal contribution, it is now common wisdom that delegating monetary policy to a central bank which is more inflation-averse than the social planner, i.e. to a "conservative central banker", can raise welfare. In the class of models Rogoff had in mind, a conservative central banker corrects the problem of an average inflation bias. In a more recent generation of micro-founded general equilibrium models for monetary policy analysis, the inflation bias is absent. Nevertheless, these models still motivate the appointment of a conservative central banker. The reason is that monetary policy under discretion gives rise to inefficient inflation stabilization - a stabilization bias emerges - which can be corrected by a hawkish central banker. Hence, the case for central bank conservatism is still compelling, although the basic rationale has changed.3 In an important paper, Giannoni (2002) asks: "Does model uncertainty justify caution?". He analyzes whether the Brainard (1967) result carries over to robust 2

Researchers routinely refer to Alan Greenspan’s quote from the 2003 Jackon Hole meeting, which has gained widespread attention: "Uncertainty is not just an important feature of the monetary policy landscape; it is the defining characteristic of that landscape." 3 See Dennis and Söderström (2006) for a thorough quantitative analysis of the stabilization bias of discretionary policy. The find a sizeable gain from precommitment. Levine, McAdam, and Pearlman (2008) quantify the welfare gains from commitment within a fully-specified Dynamic General Equilibrium Model of the Euro area.

2

policy in a New Keynesian model of monetary policy.4 Brainard argued that multiplicative uncertainty should lead to attenuated adjustment of the policy instrument. Giannoni shows that model uncertainty does no longer justify a cautious monetary policy response. In this paper we revisit another prominent and influential result of the literature, which shaped both the academic thinking about monetary policy and the actual central banking landscape very much like Brainard’s finding did. The question is: "Does model uncertainty justify conservatism?".5 We assess how Rogoff’s delegation of monetary policy to a conservative central banker is affected by uncertainty of the central bank about the underlying model. Does model uncertainty strengthen the case for conservatism or does uncertainty give rise to a less inflation-averse - a liberal - central banker? Put differently, this paper quantifies the stabilization bias under model uncertainty. We approach model uncertainty within the robust control framework laid out by Hansen and Sargent (2008). In this environment, the policymaker desires to design a policy rule that performs well even if the worst possible outcome realizes. Central to Hansen and Sargent’s robust control approach is the distinction between the policymaker’s reference model and the approximating model. The reference model provides the most likely description of the economy. In the absence of model misspecifications, this model generates the conventional rational expectations solution. Under robust control, however, the policymaker believes the model to be misspecified to a certain degree. He formulates a policy rule which is robust to these model distortions and shields the economy from the worst possible misspecification. The approximating equilibrium results if the central bank follows its robust policy in the undistorted reference model. In contrast, the worst-case equilibrium corresponds to the case in which the central bank applies the robust rule in the fully distorted model. We find that the optimal weight attached to output gap fluctuations is lower than society’s but greater than zero. Moreover, the degree of central bank conservatism increases with the degree of model uncertainty, that is, with the variety of model misspecifications against which the central banker wants to be robust. The logic behind these results stems from the effect of model uncertainty on the variances 4

Tetlow and von zur Mühlen (2001) find that uncertainty alone cannot explain the observed attenuated policy. 5 Blinder (1999, p. 46) argues that "... in the real world, the noun ’central banker’ practically cries out for the adjective ’conservative’ ".

3

of output and inflation. If uncertainty increases inflation variance, the case for appointing a conservative central banker is strengthened, since the stabilization bias is aggravated. Allowing the central bank to focus more on the objective of stabilizing inflation implies that the policymaker partly circumvents the trade-off between inflation and output volatility. Model distortions affect this trade-off and make output stabilization more costly in terms of inflation stabilization. Hence, the presence of model uncertainty motivates the delegation of policy to a central banker who attaches a larger relative weight on inflation stabilization. This paper is organized as follows. The next section briefly discusses the stabilization bias under model uncertainty, while section three presents reference model, which the central bank considers as being the most likely description of the economy. Section four formulates the discretionary optimization problem under robustness, while section five evaluates the policy outcomes under different degrees of conservatism and uses the social planner’s welfare function to assess the optimal degree of inflation-aversion of the central banker. Finally, section six, draws some conclusions.

2

The stabilization bias under uncertainty

In the standard New Keynesian model, the rationale for appointing a conservative central banker is not an average inflation bias generated by an overly ambitious central banker who wants to steer unemployment below the natural rate. Instead, the motivation for appointing a central banker whose relative weights differ from those of the planner is known as the stabilization bias. Clarida, Galí, and Gertler (1999), among others, have noted that under discretion, inflation is inefficiently stabilized. As a result, the variance of inflation is higher than under the commitment benchmark. Appointing a Rogoff-conservative central banker, who smooths inflation volatility, can raise welfare. Because the public knows inflation will respond less to a cost-push shock, future expected inflation rises less. Stabilizing inflation becomes less costly in terms of future output. Hence, appointing a Rogoff-conservative provides a (second-best) solution to the stabilization bias. For a given policy, model uncertainty generally raises inflation and output volatility. This is because the evil agent, a metaphor for the central bank’s concern about misspecifications, adds to the persistence of the shock processes, which raises volatility and induces a greater welfare loss. In addition, however, model uncertainty generally makes monetary policy more aggressive, which leads to a better inflation

4

stabilization. Whether the stabilization bias increases or decreases as uncertainty increases depends on the net effect of these two forces. Whenever the increase in policy aggressiveness is not sufficient to fully offset a more volatile inflation rate, the stabilization bias increases with the degree of model uncertainty. Note that this rationale strengthens as the persistence of the cost-push shock process increases. A larger persistence eventually translates into higher volatility and aggravates the stabilization bias. In the standard model without persistence of the cost-push shock, the motivation for conservatism disappears. Below we will see that a high degree of model uncertainty provides a rationale for conservatism even in this case of uncorrelated shocks. Four recent papers examine the delegation problem in related models. Kilponen (2003) analyzes weight-conservatism in a robust control environment. However, he uses a backward-looking model with an average inflation bias generated by an overly ambitious central bank that wants to push output above its natural rate. He finds that the larger the model misspecifications, the more inflation-averse the central banker should be. Gaspar and Vestin (2004) also analyze the rationale for delegation under uncertainty. However, they assume that the central bank knows the true structural relationships of the underlying model but cannot reliably observe potential output. Moreover, they also include endogenous private sector expectations through learning. The central finding is that "the optimal delegation degree of conservatism increases as the quality of knowledge and information declines". In contrast to the first paper, we examine the delegation problem in a small forwardlooking monetary model which constitutes the benchmark New Keynesian framework. In contrast to the latter paper, we embed the question in a robust control framework, in which the central bank faces uncertainty not only about particular data series, but about the model structure as a whole. Two papers study optimal monetary policy in an environment, in which the public uses a potentially misspecified model to form expectations. Dennis (2007) finds that as uncertainty becomes larger, the stabilization bias shrinks (since policy becomes more aggressive). Owing to this welfare improvement, it could even be optimal to allow for a specific degree of uncertainty if the model is in fact certain. Woodford (2005), on the contrary, finds that the size of the stabilization bias increases as uncertainty become larger.

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3

A simple New Keynesian framework

We adopt a standard forward-looking monetary model of the business cycle as a laboratory.6 The IS curve (1) and the forward-looking Phillips curve (2) represent log-linearised equilibrium conditions of a simple sticky-price general equilibrium model yt = Et yt+1 − σ −1 (it − Et π t+1 ) + ut

(1)

π t = βEt π t+1 + γyt + et

(2)

where π t is the inflation rate, yt the output gap, it the risk-free nominal interest rate controlled by the central bank, and Et is the expectations operator. All variables are expressed in percentage deviations from their respective steady state values. The parameters β, γ, and σ are positive and γ depends on the deep parameters of the underlying microeconomic structure. The discount factor is denoted by β < 1, σ is the inverse of the elasticity of intertemporal substitution, and γ, the slope coefficient of the Phillips curve, depends negatively on the degree of price stickiness. The processes driving the cost-push shock et and the demand shock ut with standard deviations σ u and σ e are given by ut = ρu ut−1 + σ u εut et = ρe et−1 + σ e εet

with 0 ≤ ρu < 1, εut ∼ i.i.d. N (0, 1)

with 0 ≤ ρe < 1, εet ∼ i.i.d. N (0, 1)

This model can be framed in standard matrix form to yield ⎡ ⎤ ⎤⎡ ⎤⎡ ⎡ ρu 0 ut+1 1 0 0 0 0 0 ⎢ 0 ρ ⎢ ⎥ ⎢ 0 1 0 0 ⎥⎢ e 0 0 ⎥ ⎢ ⎥⎢ ⎥⎢ ⎢ t+1 ⎥ e = ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎣ −1 0 ⎣ 0 0 1 σ −1 ⎦ ⎣ Et yt+1 ⎦ 1 0 ⎦⎣ 0 0 0 β Et π t+1 0 −1 −γ 1 ⎡ ⎤ σu 0 # " ⎢ 0 σ ⎥ εu ⎢ e ⎥ t+1 +⎢ ⎥ ⎣ 0 0 ⎦ εet+1 0 0

ut et yt πt

⎤

⎡

0 0

⎥ ⎢ ⎥ ⎢ ⎥ + ⎢ −1 ⎦ ⎣ σ 0

⎤

⎥ ⎥ ⎥ it ⎦

Premultiplying the system with the inverse of the first matrix on the left-hand side gives the compact conventional notation of forward-looking rational expectations 6

See, among others, Clarida, Galí, and Gertler (1999) and Woodford (2003) for a deeper analysis and the complete derivation of the model.

6

models

"

x1t+1 Et x2t+1

#

=A

"

x1t x2t

#

+ Bit + Cεt+1

(3)

where C0 = [C1 , 02×2 ]. The vector x0t = [x01t , x02t ] summarizes both the state and the jump variables. The 2 × 1 vector x1t collects the predetermined variables ut and et with x10 given, and x2t is a 2 × 1 vector containing the forward-looking variables yt and π t . Finally, the 2 × 1 vector εt+1 contains the white-noise innovations εut+1 and εet+1 . Monetary policy is assumed to minimize the following loss function, which is quadratic in deviations of the inflation rate, the output gap, and the interest rate from its respective targets, which are set to zero minE0 it

∞ X t=0

¡ ¢ β t π 2t + λy yt2 + λi i2t

with λy , λi ≥ 0

(4)

The central bank’s loss function differs from the social welfare function only with respect to the weight it attaches to minimizing the variance of the output gap. The central bank penalizes the variance of the output gap with a coefficient λy , . A conservative central banker in while society’s welfare is calculated using λsocial y the sense of Rogoff (1985) exhibits a smaller weight on yt2 than society, i.e. λy < . The purpose of the paper is to find the optimal value of λy and to derive λsocial y the impact of the degree of model uncertainty on the choice of the optimal output gap coefficient in the central bank’s objective function. Note that the weight on inflation stabilization is normalized to one. Hence, the weights on output gap and interest rate stabilization measure the relative emphasis the central bank puts on these conflicting arguments.7 Throughout the paper we assume that the central bank and the social planner attach the same weight to interest rate rate fluctuations, i.e. . λi = λsocial i

4

Optimal robust policy

Minimizing (4) subject to the constraint (3) gives a set of first-order conditions, from which the optimal policy response to shocks can be computed. However, the resulting targeting rule is not robust to misspecifications of the underlying model 7

Woodford (2003) shows the conditions under which minimizing a loss function such as (4) corresponds to maximizing the welfare of the representative agent.

7

represented by (1) and (2). To analyze a policy that is robust to model distortions, we apply robust control techniques proposed in a series of contributions by Hansen and Sargent (2008). The following exposition draws heavily on the survey of Giordani and Söderlind (2004). The central banker considers the model presented in the previous section as the reference model, which represents the most likely description of the economic structure. However, the policymaker knows that this model could be subject to a wide range of distortions. The task is to reformulate the central bank’s optimization problem such that the resulting policy rule performs well even if the model deviates from the reference model. A policy that is optimal in the reference model but does not take account of possible misspecifications can turn out to be disastrous if the misspecifications realize. Under robust control, in contrast, the resulting policy rule performs sufficiently well even if the underlying economic structure does not coincide with the policymaker’s reference model. We transform the minimization problem given by (4) into a min-max problem. The central bank wants to minimize the maximum welfare loss due to model misspecifications by specifying an appropriate policy. To illustrate the problem, we introduce a fictitious second rational agent, the malevolent or evil agent, whose only goal is to maximize the central bank’s loss. The evil agent chooses a model from the available set of alternative models and the central bank chooses its policy optimally. Hence, the equilibrium is the outcome of a two-person game. Note that the evil agent is a convenient metaphor for the planner’s cautionary behavior. Therefore, the evil agent shares the same reference model that the central bank entertains and optimizes the same objective function. The only difference is that the evil agent wants to maximize rather than minimize the resulting loss. The set of potential misspecifications, the control vector of the evil agent, takes the form of error terms. However, these shocks are not mere additional exogenous random innovations. Let vt+1 denote the evil agent’s (2 × 1) control vector, which is allowed to feed back on the history of the economy’s state variables xt vt+1 = ft (xt , xt−1 , ...)

(5)

where ft is a sequence of functions. In fact, misspecifications can distort the model parameters, the autocorrelation properties of the error terms, and can introduce non-linearities.8 The only constraint imposed upon the fictitious evil agent is his 8

See Hansen and Sargent (2008) for details.

8

budget constraint requiring E0

∞ X t=0

0 β t vt+1 vt+1 ≤ η

(6)

Hence, the parameter η measures the amount of misspecification the evil agent has available.9 This formulation greatly simplifies the problem, because the degree of uncertainty manifests in a single parameter. The complete optimization problem thus becomes ∞ X ¡ ¢ max E β t π 2t + λy yt2 + λi i2t (7) min 0 ∞ ∞ s.t.

"

{i}0 {v}1

x1t+1 Et x2t+1

#

t=0

=A

E0

∞ X t=0

"

x1t x2t

#

+ Bit + C (εt+1 +vt+1 )

0 β t vt+1 vt+1 ≤ η

Note that the control variables vt+1 of the evil agent are masked by the shock vector εt+1 . If there were no shocks hitting the economy, the policymaker would be able to perfectly observe model distortions.10 To simplify the analysis, we assume that the behavior of the evil agent mirrors that of the central bank in that they optimize at the same point in time and play a Nash game. The standard rational expectations solution for optimal monetary policy corresponds to η = 0, such that the evil agent’s budget is empty. In this case, the maximization part becomes irrelevant. If η > 0, on the other hand, the central bank faces model distortions. The constraint can be inserted to obtain the Lagrangian max E min ∞ ∞ 0

{i}0 {v}1

s.t.

"

∞ X t=0

x1t+1 Et x2t+1

¡ 0 £ ¢¤ β t π 2t + λy yt2 + λi i2t − θ vt+1 vt+1

#

=A

"

x1t x2t

#

(8)

+ Bit + C (εt+1 +vt+1 )

The Lagrange parameter θ is inversely related to η. Hence, the rational expectations case corresponds to θ → ∞.11 In the following, we will loosely refer to θ as the 9

Giordani and Söderlind (2004) note that the evil agent’s control vector is indexed t+1 although the distortions are known in t. This convention is supposed to stress the fact the the distortions are masked by the shock processes, i.e. the cost-push and the demand shock. 10 Intuitively, misspecifications create more damage when the variance of forecast errors is large. 11 In this case, the evil agent maximizes the welfare loss by choosing vt+1 = 0.

9

degree of robustness or the degree of uncertainty, respectively. A lower θ means that the central bank designs a policy which is appropriate for a wider set of possible misspecifications. Therefore, a lower θ is equivalent to a higher degree of robustness. The loss function and the law of motion for the forward-looking model can be redefined to formulate the optimization program in standard state-space form. This yields max E min ∞ ∞ 0

{i}0 {v}1

s.t.

"

x1t+1 Et x2t+1

∞ X

#

t=0

¡ ¢ β t x0t Qxt + h0t Rht

=A

"

x1t x2t

with h0t =

h

i

h

i

#

(9)

ˆ t + Cεt+1 + Bh "

λi 01×2 ˆ = B C ,R = it vt+1 , B 02×1 −θI2 " " # # 02×2 02×2 λy 0 Q= , Qbb = 02×2 Qbb 0 1

#

Due to the fact that the first order conditions for a minimum are the same as for a maximum, the optimization problem can be solved using standard solution algorithms designed for evaluating optimal policy in rational expectations models.12 As in other rational expectations models, the forward-looking variables, the policy instrument, and the evil agent’s and the central bank’s control vectors will be linear functions of the predetermined variables in xt ⎡ ⎤ ⎡ ⎤ x2t N ⎢ ⎥ ⎢ ⎥ (10) ⎣ it ⎦ = ⎣ −Fi ⎦ xt vt+1 −Fv

The equilibrium dynamics of the model are found by combining this solution with the reference model. If the full amount of possible misspecifications realizes, we refer to the resulting model as the worst-case model, which is formally obtained by inserting (10) in the law of motion for the reference model " # # " x1t x1t+1 (11) = (A − BFi − CFv ) + Cεt+1 Et x2t+1 x2t 12

In this paper, the solution follows the procedures and the MATLAB software outlined in Söderlind (1999) and Giordani and Söderlind (2004).

10

If, on the other hand, the reference model turns out to be undistorted, we refer to the resulting model as the approximating model, which is obtained by setting Fv = 0 in (11) " " # # x1t+1 x1t (12) = (A − BFi ) + Cεt+1 Et x2t+1 x2t A central bank concerned with robustness designs policy based on the fully distorted model. Once policy is formulated, however, the central bank acts as if there were no longer any model uncertainty. We use the resulting solutions in this approximating equilibrium to evaluate the delegation problem of monetary policy.

5

Delegating monetary policy

The optimization problem from the perspective of a welfare maximizing social planner is the following. He chooses the optimal weight in the central bank’s objective function by minimize welfare loss, which is given by the present-value of the loss function evaluated for a given λy appr

min L λy

= (1 − β) E0 = (1 − β) E0

∞ X t=0 ∞ X t=0

β t ( Lappr | λy ) t βt

n¡

¯ ¢ ¡ 2¯ ¢ ¡ ¯ ¢o π 2t ¯ λy + λsocial yt ¯ λy + λi i2t ¯ λy y

(13)

Hence, the optimal output weight λy of the central bank, which is inversely related to the degree of conservatism, is chosen such that the welfare function is maximized. ¡ ¯ ¢ The government chooses the degree of conservatism given the outcomes π 2t ¯ λy , ¡ 2¯ ¢ ¡ 2¯ ¢ yt ¯ λy , and it ¯ λy that the central banker with λy achieves under model uncertainty in the approximating model. Note that the social planner delegates policy to an uncertainty averse central bank. The planner himself, however, is not concerned about model misspecifications and takes the outcome in the approximating equilibrium the central banker delivers as given. Alternatively, one could also argue that the social planner shares the central bank’s concern for robustness. As a consequence, the planner takes his delegation decision on the basis of the worst-case outcome Lworst instead of the outcome in the approximating equilibrium when both the planner and the central bank have the same degree of model uncertainty. Below, we report a separate set of results for this case of delegation based on the worst-case outcome. 11

The solution algorithm provides the variances needed to evaluate the welfare function. We compute the welfare loss for a given combination of θ and λy and search for the lowest loss over a grid of plausible values of λy . The parameter values for calibrating the solutions and the variances of the output gap, the inflation rate, and the interest rate are reported in table (1). Table 1: Calibrated parameter values reference model loss function shock processes social β γ σ λi λy ρe ρu σe = σu I

0.99

0.024

0.16

0.00

0.25

0.35

0.35

0.30

II

0.99

0.024

0.16

0.10

0.25

0.35

0.35

0.30

III

0.99

0.024

0.16

0.24

0.05

0.35

0.35

0.30

IV

0.99

0.024

0.16

0.00

0.25

0.00

0.00

0.60

The choice of the discount factor β = 0.99 is standard in the literature and is consistent with an annual real interest rate of 4 percent. All other model parameters are set following the work of e.g. Giannoni (2002) and Woodford (2003), i.e. γ = 0.024 and σ = 0.16. The first specification exhibits no interest rate stabilization = 0.25, we employ a both plausible and widely objective, i.e. λi = 0. With λsocial y used calibration. Specification II allows for a small weight in interest rate stabilization in the central bank’s loss function. In specification III, we apply the calibration of Giannoni (2002) and Woodford (2003), which implies a large penalty for interest rate variation with an output gap weight of only 0.05. In specification IV, we set the degree of persistence in the shock processes to zero but increase the standard deviation of the shock process to 0.60. The shock processes are also standard. We evaluate the welfare loss and the rationale for appointing a conservative central banker under model uncertainty over a grid of plausible values of the robustness parameter θ. This parameter, however, is bounded only by θ > 0 with rational expectations corresponding to θ → ∞. Hence, we have no reasonable a priori range over which we should compute welfare. To overcome the problem of specifying a range for θ, we follow Hansen and Sargent (2008, chapter 9) and employ what they 12

0.5 0.45

detection error probability

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 -0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-1/θ

Figure 1: Detection error probabilities for specification I based on 1000 simulations for a sample of 60 observations refer to as a detection error probability approach. Zero robustness, i.e. the rational expectations case, corresponds to a probability of 0.5. We calculate this probability and invert it to obtain a context-specific value of θ. The resulting relationship between detection probabilities and θ is presented in figures (1) to (4). A detection error probabilities of 0.30 roughly corresponds to values of θ = 4. We report results for θ = 4, θ = 10, θ = 20, and θ = ∞ thus covering a range from a (mild) degree of uncertainty to the rational expectations case.13 Since the level of the welfare loss in absolute terms contains no direct economic meaning, we facilitate the interpretation and calculate Jensen’s (2002) metric to express the welfare effect of appointing a central banker with an optimal degree of conservatism in terms of an equivalent permanent increase in inflation, the socalled "inflation equivalent". The inflation equivalent π equiv describes a permanent deviation of inflation from target that in welfare terms is equivalent to welfare loss foregone by not delegating monetary policy optimally. A permanent deviation of inflation from target of π equiv percent results in an increase in the objective function, ¡ ¡ equiv ¢2 ¢2 P t i.e. a welfare loss, of (1 − β) E0 ∞ = π equiv . Thus, the inflation t=0 β π 13

We choose to base the analysis on a modest degree of model uncertainty. Allowing for stronger robustness would strengthen the results. Note that infinity here refers to θ = 1000.

13

0.5

detection error probability

0.45

0.4

0.35

0.3

0.25 -0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-1/θ

Figure 2: Detection error probabilities for specification II based on 1000 simulations for a sample of 60 observations

0.5 0.45

detection error probability

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 -0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-1/θ

Figure 3: Detection error probabilities for specification III based on 1000 simulations for a sample of 60 observations

14

0.5

detection error probability

0.45

0.4

0.35

0.3

0.25 -0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-1/θ

Figure 4: Detection error probabilities for specification IV based on 1000 simulations for a sample of 60 observations

29.4

25.8

θ=4

29.2 29

25.4

28.8

25.2

28.6

25

28.4

24.8

0.1

0.2 0.3 weight λ

0.4

0.5

25.2

0.1

0.2 0.3 weight λ

0.4

0.5

24.6

θ=20

25

24.2

24.6

24

24.4

23.8 0.1

0.2 0.3 weight λ

0.4

θ=inf.

24.4

24.8

24.2

θ=10

25.6

23.6

0.5

0.1

0.2 0.3 weight λ

0.4

0.5

Figure 5: Welfare loss in specification I as a function of the output weight λy

15

29.8

25.7

θ=4

θ=10

29.7

25.6

29.6

25.5

29.5

25.4

29.4

0.1

0.2 0.3 weight λ

0.4

25.3

0.5

25

0.1

0.2 0.3 weight λ

0.5

24.3

θ=20

θ=inf.

24.9

24.2

24.8

24.1

24.7

24

24.6

0.4

0.1

0.2 0.3 weight λ

0.4

23.9

0.5

0.1

0.2 0.3 weight λ

0.4

0.5

Figure 6: Welfare loss in specification II as a function of the output weight λy equivalent satisfies ¡ equiv ¢2 = Lappr | λsocial Lappr | λopt y + π y

or

π equiv =

6

q

Lappr | λsocial − Lappr | λopt y y

(14)

(15)

Results

The welfare results for the four alternative sets of parameter values are presented in figures (5) to (8) in the appendix. Each figure plots the welfare loss as a function of the output gap weight λy for four different degrees of model uncertainty θ. A first glance at this figures reveals that the degree of model uncertainty, measured by (the inverse of) θ, has a strong impact on social welfare. The lower θ, the more welfare deteriorates. Table (2) reports the inflation equivalent for each model specification. We find that π equiv increases if θ falls. A higher degree of model uncertainty makes the delegation of monetary policy more important. In terms of this inflation equivalent, an optimal monetary policy delegation realizes sizeable welfare gains. For example, under θ = 4 in specification I, we realize a gain corresponding to a reduction of the 16

inflation rate of 0.3 percentage points. This gain reaches a maximum in specification III. Result 1: The welfare gain from delegating monetary policy to a policymaker with the optimal output weight increases as the uncertainty over the true model of the economy grows.

Table 2: Welfare gain from optimal delegation in the approximating model model robustness Lappr | λopt Lappr | λsocial π equiv y y I θ=4 28.475 28.564 0.298 θ = 10 24.921 24.944 0.152 θ = 20 24.215 24.234 0.138 θ=∞ 23.627 23.642 0.122 II

θ θ θ θ

=4 = 10 = 20 =∞

29.495 25.372 24.603 23.967

29.646 25.406 24.630 23.989

0.388 0.184 0.164 0.148

III

θ θ θ θ

=4 = 10 = 20 =∞

30.219 25.695 24.865 24.182

31.200 25.875 25.007 24.302

0.990 0.424 0.377 0.346

IV

θ θ θ θ

=4 = 10 = 20 =∞

40.111 36.959 36.217 35.570

40.114 36.959 36.217 35.570

0.055 0.000 0.000 0.000

How does the degree of robustness affect the optimal delegation? Those values of λy that minimize the welfare loss clearly fall as robustness increases, i.e. as θ decreases. In all specifications, the welfare loss reaches a minimum for values λy that are smaller than λsocial . Hence, the central banker should attach a larger weight y to inflation stabilization than the social planner. 17

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31.5

25.9 31 25.8 30.5 30

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0.5

Figure 7: Welfare loss in specification III as a function of the output weight λy Result 2: The optimal output weight of the central bank is equal or smaller than the social planner’s weight. Hence, there is a case for a conservative central banker. In all specifications, the degree of optimal conservatism increases, if monetary policy becomes more robust. The mechanism behind this finding is the fact that an increase in robustness amplifies inflation volatility and, hence, exacerbates the stabilization bias of discretionary policy. To underline this rationale, figures (9) to (12) plot the variances of the interest rate, the inflation rate, and the output gap in the approximating model as a function of the delegation weight for different degrees of robustness. If the degree of robustness increases, inflation becomes much more volatile while the variance of the output gap remains more or less unchanged. Note that the interest rate variance also increases. Interest rate adjustment is more vigorous than under certainty. This finding supports Giannoni’s (2002) anti-attenuation result, which challenges the seminal Brainard principle. Under these circumstances, an increased preference for robustness strengthens the case for delegating policy to a Rogoff-conservative. Obviously, putting less weight on output gap stabilization induces the policymaker to adjusts interest rates stronger in order to fight inflation. This raises the interest rate variance. For the delegation-result it is crucial that the stronger aggressiveness of interest rate setting cannot compensate the increased inflation volatility. Hence, 18

42

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40

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Figure 8: Welfare loss in specification IV as a function of the output weight λy the net effect is higher inflation variance under uncertainty and, consequently, a larger stabilization bias. The larger this bias, the stronger the case for delegating policy to a conservative central banker.The size of the stabilization bias is known to depend on the persistence of supply shocks hitting the economy. In specification III, we therefore set the persistence to zero. Interestingly, we still find a case for policy conservatism. The increased inflation volatility under model uncertainty generates a stabilization bias and requires a larger weight on inflation stabilization in the central bank’s loss function. Results 3: Even in the absence of persistence, monetary policy should be delegated to a conservative central banker, if the degree of uncertainty is large. Even if the persistence of shocks is zero, i.e. even if the standard New Keynesian model under rational expectations provides no rationale for a conservative central banker, a high degree of robustness requires a weight on output gap stabilization lower than that of the social planner. Hence, high model uncertainty provides an alternative motivation for delegating monetary policy even in the absence of persistence. The figures reveal an interesting asymmetry. Raising the degree of model uncertainty steadily lowers the optimal output weight. Once the central bank put too 19

inflation variance

3.5 3 2.5 2 0.05

0.1

0.15

output variance

0.8

0.25

0.3

θ=4 θ = 10 θ = 20 θ = inf

0.6 0.4 0.2 0 0.05

interest rate variance

0.2 weight λ

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weight λ

1 0.8 0.6 0.4 0.2 0.05

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Figure 9: Variances of inflation, output, and the interest rate in specification I as a function of the output weight λy

20

inflation variance

3.5 3 2.5 2 0.05

0.1

0.15

output variance

0.4

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θ=4 θ = 10 θ = 20 θ = inf

0.2

0 0.05 interest rate variance

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Figure 10: Variances of inflation, output, and the interest rate in specification II as a function of the output weight λy

21

inflation variance

4

3

2 0.05

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0.1

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θ= θ= θ= θ=

0.05

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Figure 11: Variances of inflation, output, and the interest rate in specification III as a function of the output weight λy

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inflation variance

1.5

1

0.5 0.05

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output variance

0.4

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θ=4 θ = 10 θ = 20 θ = inf

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0 0.05 interest rate variance

0.2 weight λ

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weight λ

0.08 0.06 0.04 0.02 0.05

0.1

0.15 weight λ

Figure 12: Variances of inflation, output, and the interest rate in specification IV as a function of the output weight λy

23

much weight on inflation stabilization, however, welfare sharply deteriorates. Result 4: Appointing a central banker who is too conservative, i.e. λy < λopt y , is more costly in term of welfare than appointing a central banker who is too liberal, i.e. λy > λopt y . Consider for example specification I in figure (5) under θ = 4. Moving from the optimal output weight of around 0.08 to a slightly more conservative central banker with, for example, λy = 0.05 entails a much larger additional welfare loss than moving to a slightly more liberal central banker with λy = 0.10.14 Welfare sharply deteriorates if λy lies below its welfare maximizing value but only mildly decreases if λy lies above the optimal weight. The results presented thus far are based on the assumption that the central banker faces model uncertainty while the social planner is not concerned about robustness. As a consequence, the social planer bases the delegation decision on the outcome under the approximating model, in which the central bank implements a robust policy but model distortions never fully realize. Alternatively, the social planner could share the central bank’s degree of uncertainty. In this case, the planner would base the delegation decision on the outcome of the worst-case, i.e. the fully distorted model. We refer to this case as robust delegation. Table (3) present welfare gain under robust delegation. Figure (13) depicts the welfare loss as a function of the output weight in parameterization I. Of course, the welfare loss is larger under the worst-case model than under the approximating model. Result 5: Under robust delegation, the welfare gain from optimal delegation increases and the optimal output weight decreases relative to optimal delegation under the approximating model. = 0.11 In specification I, for example, optimal delegation would require λopt y opt under worst-case delegation but λy = 0.12 under delegation in the approximating model. Put differently, the central banker should be slightly more conservative if the social planner is also concerned about model uncertainty. Hence, allowing for robust delegation lends further support to the central finding of this paper. 14

This corresponds to the findings of Gaspar and Vestin (2004).

24

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Figure 13: Welfare loss in specification I as a function of the output weight λy

7

Conclusions

This paper contributes to the growing literature on optimal monetary policy under model uncertainty. Specifically, we analyzed which relative weight the central bank should optimally attach to fluctuations in the output gap as opposed to fluctuations in inflation and the interest rate. We found that the presence of model uncertainty makes this question even more important as the optimal degree of conservatism, i.e. the relative weight on inflation stabilization, increases with growing uncertainty. Hence, this paper showed that the seminal result of Rogoff (1985) - that a central banker could be optimal which places a larger weight on inflation stabilization than the social planner - still holds under uncertainty about the structure of the economy. This paper answered this question in a standard forward-looking monetary policy model without an average inflation bias and used robust control techniques to derive optimal policy. Given that model uncertainty and the desire for robustness are among the striking features of contemporary monetary policy, these results lend further support to the robustness of the classic Rogoff-result itself. This paper leaves a number of questions unanswered which affect optimal monetary policy under model uncertainty and, in particular, the design of monetary institutions. For example, the social planner and the central bank are assumed to share the same degree of robustness. It might be interesting to allow for discrep25

Table 3: Welfare gain from optimal delegation in the worst-case model ¯ ¯ model robustness Lworst ¯ λopt Lworst ¯ λsocial π equiv y y I θ=4 31.644 31.811 0.409 θ = 10 25.609 25.637 0.167 θ = 20 24.517 24.537 0.141 θ=∞ 23.632 23.647 0.122 II

θ θ θ θ

=4 = 10 = 20 =∞

33.077 26.101 24.920 23.973

33.359 26.141 24.948 23.994

0.694 0.200 0.167 0.145

III

θ θ θ θ

=4 = 10 = 20 =∞

33.860 26.439 25.194 24.187

35.691 26.656 25.343 24.308

1.353 0.466 0.386 0.348

IV

θ θ θ θ

=4 = 10 = 20 =∞

40.792 37.018 36.229 35.571

40.797 37.018 36.229 35.571

0.071 0.000 0.000 0.000

ancies in the awareness with respect to model uncertainty. Furthermore, this paper follows the literature and treats the degree of robustness as an exogenous parameter. A fruitful extension should explore the trade-off that could determine an optimal degree of robustness. We leave these questions for further research.

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[16] Tetlow, R. J. and P. von zur Muehlen (2001): "Robust monetary policy with misspecified models: Does model uncertainty always call for attenuated policy?", Journal of Economic Dynamics and Control 25, 911-949. [17] Woodford, M. (2003): Interest and Prices, Princeton University Press: Princeton. [18] Woodford, M. (2005): "Robustly Optimal Monetary Policy with Near-Rational Expectations", unpublished, Columbia University.

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