Central Bankers Preferences and Attitudes Towards Uncertainty: Identification by Means of Asset Prices

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1 Central Bankers Preferences and Attitudes Towards Uncertainty: Identification by Means of Asset Prices Anna Orlik February 1, 2018 Abstract Can we identify preferences of a central bank and her attitude towards model uncertainty? Not if we restrict our attention to commonly used observable macro time series (no matter how many; in the context of monetary models, say, interest rates set by the central bank, inflation and unemployment (or growth)). Yes, if we make an informed use of asset prices. This paper constructs an observational equivalence result between standard optimal control problems based on rational expectations paradigm and robust control theory in which a decision maker is assumed to be faced with model uncertainty: According to the finding, the same optimal policy rule will be chosen by a central banker with rational expectations who fully trusts his model as well as by a robust policy maker who acknowledges model uncertainty and whose preferences are appropriately augmented to offset his fears about model misspecification. In an example application of our result, the introduction of model uncertainty is shown to bridge the difference between empirical values of central bank preferences for inflation Preliminary and incomplete. The views expressed herein are those of the authors and do not necessarily reflect the position of the Board of Governors of the Federal Reserve or the Federal Reserve System. Board of Governors of the Federal Reserve System. Anna.A.Orlik@frb.gov 1

2 stabilization and its theoretical values derived in a microfounded New Keynesian model as it is shown to be explicitly equivalent to augmenting central bank s preference towards less favorable inflation - output gap trade-off (higher λ). Since for every preferencefor-robustness characteristic θ we can identify such λ that leaves the interest rate rule unaltered we conclude that it is impossible to pin down at the same time the central bank s intrinsic preference for inflation stabilization and its attitude towards model uncertainty from the observations of interest rates, inflation and growth, the time series typically used in estimations of New Keynesian model. What breaks the equivalence and restores identification is the use of asset prices as two economies with identical macro outcomes are shown to be associated with different underlying asset prices depending on central banker s attitude towards model uncertainty. This result reopens the discussion on the desirability of targeting asset prices when setting monetary policy as pioneered by Bernanke and Gertler (2000). Keywords: robust control, model misspecification, New Keynesian monetary policy JEL Classification: E52, E58, D81 1 Introduction Can we identify preferences of a central bank and her attitude towards model uncertainty? Not if we restrict our attention to commonly used observable macro time series (no matter how many; in the context of monetary models, say, interest rates set by the central bank, inflation and unemployment (or growth)) and have a very precise knowledge about central bank s attitude towards model uncertainty. Yes, if we make use of asset prices. Within only the last decade the issue of robustness in economic policy design has attracted a large amount of attention in economic theory and in policy-making circles. Two seemingly contradicting approaches - standard optimal control based on rational expectations paradigm and robust control - guide the decision maker faced with uncertainty concerning his model in

3 designing reliable policy rules. If an agent is set in the rational expectations scenario she is bound to have either a perfect knowledge of the modelling environment or probability distributions over all possible realizations of uncertain variables. Then, being a Bayesian, she is able to reduce all the uncertainty to calculated risk. Therefore, Hansen and Sargent (2000) suggest that the rational expectations hypothesis imposes a communism of models : standard optimal control theory tells the policy maker how to derive optimal decisions when his model is the correct one. Contrary to the latter approach robust control is set on the assumption that the model constructed by the decision maker only approximates the true model. In this sense a robust policy making can be viewed as a willingness to manage model uncertainty with the aim of making a policy as reliable as possible given the information provided. To acknowledge the possibility of model misspecification Hansen and Sargent (2000) design the following framework. There exists a set of available perturbated models around the true (reference) model. Being uncertain about her model a robust policy maker tries to design a rule that performs best for the worst possible scenario, i.e. for the situation in which the prior distribution would be the worst distribution among all possible distributions. On top of that, the decision maker s preferences are augmented by adding the preference-for-robustness parameter so as to enhance risk sensitivity. Then, computing the robust decision rules comes down to solving for an equilibrium of a zero-sum game between the policy maker and the evil agent as a dynamic version of max-min game of Milnor (1954). The first applications of robust control techniques bring a fresh look at the matter of the response of monetary authority to increased uncertainty. Contrary to Brainard (1967) who showed that under parameter uncertainty it would be optimal for the central bank to act less vigorously than it would under perfect foresight robust policy design literature concludes in favor of a more aggressive response as compared with the rational expectations benchmark (Hansen and Sargent (2001, 2004), Giannoni (2002), Giordani and Söderlind (2003)). In light of the above results the question posed in this paper may seem intriguing. Is it possible that the observed optimal policy is the same for rational and robust decision makers? Can it 3

4 be therefore independent of the central banker s attitude towards model uncertainty? To our knowledge the issue of the equivalence between standard optimal and robust control problems has been taken up in two studies. Hansen, Sargent and Tallarini (1999) show that the risk-sensitive version of the permanent income model may be observationally equivalent (with respect to the same consumption and savings allocation rules) to the rational expectations benchmark provided that the agents discount factor is appropriately smaller than in the benchmark. As an explanation for this phenomenon they suggest a type of precautionary savings interpretation: activating a concern about robustness induces an increase in savings in the same way as increasing the time discount factor would. We ask the reader to bear in mind that the robust permanent income model is a pure forecasting model with no control and one state variable only. Kasa (1999) constructs an observational equivalence result for the linear optimal regulator problem and its robust counterpart for the case of a single-state and single-control model where the analytical solution can be found in a very straightforward way. He shows that for a given H decision rule there is a strictly convex function relating values of the H -norm to the variable summarizing the relative cost of state versus control variability. He concludes by stating that even though his result is highly specific to the scalar case it is generally impossible to obtain the same feedback matrix for a standard and robust control in a multivariate case. Our results are generalization of the above result. First, we prove that for every rational agent who solves a standard optimal control problem there exists another robust decision maker who is uncertain about his model and yet chooses exactly the same optimal policy rule as his rational counterpart. This result is used to show that the same interest rate will be set by the rational and robust central bankers provided that the latter agent has appropriately stronger preferences towards inflation stabilisation (i.e lower weight attached to output stabilisation and/or interest rate smoothing) than her rational counterpart. The mechanism operating behind this finding is the following. Ceteris paribus, in the context of optimal monetary policymaking, an increased model uncertainty calls for more aggressive behavior. The paper shows 4

5 that this tendency can be completely offset by an appropriate increase in the costs of variation the control variable. This result help explain the difference between empirical values of central bank preferences for inflation stabilization and its theoretical values derived in a microfounded New Keynesian model as it is shown to be explicitly equivalent to augmenting central bank s preference towards less favorable inflation - output gap trade-off (higher λ). Since for every preference-for-robustness characteristic θ we can identify such λ that leaves the interest rate rule unaltered we conclude that it is impossible to pin down at the same time the central bank s intrinsic preference for inflation stabilization and its attitude towards model uncertainty from the observations of interest rates, inflation and growth, the time series typically used in estimations of New Keynesian model. What breaks the equivalence and restores identification is the use of asset prices as two economies with identical macro outcomes are shown to be associated with different underlying asset prices depending on central banker s attitude towards model uncertainty, ceteris paribus. This result reopens the discussion on the desirability of targetting asset prices when setting monetary policy as pioneered by Bernanke and Gertler (2000). The remainder of the paper is organized as follows. In Section 2 we discuss the robust control terminology and derive the general observational equivalence result. As an application of our proposition Section 3 sets up the monetary policy problem in New Keynesian framework with the rational and robust central banker respectively. We show how the use of fundamental asset prices restores identification of central bank s intrinsic preference towards inflation stabilization and allows us to identify, on top of that, central bank s preference for robustness. The last section concludes and outlines the direction of further research that the author intends to pursue. 5

6 2 Robust Control as an Optimal Control Problem In what follows we study the optimum and equilibrium properties of linear-quadratic rational and robust expectations models, respectively. Therefore, let x t be a vector of state variables of dimension n 1 and u t a vector of controllers of dimension k 1. Furthermore let Q [k k] be a symmetric positive definite matrix and R [n n] be some symmetric positive semidefinite matrix. Then the benchmark model we study is cast in the following form Let the one-time period loss be r (x t, u t ) = (x trx t + u tqu t ) max E 0 β t r (x t, u t ) {u t} t=0 t=0 s.t. x t+1 = Ax t + Bu t + Cɛ t+1 and x 0 given where 0 < β < 1 and ɛ t+1 is an i.i.d Gaussian vector process with N (0, I). Provided that some regularity conditions are fullfilled (see e.g. Hansen and Sargent (2006)) the state feedback rule is u t = F x t with F given as F = β(q + βbp B) 1 BP A Matrix P denotes the stabilizing solution of the corresponding algebraic matrix Riccati equation P = T (P ) or P = R + βa P A β 2 A P B(Q + βb P B) 1 BP A To acknowledge the possibility of model misspecification Hansen and Sargent surround the unreliable underlying model with the set of available perturbated data generated processes that lie within a certain neighborhood of the approximated model. The law of motion for the state variables becomes x t+1 = Ax t + Bu t + C (ɛ t+1 + w t+1 ) 6

7 where w t+1 is the term reflecting dynamic misspecification chosen by the evil agent. Thus, the model uncertainty is reflected by augmenting the conditional mean of the exogenous shock to unknown value w t+1 rather than 0. Notice that the conditional volatility of the shock process, as parametrized by C, remains unchanged 1. If the planner was a Bayesian one she would form the probability distributions over all possible realizations of uncertain variables and, hence, reduce all the uncertainty to calculated risk. But once we abstract from priors and rational expectations paradigm we need other tools at hand. First of all, since the robust agent s criterion function cannot involve, by definition, any well defined probabilistic statements Hansen and Sargent (following Gilboa and Schmeidler (1989)) suggest adoption of the dynamic version of the so called max-min approach. In this seetting a policy maker tries to design a rule that performs best for the worst possible scenario, i.e. for the situation in which the prior distribution would be the worst distribution among all possible distributions. A very useful way of setting up and solving the max-min problems is by means of a zero-sum two-agents game: decision maker versus a fictious evil agent. While nature chooses the model from the available set of models so as to maximize the planner s loss the latter agent chooses an optimal robust policy function. Since the evil agent is just a metaphor of policy maker s worst fears about model misspecification the two players share the same reference model given by { max min E 0 β ( ) } t r (x t, u t ) + θw t+1w t+1 {u t} t=0{w t} t=0 t=0 s.t. x t+1 = Ax t + Bu t + C (ɛ t+1 + w t+1 ) and x 0 given max {u t} t=0{w t} t=0 min E 0 β { } t r (x t, u t ) + θw t+1w t+1 t=0 s.t. x t+1 = Ax t + Bu t + C (ɛ t+1 + w t+1 ) and x 0 given 1 This assumption is set for illustration purposes. Hansen and Sargent (2006) show that the worst-case value of w t+1 remains unchanged if we allow the volatility matrix to varry in the approximating and perturbated models. So will we in the analysis which follows. 7

8 The penalty parameter θ θ, ) (sometimes called a preference-for-robustness parameter) restrains the minimizing choice of the w t+1 sequence since it is associated with a Lagrange multiplier on the evil agent s budget constraint E 0 β t w t+1w t+1 η 0 t=0 In the above formulation η 0 can be defined as the set of alternative models (misspecifications) considered by the decision maker. The standard rational expectations solution is achieved by setting η 0 = 0 or, equivalently, by letting θ. Since the evil agent s constraint is always binding in a linear-quadratic framework choosing the degree of robustness is crucial. For calibrating θ Hansen and Sargent suggest a detection error probability procedure using a likelihood ratio test. Zero robustness corresponds to a detection error probability of 50 percent. By analogy to model (??) its robust version (??) has a solution in form of a robust decision rule for control u t = F x t while w t+1 = Kx t is the formula for the worst-case shock where the optimal feedback coefficients are F = β(q + βbd(p )B) 1 BD(P )A K = 1 ( I 1 1 θ θ C P C) C P (A BF ) and where D(P ) = P + P C(θI C P C) 1 C P. One can easily verify that, as mentioned above, with θ we have K = 0 and we are back to the rational expectations state feedback policy rule. The main question posed in this paper is whether it is possible to derive the robust optimal decision rule which will be equivalent to the standard rational expectations (non-robust) optimal control policy rule for any θ θ, ). To show this we first present an alternative way of solving the game defined in (??). Consider the standard (non-robust) optimal regulator problem (w t+1 = K x t ) with the current period criterion x trx t + u tqu t βθ (K x t ) (K x t ) 8

9 subject to the law of motion x t+1 x t+1 = A  0 A x t x t + B 0 u t where  = CK and A = A BF + CK. Hansen and Sargent (2006) prove that for this problem the optimal value function is given by x 0 P P x 0 P P x 0 x 0 where P = P P and P = P P with P and P being a stabilizing solution of the Riccati equation for the standard and robust optimal control problems respectively. The direct implication of the above follows. Lemma 1 Every robust control problem is an augmented regulator problem as defined by Anderson et. al. (1996). Therefore, it can be broken down into two subproblems the first of which is a non-robust version of the decision maker s problem with the optimal policy rule u t = F x t. The second subproblem contains the part directly responsible for promoting robustness F such that F = ( Q + βb P B ) 1 ( βb P A ) F = ( Q + βb P B ) ( ) 1 βb P  + βb P A where u t = F x t is the control law for the robust problem in which F = F + F. Proof. See Hansen and Sargent (2006), p (in the version of 12 September 2006). We will use the above lemma to prove the core result of our paper as given by the proposition below. Fix all parameters except for (R, Q, θ). Then, for the robust planner with 0 < θ < θ < whose preferences (R 1, Q 1 ) satisfy ( Q1 + βb P 1 (R 1 ) B ) 1 [ ] βb P (R 1 ) (A BF1 + CK ) + βb P 1 (R 1 ) BF1 = F 2 9

10 Proposition 1 the optimal feedback control law F 1 is the same as chosen by the rational decision maker with θ = and preferences given by (R 2, Q 2 ), i.e.f 1 = F 2. Proof. By contradiction. Assume that there exists a robust policy maker whose preferences (R 1, Q 1 ) obey eq. (??) but, contrary to the proposition result, there exists other candidate stabilizing robust controller of her problem F c. In other words, there exists such F c F 2 that ( Q1 + βb P 1 (R 1 ) B ) 1 [ ] βb P (R 1 ) (A BFc + CK ) + βb P 1 (R 1 ) BFc = F 2 If F c is the stabilizing controller of the robust control problem then by the Lemma we know that we can break down the problem such that F c = F c + F c where F c = ( Q + βb P (R 1 ) B ) 1 ( βb P (R 1 ) A ) and F c = ( Q + βb P (R 1 ) B ) 1 ( βb P (R 1 ) Â + βb P (R1 ) A ). Hence, the consecutive rearrangements of the equation F c = ( Q + βb P (R 1 ) B ) 1 [ βb P (R 1 ) A + βb P (R 1 ) Â + βb P (R1 ) A ] yield ( Q 1 + βb P 1 (R 1 ) B ) 1 [ βb P (R 1 ) (A BF c + CK ) + βb P 1 (R 1 ) BF c ] = F c. But at the same time by eq. (??) it must be that F c = F 2. QED. The above proposition states that ex post it is possible to interpret the decision taken by a robust policy maker who is uncertain about her model as a decision of a rational agent who fully trusts her model provided that the preferences of a former agent are designed according to eq. (??). Thus, the robust policy rule is observationally equivalent to standard optimal (rational) rule. The direct implication of our finding is that it is impossible to pin down at the same time the inherit preferences of the robust planner, (R 1, Q 1 ), and her need for robustness as measured by θ. We do realize that our proposition establishes a certain existence type of an argument. It does not, however, provide any intuition behind the equivalence result. In particular, we would be interested in the character of the trade-off between (R 1, Q 1 ) and θ. Therefore, in what follows we solve analytically the rational expectations and robust versions of the monetary New Keynesian model. 10

11 3 Application: New Keynesian Model of Monetary Policy The model is set in the New Keynesian (NK) framework for monetary analysis. The popularity of the NK models is due to the fact that they summarize the behavior of optimizing agents (households and firms) in two equations: New Keynesian Philips curve (NKPC) which represents the supply side of the economy and forward-looking (expectational) IS equation that corresponds to the demand side. We refer to the standard model by Clarida, Galí and Gertler (1999) as given by π t = κx t + βe t π t+1 + u t x t = E t x t+1 ϕ (i t E t π t+1 ) + g t where π t, x t and i t are the log-deviations of inflation, output gap and interest rate from their steady-state values, respectively. The disturbance terms, u t (inflation/cost-push shock) and g t (demand shock) evolve according to the following AR(1) specification u t = ρu t 1 + ε u,t g t = µg t 1 + ε g,t with 0 < ρ < 1, 0 < µ < 1 and ε i,t iid(0, σ 2 i ) for i = u, g. The first of the equations of the model may be interpreted as the NKPC which represents the supply side of the economy, whereas the expectational (forward-looking) IS curve (eq. (??)) corresponds to the demand side. Since both equations have been derived from private agents optimization problems together they constitute the equilibrium conditions for a well-specified general equilibrium model. Unlike the traditional Philips curve, the NKPC implies that the correct driving force for an inflation process is a real marginal cost. Deviations in real marginal cost are, in turn, due to changes in output gap and technology. The two objectives of monetary policy which are commonly assumed are: low and stable average rate of inflation and stabilizing output around the full employment. Therefore, we 11

12 assume a policy objective function that minimizes a weighted (by λ) sum of the unconditional variances of the inflation rate and of the output gap 2 [ E t β [ s (π t+s ) 2 + λ (x t+s ) 2]] s=0 Later on we will consider a more general form of the central bank s loss function which accounts for the costs associated with adjusting the interest rate. 3.1 Discretionary Monetary Policy Scenario I. Rational Expectations Since under discretionary monetary stance the period-t decisions of the central banker are not binding in any other period we take expectations as given in the optimization and the decision problem becomes a single-period problem of minimizing the social welfare loss function subject to NKPC equation. In the RE scenario this is equivalent to minimizing (??) subject to eq. (??). { } min π 2 t + λx 2 t x t s.t. NKPC From the FOC we derive the condition relating output gap and inflation x t = κ λ π t. Substituting it into NKPC one obtains π t = βe t π t+1 κ2 λ π t + u t Using the educative guess for the solution to the above expectational difference equation, π t = Mu t, and knowing that with rational expectations the following relation holds: u t+1 = ρu t, 2 For discussions concerning this type of monetary policy objective function see e.g. Clarida Gali and Gertler (1999) or Walsh (2003). 12

13 ( ) 1. we solve for underlying coefficient to be equal to M = 1 βρ + κ2 λ Hence, the optimal deviations in inflation and output gap are given by π t = ) 1 (1 βρ + κ2 u t λ x t = κ λ (1 βρ) + κ u 2 t Finally, to obtain the implied instrument rule for rational expectations scenario substitute the results obtained above into expectational IS curve equation to get i RE t = λρ + (1 ρ) κϕ 1 λ (1 βρ) + κ u 2 t + ϕ 1 g t = P u t + ϕ 1 g t Scenario II. Robust Control The robust decision maker fears that her model is misspecified and seeks robustness. The dynamic misspecification v t+1 = [v 1,t+1, v 2,t+1 ] manifests itself in augmenting the law of motion for the exogenous disturbances u t+1 = ρu t + ε 1,t+1 + v 1,t+1 g t+1 = µg t + ε 2,t+1 + v 2,t+1 In principle, we allow for different level of uncertainty across the model equation. However, as shown in Orlik (2006a) the optimal misspecification of the expectational IS curve in the New Keynesian model with the absence of costs associated with adjusting the interest rate is always equal to zero. To see how the model is misspecified rewrite the NKPC equation as π t = κx t + βe t π t+1 + ρ 1 (u t+1 ε 1,t+1 v 1,t+1 ) As mentioned, to solve for the robust optimal policy under discretion we bring to life another player - evil nature - and construct the following zero-sum game. 13

14 Definition 1 An equilibrium of the Markov perfect multiplier game is a pair of strategies ( ) x t, v1,t+1 such that (a) Given v 1,t+1, x t minimizes the robust agent s expected loss function as given by π 2 t + λx 2 t θβ (v 1,t+1 ) 2 s.t. π t = κx t + βe t π t+1 + ρ 1 (u t+1 ε 1,t+1 v 1,t+1 ). (a) Given x t, v 1,t+1 maximizes the robust agent s expected loss function s.t. the above eq. From the FOCs we obtain the following optimality conditions relating inflation, output gap and misspecification to each other x t = κ λ π t v 1,t+1 = (βθρ) 1 π t Eq. (??) relates the worst-case misspecification to the deviations in inflation through the agent s preference for robustness θ. One salient fact about robust control problems should be discussed before we proceed to find the closed-form solution to our problem, namely a modified certainty equivalence principle. Hansen and Sargent show that despite the fact that the state-control feedback matrix is a function of the shock volatility the solution to the problem remains the same if we set ε i,t+1 = 0. We guess that the equilibrium misspecification chosen by the evil agent evolves optimally with the exogenous predetermined variable, i.e. v 1,t+1 = F u t. At the same time we guess that, as in the rational expectations case, inflation reacts to changes in exogenous state variable accordingly, π t = Nu t, such that N M, (1 ρ) βθρ.hence, using eq. (??) F = N (βθρ) 1.To verify our guess we first take advantage of the modified certainty equivalence property and ) rewrite the disturbance process as u t+1 = u t (ρ + N. Furthermore substituting our guess βθρ into eq. the following condition holds at equilibrium ) N 2 (θρ) 1 N (1 βρ + κ2 λ + 1 = 0 14

15 Notice that the solution to the above equation with θ is N = M if λ = λ, i.e we reproduce the rational expectations solution. However, if the policy maker mistrusts his model, i.e. θ θ, ), and λ λ than the above quadratic equation has the following positive real roots given by N = θρ 2 N < (1 ρ) βθρ. Since N θ will be given by θ= ) (1 βρ + κ2 λ ( ) 2 1 βρ + κ2 λ 4 (θρ) 1 < 0 then the lower bound on preference-for-robustness parameter The last thing is to compute the robust implied interest rate rule from the expectational IS curve as i t = 1 κ ϕ λ N ( ρ + N ) βθρ u t + 1 κ ϕ λ Nu t + N The next proposition shows the conditions under which i t = i RE t. ( ρ + N ) u t + ϕ 1 g t βθρ Proposition 2 Observational equivalence result for rational and robust policies under discretionary monetary stance. Fix all parameters, excluding (λ, θ). Then there exists such λ that the same optimal implied interest rate rule is pursued by a rational monetary authority (λ, θ = ) and by a robust decision maker characterized by (λ, θ ) and where λ < λ is given by λ = κ 2 N N 2 θρ N (1 βρ) + 1 where N is the positive root real (satisfying the stability condition) of the following polynomial [ ] N θ 2 ρ 2 βϕκ + N 2 (1 βρ) (1 ρ) + + βθρ βθρϕκ θρϕκ [ (1 ρ) (1 βρ) +N ρ 1 ] + 1 ρ ϕκ βθρϕκ ϕκ P = 0 Proof. See Appendix A. The graphical illustration of the above proposition follows. 15

16 Figure 1. Socially optimal inflation-output gap trade-off. The value of λ has been set up to 0.5. The figure shows that, ceteris paribus, the higher the model uncertainty (i.e. the lower the θ) the higher λ should be in order for the rational and robust policy rules to be observationally equivalent. This is due to the fact that the higher the model uncertainty the more aggressive the response of the robust planner will be. This effect will be totally offset in case of the central banker who attaches an appropriately higher weight to output gap deviations. Notice that with θ all the model uncertainty disappears and λ λ = Optimal Unrestricted Commitment In what follows we present the situation in which the decision maker credibly commits to optimal policy rule Scenario I. Rational Expectations As before we start our analysis of misspecified monetary dynamics with commitment policy by solving for the rational expectations equilibrium. We assume an unrestricted commitment mechanism in the sense that we don t apply any specific form of it as it would be the case with, e.g., the Taylor-type rules. In this framework the policymaker s objective is to choose sequences of i t+i, π t+i and x t+i to minimize: [ 1 E t β i 2 (π t+i) ] 2 λ (x t+i) 2 i=0 κ 1,t+i (π t+i βπ t+i+1 κx t+i u t+i ) κ 2,t+i (x t+i x t+i+1 ϕπ t+i+1 + ϕi t+i g t ) The FOC with respect to i t is given by ϕe t (κ 2,t+i ) = 0 for i 0 16

17 This condition reflects the fact that the expectational IS curve equation imposes no real constraint on the central bank as long as there are no restrictions or costs associated with adjusting the nominal interest rate. Taking it into account, the further FOCs take the following form π t + κ 1,t = 0 E t (π t+i + κ 1,t+i κ 1,t+i 1 ) = 0 for i > 0 E t (λx t+i κκ 1,t+i ) = 0 for i > 0 The fully optimal commitment policy (see Walsh (2003)) does not honour any past commitment. If we set the Lagrangian multipliers the problem becomes equivalent to discretionary solution Scenario II. Robust Control with Commitment In what follows we assume that also the evil agent has access to a certain commitment mechanism. This assumption is very intuitive considering the fact that the evil agent is just a technical metaphor used for expressing the policymaker s fears about model misspecification. From this perspective the evil agent should reoptimize only once the central banker does so. Hence, the only difference with respect to the definition of the robust game presented in the previous section is in the definition of the strategies: at time zero the evil agent chooses and credibly commits to the sequence of perturbations {v t } t=0 the optimal policy forever 3. while the planner commits to implementing Definition 2 An equilibrium of the robust multiplier game in sequences is a pair of sequences {v t } t=0 and {x t } t=0 such that (a) Given {vt } t=0, {x t } t=0 minimizes the robust agent s expected loss function as given by [ 1 β i 2 (π t+i) λ (x t+i) 2 1 ] 2 βθ (v 1,t+i+1) 2 Et R i=0 3 Giordani and Söderlind (2004) show that if the evil agent commits to a non-stationary process of misspecification the policy maker s loss function becomes unbounded. 17

18 subject to NKPC and IS curve and where u t+i and g t+i are misspecified according to eq. u t+i+1 = ρu t+i + ε 1,t+i+1 + v 1,t+i+1 g t+i+1 = µg t+i + ε 2,t+i+1 + v 2,t+i+1 (b) Given {x t } t=0, {v t } t=0 above. maximizes the robust agent s expected loss function as given the Hence, the Lagrangian for the problem is given by maxmin v x E R t {[ 1 β i 2 (π t+i) ] 2 λ (x t+i) 2 1 } 2 βθ (v 1,t+i+1) 2 + i=0 +ψ 1,t+i (π t+i βπ t+i+1 κx t+i u t+i ) + +ψ 2,t+i (x t+i x t+i+1 ϕπ t+i+1 + ϕi t+i g t ) + +ψ 3,t+i (γg t+i + ɛ g,t+i+1 g t+i+1 ) + +ψ 4,t+i (ρu t+i + ɛ u,t+i+1 + v t+i+1 u t+i+1 ) (1) As in the RE scenario the FOC for interest rate again reflects the no real constraint property, since E t ( ψ2,t+i ) = 0 for i 0. Therefore, the remaining FOCs are given by π t + ψ 1,t = 0 (2) E R t ( πt+i + ψ 1,t+i ψ 1,t+i 1 ) = 0 for i 0 (3) E t ( λxt+i κψ 1,t+i ) = 0 (4) ψ 1,t+i + ρψ 4,t+i 1 β ψ 4,t+i 1 = 0 (5) βθv 1,t+i+1 + ψ 4,t+i = 0 (6) Again, assume that also evil agent ignores the past commitments. As before the robust solution with commitment follows as in the discretionary case. 18

19 3.3 General Case with Costs of Adjusting the Nominal Interest Rate Scenario I. Rational Expectations [ 1 E t β i 2 (π t+i) λ (x t+i) ] 2 λ i (i t+i ) 2 i=0 +κ 1,t+i (π t+i βπ t+i+1 κx t+i u t+i ) + +κ 2,t+i (x t+i x t+i+1 ϕπ t+i+1 + ϕi t+i g t ) The FOCs are E t (π t+i + κ 1,t+i κ 1,t+i 1 = 0 for i 0 E t (λx t+i κκ 1,t+i + κ 2,t+i 1 ) β κ 2,t+i 1 = 0 for i 0 The additional condition governing the costs of adjusting the nominal interest rate is given by E t (λ i i t+i + ϕκ 2,t+i ) = 0 for i 0 We combine the first-order conditions under the assumption of fully optimal commitment policy. Hence, λx t + κπ t λ i ϕ i t = 0 (7) To generate the candidate equilibrium guess that the solution to the problem takes the form: π t = Su t + Kg t. Details of the algorithm for computing the response coefficients S and K can be found in the appendix. Then, it can be shown that the implied interest rate rule takes the following form: i t = ϕ λ i (Au t + Bg t ) where A = λ (S βρs 1) + κs and B = κ λ (K βkγ) + κk. κ 19

20 3.3.2 Scenario II. Robust Control [ 1 maxminet R β (π i 2 t+i) 2 + 1λ (x 2 t+i) 2 + 1λ 2 i (i t+i ) 2] v i 1βθ (v 2 1,t+i+1) 2 1βθ (v 2 2,t+i+1) 2 i=0 +µ 1,t+i (π t+i βπ t+i+1 κx t+i u t+i ) + +µ 2,t+i (x t+i x t+i+1 ϕπ t+i+1 + ϕi t+i g t ) + +µ 3,t+i (ρu t+i + ɛ u,t+i+1 + v 1,t+i+1 u t+i+1 ) + +µ 4,t+i (γg t+i + ɛ g,t+i+1 + v 2,t+i+1 g t+i+1 ) + for i 0 E R t ( ) Et R πt+i + µ 1,t+i µ 1,t+i 1 = 0 ( λx t+i κµ 1,t+i + µ 2,t+i 1 ) β µ 2,t+i 1 = 0 E R t ( λi i t+i + ϕµ 2,t+i ) = 0 Optimal commitment policy is characterized by µ 1,t+i + ρµ 3,t+i + 1 β µ 3,t+i 1 = 0 µ 2,t+i + γµ 4,t+i + 1 β µ 4,t+i 1 = 0 θv 1,t+i+1 + µ 3,t+i = 0 θv 2,t+i+1 + µ 4,t+i = 0 λx t + κπ t λ i ϕ i t = 0 (8) v 1,t+1 = 1 βθρ π t (9) v 2,t+1 = λ i γβθϕ i t (10) This time we generate our candidate solutions as π t = Cu t + Dg t. Furthermore guess that v 1,t+1 = F 1 u t + F 2 g t and v 2,t+1 = F 3 u t + F 4 g t. Then, the exogenous state variables evolution will be determined as u t+1 = (ρ + F 1 ) u t + F 2 g t and g t+1 = (γ + F 4 ) g t + F 3 u t. 20

21 4 Conclusion TBC Appendix A Observational equivalence result for discretionary monetary stance Proposition 1 Observational equivalence result for rational and robust policies under discretionary monetary stance. Fix all parameters, excluding (λ, θ). Then there exists such λ that the same optimal implied interest rate rule is pursued by a rational monetary authority (λ, θ = ) and by a robust decision maker characterized by (λ, θ ) and where λ < λ is given by N 2 θρ κ 2 N = λ N (1 βρ) + 1 where N is the positive root real of the following third-degree polynomial [ ] N θ 2 ρ 2 βϕκ +N 2 (1 βρ) (1 ρ) + + +N βθρ βθρϕκ θρϕκ and P = λρ+(1 ρ)κϕ 1 λ(1 βρ)+κ 2. [ (1 ρ) (1 βρ) ρ 1 ϕκ βθρϕκ ] + 1 ρ ϕκ P = 0 Proof. Rewrite consecutively the latter equation N 2 βθρ N [ ] N 2 N (1 βρ) [ ] N 2 N (1 βρ) Nρ ρ [ N 2 βθρϕκ θρ ϕκ θρ ϕκ θρ N 2 ( 1 1 ) ( κ 1 κ βθρ ϕ λ + N ϕ λ + ρ ρ 1 κ ϕ λ 1 ( κ ϕ λ N ρ + N ) u t + 1 ( κ βθρ ϕ λ Nu t + N ρ + N βθρ to obtain i t = i RE t. QED. ] N (1 βρ) + 1 P = ) P = ) u t + ϕ 1 g t = 21

22 B General Case with Costs of Adjustment of the Interest Rate TBC 22

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