Robust control with commitment: A modification to Hansen-Sargent

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Journal of Economic Dynamics & Control 32 (2008) 2061–2084 www.elsevier.com/locate/jedc

Robust control with commitment: A modification to Hansen-Sargent Richard Dennis Economic Research, Mail Stop 1130, Federal Reserve Bank of San Francisco, 101 Market St, CA 94105, USA Received 7 December 2006; accepted 20 August 2007 Available online 8 September 2007

Abstract I examine the Hansen and Sargent [2003. Robust control of forward-looking models. Journal of Monetary Economics 50, 581–604] formulation of the robust Stackelberg problem and show that their method of constructing the approximating equilibrium is generally invalid. I then turn to the Hansen and Sargent [2007. Robustness, manuscript (version dated March 22, 2007)] treatment, which, responding to the problems raised in this paper, changes subtly, but importantly, how the robust Stackelberg problem is formulated. In the context of Hansen and Sargent [2007. Robustness, manuscript (version dated March 22, 2007)], I prove, first, that their method for obtaining the approximating equilibrium is now equivalent to the one developed in this paper, and, second, that the worst-case specification errors are not subject to a time-consistency problem. In the context of the Erceg et al. [2000. Optimal monetary policy with staggered wage and price contracts. Journal of Monetary Economics 46, 281–313] sticky wage/sticky price model, I find that a robust central bank will fear primarily that the supply side of its approximating model is misspecified and that robustness affects importantly central bank promises about future policy. r 2007 Elsevier B.V. All rights reserved. JEL classification: E52; E62; C61 Keywords: Robust control; Robust Stackelberg games; Approximating equilibrium

E-mail address: [email protected] 0165-1889/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jedc.2007.08.003

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1. Introduction Uncertainty of one form or another is a problem that all decisionmakers must confront and it is an issue that is clearly important for policy institutions such as the Federal Reserve. In the words of former Federal Reserve chairman Alan Greenspan, ‘Uncertainty is not just an important feature of the monetary policy landscape; it is the defining characteristic of that landscape’ (Greenspan, 2003). Recognizing the impact that uncertainty can have on behavior, Hansen and Sargent and coauthors show how economic agents can use robust control theory to make decisions while guarding against the uncertainty that they fear.1 Building on and extending rational expectations, robust control theory allows a concern for model misspecification, or pessimism, to distort how expectations are formed and thereby alter decisions. Although decisionmakers may possess a ‘good’ model of the economy (their approximating model), fears of misspecification are operationalized by assuming that expectations are formed and decisions are made in the context of a model distorted strategically by specification errors. The distorted model twists the approximating model according to the worst fears of that particular agent.2 Importantly, although it is common to assume that all agents share the same approximating model, there can be as many distorted models as there are agents in the economy, with each distorted model reflecting the concerns of that particular agent. It is important to appreciate, however, that although agents may fear that their model is misspecified, specification errors need not actually be present. For this reason, the concept of an approximating equilibrium is central to any robust control exercise. In the approximating equilibrium, although the approximating model is correctly specified, agents who seek robustness apply decision rules that guard against their worst-case specification error. The approximating equilibrium is distinct from the rational expectations equilibrium, in which agents do not fear misspecification, and from an agent’s worst-case equilibrium, in which that agent’s worst fears materialize. For robust Stackelberg problems, Hansen and Sargent (2003) describe a procedure to recover the approximating equilibrium from a pessimistic agent’s worst-case equilibrium. However, complicating the analysis of robust Stackelberg problems is the fact that the leader’s commitment to its policy means that shadow prices, or Lagrange multipliers on implementability conditions, enter the solution.3 In this paper, I show that, although the method that Hansen and Sargent (2003) use correctly locates the approximating equilibrium for planning problems in which 1

In their manuscript Robustness, Hansen and Sargent (2007) provide a sweeping analysis of how robust control can be applied to economic decisionmaking. 2 Insomuch as in equilibrium they each involve probabilities being distorted, or slanted, toward unfavorable payoffs, there are ties between robust control, risk-sensitive control (Whittle, 1990), uncertainty aversion (Epstein and Wang, 1994), disappointment aversion (Gul, 1991), and the latter’s generalization (Routledge and Zin, 2003). 3 In this respect, there is a clear parallel between the solution of robust Stackelberg games and nonrobust Stackelberg games, in which shadow prices also serve as state variables (Kydland and Prescott, 1980).

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private agents are not forward-looking (Sargent, 1999; Hansen and Sargent, 2001a), it fails for the very class of robust Stackelberg problems that they sought to analyze. After demonstrating why their method fails, I show how to recover the approximating equilibrium correctly. Unlike Hansen and Sargent (2003), when recovering the approximating equilibrium, I recognize that the shadow prices that arise in the solution when followers are forward-looking are a component of the optimal policy; they reflect the history dependence of the leader’s policy and are intended to induce a particular behavior from the followers. Therefore, when constructing the approximating equilibrium, both the leader’s decision rule and the law of motion for the shadow prices should reflect the leader’s fear of misspecification. Responding to these problems,4 Hansen and Sargent (2007, chapter 16) modified the class of robust Stackelberg problems that they analyze; this is an important contribution of this paper. In addition to specification errors residing in the predetermined block of the model, they now allow the Stackelberg leader to fear distortions to private-agent expectations, similar to Woodford (2005). For this alternative formulation of the robust Stackelberg problem, I prove that the worstcase distortions are unaffected by the Stackelberg leader’s promises about future policy. I also prove that, conditional on the Stackelberg leader’s robust decision rule, the worst-case distortions do not affect the worst-case law of motion for the shadow prices. As a consequence, simply setting the worst-case distortions to zero does not alter the law of motion for the shadow prices, and, for this reason, the approximating equilibrium that emerges is equivalent to the one obtained using my procedure. It follows that, whether interest centers on the robust Stackelberg problem analyzed in Hansen and Sargent (2003) or on the one analyzed in Hansen and Sargent (2007, chapter 16), the method I propose returns the correct approximating equilibrium. To understand the nature of robust monetary policy, I employ the sticky price/ sticky wage model developed by Erceg et al. (2000), which provides the foundation for many modern new Keynesian business cycle models. Introducing the specification errors as per Hansen and Sargent (2003), I show that a pessimistic central bank will fear primarily that its approximating model understates the persistence of the technology shock. Allowing the central bank to fear distortions to private-agent expectations, as per Hansen and Sargent (2007), I find that the central bank’s desire for robustness primarily affects the promises it makes about future policy, as the central bank attempts to manipulate private-agent expectations to guard against the specification errors that it fears. The remainder of the paper is organized as follows. In the following section, I summarize some related literature. In Section 3, I introduce the general structure of 4 After writing the first version of this paper, I sent copies to Lars Hansen and Tom Sargent. Subsequently, they acknowledged the error, changed the nature of the distortions that the Stackelberg leader fears, and revised the relevant chapters of their manuscript. As a consequence, I do not believe that the treatment that Hansen and Sargent currently provide in their manuscript, Robustness, is in error. Indeed, I show that their method of constructing the approximating equilibrium is now equivalent to mine.

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the approximating model and outline how Hansen and Sargent (2003) formulate the optimization problem confronting a Stackelberg leader who fears misspecification. Section 4 shows how control methods developed by Backus and Driffill (1986) can be employed to solve for the worst-case equilibrium.5 Two methods for constructing the approximating equilibrium are presented and explained in Section 5. Section 6 clarifies how and why the approximating equilibrium developed in Section 5 differs from Hansen and Sargent (2003). Section 7 describes and analyzes the Hansen and Sargent (2007) revised formulation of the robust Stackelberg problem. Section 8 examines robust monetary policy in a sticky wage/sticky price new Keynesian business cycle model. Section 9 concludes. Appendices contain technical material.

2. Related literature Prominent applications of robust control have examined how uncertainty affects asset prices. For example, Hansen et al. (1999) study consumption behavior and the price of risk using a model in which a representative consumer, facing a linear production technology and an exogenous endowment process, has a preference for robustness. They show that a preference for robustness operates on asset prices like a change in the discount factor, with expressions for consumption and investment unchanged from the usual permanent income formulas. Hansen et al. (2002) build on Hansen et al. (1999) by assuming that part of the state vector is unobserved and by introducing robust filtering. Exploiting a separation principle that allows consumers to first estimate the state and then apply robust control as if the state were known, they show that the consumer’s filtering problem does not add to the price of risk for a given detection error probability. Cagetti et al. (2002) extend the analysis in Hansen et al. (2002) to a continuous-time nonlinear stochastic growth model in which technology follows a two-state Markov process. Where the studies outlined above all employ single-agent robust decision theory, Hansen and Sargent (2003) consider a multi-agent environment, focusing on robust Stackelberg problems in which the leader and the follower(s) may each express a preference for robustness separately. To illustrate their approach, Hansen and Sargent (2003) solve a model in which a monopoly leads a competitive fringe. Naturally, multi-agent economies in which a fiscal or a monetary authority seeks to conduct policy while expressing distrust in its model can also be modeled as a robust Stackelberg problem. Taking this path, Giordani and So¨derlind (2004) analyze monetary policy in an economy consisting of households, firms, and a central bank. They study robust Stackelberg problems (commitment equilibria) and robust Markov-perfect Stackelberg problems (discretionary equilibria) in which the central bank formulates policy while fearing misspecification. Dennis (2007) extends the analysis of Markov-perfect Stackelberg problems to environments where the central bank fears that the specification errors also distort private agent expectations. Walsh 5

Backus and Driffill (1986) build on earlier work by Oudiz and Sachs (1985). See also Currie and Levine (1985, 1993).

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(2003) relates robust control policies for central banks to policies motivated by other forms of uncertainty and other notions of robustness. In recent work, Hansen and Sargent (2007, chapter 15) consider Markov-perfect equilibria in multi-agent economies, allowing each agent to seek robustness.

3. The benchmark robust Stackelberg problem In this section, I describe a linear-quadratic control problem in which a Stackelberg leader commits to a policy while expressing concern that its model may be misspecified. I focus on an environment in which the Stackelberg leader fears misspecification and, consistent with Hansen and Sargent (2003), I assume that the specification errors that the Stackelberg leader fears reside in the ‘predetermined block’ of the model and do not distort private agent expectations.6 In Section 7, I consider the case where the Stackelberg leader also fears that specification errors will distort private-agent expectations, expanding on Hansen and Sargent (2007, chapter 16). The economic environment is one in which an n  1 vector of endogenous variables, zt , consisting of n1 predetermined variables, xt , and n2 ðn2 ¼ n  n1 Þ nonpredetermined variables, yt , evolves over time according to xtþ1 ¼ A11 xt þ A12 yt þ B1 ut þ C1 extþ1 , E t ytþ1 ¼ A21 xt þ A22 yt þ B2 ut ,

ð1Þ ð2Þ

where ut is a p  1 vector of policy control variables, ext i:i:d:½0; Is  is an s  1 ðspn1 Þ vector of white-noise innovations, and E t is the private sector’s mathematical expectations operator conditional upon period t information. Eqs. (1) and (2) describe the approximating model, the model that the leader and private agents (the followers) believe comes closest to describing the data generating process. The matrices A11 , A12 , A21 , A22 , B1 , and B2 are conformable with xt , yt , and ut as necessary and contain the structural parameters that govern preferences and technology. The matrix C1 is determined to ensure that ext has the identity matrix as its variance-covariance matrix. In a world with rational expectations, the control problem is for the leader to choose the control variables fut g1 0 to minimize E0

1 X

bt ½z0t Wzt þ 2z0t Uut þ u0t Rut ,

(3)

t¼0

where b 2 ð0; 1Þ is the discount factor and zt  ½x0t y0t 0 , subject to Eqs. (1) and (2). The weighting matrices, W and R, are assumed to be positive semidefinite and positive definite, respectively. 6

Note that Walsh (2003), Giordani and So¨derlind (2004), Dennis et al. (2006a, b), and many other robust control applications model the specification errors this way.

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However, the leader is concerned that its approximating model may be misspecified. To accommodate this concern, distortions, or specification errors, vtþ1 , are introduced and the approximating model is surrounded by a class of models of the form xtþ1 ¼ A11 xt þ A12 yt þ B1 ut þ C1 ðvtþ1 þ extþ1 Þ, E t ytþ1 ¼ A21 xt þ A22 yt þ B2 ut .

ð4Þ ð5Þ

Eqs. (4) and (5) describe the ‘distorted’ model, the approximating model distorted by specification errors. Because private agents know that the leader is concerned about misspecification, the distorted model can be written more compactly as e tþ1 , ztþ1 ¼ Azt þ But þ Cvtþ1 þ Ce (6) h i C  C 0 e  1 , and etþ1  extþ1 in which eytþ1  y  E t y where C  01 , C tþ1 tþ1 is a e 0 I ytþ1

Martingale difference sequence; these expectation errors are determined as part of the equilibrium. Assume that the pair ðA; BÞ is stabilizable (Kwakernaak and Sivan, 1972, chapter 6) and that the sequence of specification errors, fvtþ1 g1 0 , satisfies the boundedness condition E0

1 X

btþ1 v0tþ1 vtþ1 pZ,

(7)

t¼0

where Z 2 ½0; ZÞ. It is the satisfaction of this boundedness condition that defines the sense in which the approximating model, shown in Eqs. (1) and (2), is a ‘good’ one. Note that in the special case in which Z ¼ 0, the nonrobust decision problem is restored. To guard against the misspecifications that it fears, the leader formulates policy subject to the distorted model with the mind-set that the specification errors will be as damaging as possible, a view that is operationalized through the metaphor that fvtþ1 g1 0 is chosen by a fictitious evil agent whose objectives are diametrically opposed to those of the leader. Applying Luenberger’s (1969) Lagrange multiplier theorem, Hansen and Sargent (2001b) show that the constraint problem, in which Eq. (3) is 1 minimized with respect to fut g1 0 and maximized with respect to fvtþ1 g0 , subject to Eqs. (6) and (7), can be replaced with an equivalent multiplier problem, in which E0

1 X

bt ½z0t Wzt þ 2z0t Uut þ u0t Rut  byv0tþ1 vtþ1 ,

(8)

t¼0

y 2 ½y; 1Þ, is minimized with respect to fut g1 with respect to fvtþ1 g1 0 and maximized 0 , h i ut e  ½B C, U e  ½U 0, subject to Eq. (6). With the following definitions e ut  v , B tþ1 h i e  R 0 , the robust problem can be expressed as7 and R 0 byI min max1 E0 1

fut g0 fvtþ1 g0 7

1 X t¼0

e ut þ e e ut , bt ½z0t Wzt þ 2z0t Ue u0t Re

(9)

In moving from the constraint problem to the multiplier problem, the assumptions that the pair (A, B) e is stabilizable and the sequence fvtþ1 g1 0 is bounded are replaced with the assumptions that the pair (A, B) is

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subject to the distorted model e tþ1 . e ut þ Ce ztþ1 ¼ Azt þ Be

(10)

4. The worst-case equilibrium The problem facing a Stackelberg leader that must conduct policy while fearing that its model is misspecified is described by Eqs. (9) and (10)). The solution to this problem returns the leader’s worst-case equilibrium, the decision rules, and the law of motion for the state variables that govern the economy’s behavior according to the leader’s worst-case fears. For the control problem described above, the leader’s worst-case equilibrium can be obtained by applying the Backus and Driffill (1986) solution method, developed originally to solve problems with rational expectations, which formulates the optimization problem as a dynamic program.8 Since the control problem is linear-quadratic, the value function takes the form V ðzt Þ ¼ z0t Vzt þ d and the dynamic program can be written as e ut þ e e ut þ bE t ðz0 Vztþ1 þ dÞ, u0t Re z0t Vzt þ d  min max½z0t Wzt þ 2z0t Ue tþ1 ut

vtþ1

(11)

subject to e tþ1 . e ut þ Ce ztþ1 ¼ Azt þ Be

(12)

To keep the paper self-contained, I describe the Backus–Driffill solution method in Appendix A, where I show that in the worst-case equilibrium the law of motion for the state variables is " # " # xtþ1 xt 1 (13) ¼ TðA  BFu  CFv ÞT þ Cxtþ1 , ptþ1 pt p0 ¼ 0, where pt is an n2  1 vector of shadow prices of the non-predetermined variables, yt . These shadow prices, pt , are predetermined variables that enter the equilibrium as state variables to give private agents the necessary incentives to form expectations and to make decisions that conform with the unique stable rational expectations equilibrium that is consistent with the leader’s chosen policy. An alternative way of thinking about these shadow prices is that they encode the history dependence of the leader’s policy, a history dependence arising from the leader’s commitment to its policy. The matrix T provides the mapping between the state variables, xt and pt , and the endogenous variables, zt , and is given by (footnote continued) 0 e VC e is positive definite, where V is defined implicitly by Eq. (11). The second of stabilizable and that yI  C these two assumptions determines the lower bound, y, that y must exceed. 8 Alternatively, this robust control problem can be solved using a Lagrangian or by applying the recursive saddle-point theorem (Marcet and Marimon, 1999).

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T¼

h

I 0 V21 V22

i

, where V21 and V22 are submatrices of V, conformable with xt and yt

(see Appendix A). Together with the law of motion for the state variables, the solution for the nonpredetermined variables is " # xt 1 1 yt ¼ ½V22 V21 V22  , (14) pt while the robust decision rule(s) for the leader and the worst-case distortions are given by " # " # " # ut Fu 1 xt ¼ T . (15) pt vtþ1 Fv The leader’s worst-case fear is that the approximating model is distorted by the specification errors contained within Eq. (15).

5. The approximating equilibrium The approximating equilibrium describes the economy’s behavior under robust decisionmaking but in the absence of misspecification. Although the worst-case equilibrium and the approximating equilibrium differ, because the distorted model is a device the leader employs to achieve robustness, the approximating equilibrium and the leader’s worst-case equilibrium are closely connected. To construct the approximating equilibrium, I begin by taking the worst-case equilibrium, Eqs. (13)–(15) and writing it out in terms of states as xtþ1 ¼ M11 xt þ M12 pt þ C1 extþ1 ,

ð16Þ

ptþ1 ¼ M21 xt þ M22 pt , yt ¼ H21 xt þ H22 pt , eu1 xt þ F eu2 pt , ut ¼ F

ð17Þ ð18Þ

ev1 xt þ F ev2 pt , vtþ1 ¼ F

ð20Þ

ð19Þ

1 1 e e e e where H21  V1 V , H22  V1 22 , ½Fu1 Fu2   Fu T , ½Fv1 Fv2   Fv T , and h i 22 21 11 M12  TðA  BF  CF ÞT1 . M¼ M u v M M 21

22

With the worst-case equilibrium given by Eqs. (16)–(20), identical representations of the approximating equilibrium can be derived by working with either the approximating model or the distorted model. Building up from the approximating model, the first step is to discard the worst-case distortions (the evil agent’s decision rule), Eq. (20). The second step is to substitute Eqs. (18) and (19) into the approximating model, Eq. (1). With these substitutions, in the approximating

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equilibrium the predetermined variables, xt , evolve over time according to xtþ1 ¼ A11 xt þ A12 yt þ B1 ut þ C1 extþ1 , eu1 xt þ F eu2 pt Þ þ C1 extþ1 , ¼ A11 xt þ A12 ðH21 xt þ H22 pt Þ þ B1 ðF eu1 Þxt þ ðA12 H22 þ B1 F eu2 Þpt þ C1 extþ1 , ¼ ðA11 þ A12 H21 þ B1 F

ð21Þ

with the remainder of the approximating equilibrium given by Eqs. (17)–(19). Alternatively, simplifying down from the distorted model, substituting Eqs. (18)–(20) into the distorted model, Eq. (4), yields xtþ1 ¼ A11 xt þ A12 yt þ B1 ut þ C1 vtþ1 þ C1 extþ1 , eu1 xt þ F eu2 pt Þ þ C1 ðF ev1 xt þ F ev2 pt Þ ¼ A11 xt þ A12 ðH21 xt þ H22 pt Þ þ B1 ðF þ C1 extþ1 , eu1 þ C1 F ev1 Þxt þ ðA12 H22 þ B1 F eu2 þ C1 F ev2 Þpt ¼ ðA11 þ A12 H21 þ B1 F þ C1 extþ1 .

ð22Þ

ev1 ¼ 0 and F ev2 ¼ 0 in Eq. (22), Then, zeroing-out the distortions by setting F produces eu1 Þxt þ ðA12 H22 þ B1 F eu2 Þpt þ C1 extþ1 , xtþ1 ¼ ðA11 þ A12 H21 þ B1 F

(23)

with the remainder of the system behaving according to Eqs. (17)–(19). Eq. (23) is, of course, identical to Eq. (21). Notice that in this approximating equilibrium, the non-predetermined variables, yt , the leader’s decision variables, ut , and the shadow prices of the non-predetermined variables, pt , all respond to the state variables as they do in the worst-case equilibrium.9

6. The Hansen–Sargent approximating equilibrium In contrast to the approach described above, Hansen and Sargent (2003) derive the approximating equilibrium as follows. Given the leader’s worst-case equilibrium, Hansen and Sargent (2003, p. 592) set Fv ¼ 0 in Eqs. (13) and (15) to obtain what I will term the ‘Hansen–Sargent approximating equilibrium.’ With this simplification, the law of motion for the state variables becomes " # " # xtþ1 xt 1 (24) ¼ TðA  BFu ÞT þ Cxtþ1 ptþ1 pt 9 In effect, although the Stackelberg leader is the only agent seeking robustness, private agents are assumed to form expectations and to make decisions using the leader’s distorted model. In this sense the followers are also acting as robust agents (this is the assumption in Hansen and Sargent (2003) and Giordani and So¨derlind (2004)). An alternative is to assume that private agents acknowledge the leader’s robust policy, but use the approximating model to make decisions. Under this assumption, the approximating equilibrium is constructed by solving the system given by Eqs. (1), (2), (17), and (19) for its rational expectations equilibrium; see Dennis (2007).

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(see Hansen and Sargent, 2003, equation (3.21a)), while the non-predetermined variables and the leader’s decision rule continue to be given by Eqs. (18) and (19), respectively. According to the Hansen–Sargent approximating equilibrium, the decision rules for private agents and for the Stackelberg leader are formulated to guard against the leader’s worst-case misspecification, but the law of motion for the state variables – the predetermined variables and the shadow prices of the nonpredetermined variables – is modified because the specification errors that the leader fears are absent. On a mechanical level, the difference between the approximating equilibrium presented in Section 5 and the Hansen–Sargent approximating equilibrium described ev1 ¼ 0 and by Eq (24) is that, whereas I zero out the specification errors by setting F e Fv2 ¼ 0 in Eq. (22), Hansen and Sargent (2003) zero out the specification ev1 and F ev2 represent errors by setting Fv ¼ 0 in Eqs. (13) and (15). But, whereas F coefficients on the state variables (the predetermined variables and the shadow prices) in the evil agent’s decision rule, Fv describes how the evil agent’s decision variables relate to the predetermined variables, xt , and the non-predetermined variables, yt . It is apparent from Eq. (13) that setting Fv ¼ 0 can change the behavior of both xt and pt , allowing the absence of misspecification to alter the law of motion for the shadow prices even while the decision rules for private agents and the leader are unaltered. Unfortunately, the method that Hansen and Sargent (2003) propose will return an incorrect solution whenever setting Fv ¼ 0 changes the law of motion for the shadow prices. The method is generally inappropriate because it treats the shadow prices as if they are physical states rather than recognizing that they enter the equilibrium to induce private agents to behave in a manner consistent with the equilibrium desired by the Stackelberg leader. Since the Stackelberg leader fears misspecification, and this fear does not depend on whether the approximating model is actually misspecified, in the approximating equilibrium the shadow prices, which encode the history dependence of the leader’s robust policy, should respond to the state variables as they do in the worst-case equilibrium.

7. An alternative robust Stackelberg problem Responding to the issues raised above, Hansen and Sargent (2007, chapter 16) change how they formulate the robust Stackelberg problem; the change is subtle, but important. In this section, I describe and interpret this alternative formulation. Then I prove, first, that, conditional on the Stackelberg leader’s worst-case decision rule, the worst-case law of motion for the shadow prices of the non-predetermined variables does not depend on the worst-case distortions and, second, that the Stackelberg leader cannot influence the worst-case distortions through promises about future policy. The former result implies that Hansen and Sargent’s (2003) approach to constructing the approximating equilibrium would have been correct had it been applied to the Hansen and Sargent (2007) formulation of the problem. The latter result implies, among other things, that the trade-offs confronting the

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metaphorical evil agent are very simple, and unencumbered by promises about future policy. Recall that the approximating model is xtþ1 ¼ A11 xt þ A12 yt þ B1 ut þ C1 extþ1 , Et ytþ1 ¼ A21 xt þ A22 yt þ B2 ut ,

ð25Þ ð26Þ

where the expectations operator in Eq. (26) reflects expectations formed by private agents. By introducing the expectational error, eytþ1  ytþ1  E t ytþ1 , the approximating model can be written in terms of realized values xtþ1 ¼ A11 xt þ A12 yt þ B1 ut þ C1 extþ1 ,

(27)

ytþ1 ¼ A21 xt þ A22 yt þ B2 ut þ eytþ1 .

(28)

In recognition that in equilibrium the expectational errors will be a linear function of the shocks, eyt ¼ C2 ext , Eqs. (27) and (28) can be written more compactly as ztþ1 ¼ Azt þ But þ Cextþ1 , (29) h i C1 where C  C . With C2 yet to be determined, Hansen and Sargent (2007, chapter 2 16) introduce specification errors to Eq. (29) and write the distorted model as ztþ1 ¼ Azt þ But þ Cvtþ1 þ Cextþ1 ,

(30)

where the specification errors are, of course, constrained to satisfy Eq. (7). Conditional on C2 , the dynamic program for this robust control problem is e ut þ e e ut þ bE t ðz0 Vztþ1 þ dÞ, z0t Vzt þ d  min max½z0t Wzt þ 2z0t Ue u0t Re tþ1 ut

vtþ1

(31)

subject to ztþ1 ¼ Azt þ But þ Cvtþ1 þ Cextþ1 .

(32)

To obtain the worst-case equilibrium, the approach is to conjecture a solution for C2 and to use the method described in Appendix A to solve the dynamic programming problem. The conjecture for C2 is then updated according to 10 C2 V1 22 V21 C1 , see Eq. (A11), and the procedure is iterated until convergence. How does this formulation of the robust Stackelberg problem differ from the one described in Section 3? The difference lies in how the specification errors distort the economy. In the formulation described in Section 3, the Stackelberg leader fears specification errors that reside in the predetermined block of the model (Eq. (1). In contrast, in this section the Stackelberg leader also fears distortions to private-agent expectations, i.e., the Stackelberg leader fears that the expectations that private agents form will be slanted in a direction that makes its stabilization problem more difficult. Put differently, here the Stackelberg leader fears that the followers will use the distorted model to form expectations and formulates policy accordingly, whereas in Section 3 the Stackelberg leader believes that the followers will use the 10

Although it is not obvious that this procedure will necessarily converge, I did not encounter any difficulties with it.

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approximating model to form expectations and formulates policy accordingly. Of course, because the solution to this alternative problem involves repeated application of the solution method used in Section 4, the worst-case equilibrium still takes the form of Eqs. (13)–(15), with the worst-case specification errors distorting the conditional mean of the shock processes. Lemma 1. Conditional on the Stackelberg leader’s robust decision rule, the worst-case law of motion for the shadow prices of the non-predetermined variables does not depend on the worst-case distortions. Proof. The shadow prices of the non-predetermined variables are given by ptþ1 ¼ ½V21 V22 ztþ1 ,

ð33Þ

¼ ½V21 V22 ðAzt þ But þ Cvtþ1 þ Cextþ1 Þ.

ð34Þ

However, the coefficient on the specification errors in this law of motion is given by " # C1 ½V21 V22 C ¼ ½V21 V22  ¼ 0: & (35) V1 22 V21 C1 An immediate implication of Lemma 1 is that the law of motion for the shadow prices of the non-predetermined variables is unaffected by whether the approximating model is misspecified or not, the essential property emphasized in this paper. It follows that the procedures described in Sections 5 and 6 will deliver the same approximating equilibrium. Lemma 2. The worst-case distortions do not depend on the non-predetermined variables. As a consequence, (i) the Stackelberg leader cannot use promises about future policy to influence the worst-case distortions, and (ii) the worst-case distortions are unaffected by whether the fictitious evil agent makes decisions once-and-for-all or sequentially. Proof. The easiest way to see these important features of the solution is to reformulate the robust Stackelberg problem in terms of a Lagrangian and to exploit the relationship between the two solution methods. The Lagrangian for the robust Stackelberg problem is L ¼ E0 þ

1 X

bt ½z0t Wzt þ 2z0t Uut þ u0t Rut  byv0tþ1 vtþ1

t¼0 l0tþ1 ðAzt

þ But þ Cvtþ1 þ Cextþ1  ztþ1 Þ,

fut g1 0

ð36Þ

fvtþ1 g1 0

is chosen to minimize Eq. (36) and is chosen to maximize where 0 0 Eq. (36) and where ltþ1  ½lxtþ1 lytþ1 0 is a vector of Lagrange multipliers. Focusing on the maximization problem, the first-order condition associated with vtþ1 is11 qL 0 ¼ 2byvtþ1 þ C Et ltþ1 ¼ 0. qvtþ1 11

(37)

Since the constraints are linear and y40, this first-order condition is associated with a maximum.

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As is well known, the Lagrange multipliers in Eq. (36) are equivalent to the shadow prices on zt in the dynamic program, Eqs. (31) and (32).. Therefore,12 " x # " #" # ltþ1 xtþ1 V11 V12 ¼2 . (38) lytþ1 ytþ1 V21 V22 Now, substituting Eq. (38) into Eq. (37), this first-order condition can be written as " # " # xtþ1 V11 V12 qL 0 ¼  byvtþ1 þ C Et ¼ 0, ð39Þ qvtþ1 ytþ1 V21 V22 ¼  byvtþ1 þ ðC01 V11 þ C02 V21 ÞE t xtþ1 þ ðC01 V12 þ C02 V22 ÞE t ytþ1 ¼ 0. ð40Þ The next step is to recognize that C2 ¼ V1 22 V21 C1 in equilibrium. Therefore, Eq. (40) becomes qL 0 0 ¼ byvtþ1 þ C01 ðV11  V021 V1 22 V21 ÞE t xtþ1 þ C1 ðV12  V21 ÞE t ytþ1 ¼ 0, qvtþ1 (41) which, because V is symmetric (see Eq. (A7)), collapses to qL ¼ byvtþ1 þ C01 ðV11  V021 V1 22 V21 ÞE t xtþ1 ¼ 0. qvtþ1

(42)

If I make vtþ1 the subject, Eq. (42) implies vtþ1 ¼

1 0 C ðV11  V021 V1 22 V21 ÞE t xtþ1 , by 1

(43)

establishing that the worst-case distortions do not depend on E t ytþ1 and therefore do not depend on E t lytþ1 (E t ptþ1 ). If this last point is unclear, simply note that the state variables are given by " # " #" # xt xt I 0 ¼ , (44) lyt yt V21 V22 and that Eq. (43) can therefore also be written as " I 1 0 0 1 ½C1 ðV11  V21 V22 V21 Þ 0 vtþ1 ¼ 1 V22 V21 by

0 V1 22

#"

E t xtþ1 E t lytþ1

# :

&

(45)

Lemma 2 has several important implications. First, Eq. (43) establishes that when determining how to distort the model the fictitious evil agent need only consider the expected future natural state variables. Second, Eq. (42) shows that when distorting the model the fictitious evil agent equates the marginal cost of a change in its decision variables to the marginal benefit, where the marginal 12 Note that the ‘2’ in Eq. (38) is simply a scale factor that is sometimes accommodated by placing a ‘2’ in front of the Lagrange multipliers in Eq. (36) (which makes the Lagrange multipliers equal to half the shadow prices). Naturally, both approaches yield the same answer.

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benefit comes in two forms: a direct effect on the loss function of a change in E t xtþ1 and an indirect effect through how the change in this expectation alters the non-predetermined variables. Third, because its first-order condition holds for all tX0 and does not depend on the non-predetermined variables, the fictitious evil agent is not subject to a time-consistency problem. Therefore, the solution to the problem in which the Stackelberg leader and the fictitious evil agent both make decisions once-and-for-all is also the solution to the problem in which the Stackelberg leader makes decisions once-and-for-all, but the fictitious evil agent makes decisions sequentially.

8. Robust monetary policy In this section I employ the sticky price/sticky wage business cycle model developed by Erceg et al. (2000) to analyze robust monetary policy. I characterize how equilibrium outcomes differ depending on whether the robust Stackelberg problem is formulated according to Section 3 or Section 7 and I examine the extent to which a preference for robustness drives a wedge between the approximating equilibrium and the rational expectations (nonrobust) equilibrium. 8.1. The model Complete details of the model can be found in Erceg et al. (2000). Briefly, however, the production technology is linear in labor, and prices and wages are each subject to a Calvo-style nominal rigidity. In addition to a consumption preference shock, qt , the model contains a leisure preference shock, zt , and a serially correlated technology shock, xt . When log-linearized around its flex-price equilibrium, the model can be written as gt ¼ E t gtþ1 

1 ðit  E t ptþ1  rnt Þ sl c

(46)

pt ¼ bE t ptþ1 þ kr ðwt  mpl t Þ

(47)

wt ¼ bE t wtþ1 þ kw ðmrst  wt Þ

(48)

mpl t ¼ wnt  lmpl gt

(49)

mrst ¼ wnt þ lmrs gt

(50)

wt ¼ wt1 þ wt  pt

(51)

rnt ¼ sl c ðE t yntþ1  ynt Þ þ sl q ðE t qtþ1  qt Þ

(52)

wnt ¼ wnx xt þ wnq qt þ wnz zt

(53)

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ynt ¼ ynx xt þ ynq qt þ ynz zt

(54)

xt ¼ rxt1 þ t ,

(55)

where gt denotes the output gap, it denotes the short-term nominal interest rate, pt denotes price inflation, wt denotes the real wage, mpl t denotes the marginal product of labor, wt denotes wage inflation, mrst denotes the marginal rate of substitution between consumption and leisure, and rnt , wnt , and ynt denote the Pareto-optimal real interest rate, real wage, and level of output, respectively.13 The central bank’s objective is to set the nominal interest rate sequence, fit g1 0 , to maximize social welfare, where social welfare is represented by a second-order approximation (around the flex-price equilibrium) of the representative household’s utility function. Erceg et al. (2000) show that in this second-order approximation e , from its Pareto-optimal level, W n , is the expected deviation of social welfare, W equivalent to e Þ / ðlmrs þ lmpl Þ Varðgt Þ þ ð1 þ yp Þ ð1  bxp Þ 1 Varðpt Þ EðW n  W 2 2yp ð1  xp Þ kp ð1 þ yw Þ ð1  bxw Þ ð1  aÞ þ Varðwt Þ. 2yw ð1  xw Þ kw

ð56Þ

Because social welfare cannot exceed the Pareto-optimal level, the central bank’s optimization problem is to minimize Eq. (56) subject to Eq. (7) and Eqs. (46)–(55). According to the model’s underlying parameterization (see footnote 13), the weight on wage stabilization is 55:68, the weight on price stabilization is 24:00, and the weight on output stabilization is (approximately) 1:46. As a consequence, the welfare function assigns greater importance to wage stabilization than it does to price stabilization, and it assigns relatively little importance to output stabilization. 8.2. Simulation results I begin by formulating the robust Stackelberg problem according to Section 3. Writing the model in state-space form, it is straightforward to solve for the rational expectations equilibrium. After adding the specification errors (the controls of the fictitious evil agent), the worst-case equilibrium and the approximating equilibria are also straightforward to obtain. To determine the robustness parameter, y, I set the detection-error probability to 0:1, which implies y ¼ 4:75 (so that logðyÞ ¼ 1:558). With this parameterization of the robustness parameter, the specification errors that the central bank fears are vxtþ1 ¼ 0:107xt þ 0:001zt  0:270wt1  0:006pwt  0:001pgt þ 0:016ppt ,

(57)

13

I parameterize the model according to Erceg et al. (2000), setting s ¼ w ¼ 1:5, a ¼ 0:3, yw ¼ yp ¼ 13,

Q C zw ¼ zp ¼ 0:75, r ¼ 0:95, b ¼ 0:99, C ¼ 3:163, Q ¼ 0:3163, N ¼ 0:27, Z ¼ 0:03, l c ¼ CQ , l q ¼ CQ , N l n ¼ 1NZ ,

kw ¼

Z l z ¼ 1NZ ,

ð1zw Þð1bzw Þ , w ÞÞz ð1þwl n ð1þy w y w

L ¼ a þ wl n þ ð1  aÞsl c ,

n c wnx ¼ wl n þal L , wq ¼ 

asl q L ,

a lmpl ¼ 1a ,

wl n lmrs ¼ sl c þ 1a ,

1þwl n n n z wnz ¼ awl L , yx ¼ L , yq ¼

ð1aÞsl q , L

kp ¼

z ynz ¼ ð1aÞwl . L

ð1zp Þð1bzp Þ , zp

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vqtþ1 ¼ 0:001xt þ 0:002wt1 ,

(58)

vztþ1 ¼ 0:003xt  0:006wt1 þ 0:001ppt .

(59)

Clearly, the central bank is most concerned with specification errors that distort the technology shock, Eq. (57). In this model, because both prices and wages are subject to some rigidity, technology shocks cannot be perfectly accommodated, and the central bank faces a trade-off between stabilizing prices and wages and stabilizing output. In this respect, from a stabilization perspective, the technology shock in this model behaves similarly to adverse markup shocks in the canonical new Keynesian model (Clarida et al., 1999). Where technology shocks are costly because they force this trade-off, persistent technology shocks are even more damaging, requiring larger movements in interest rates that have larger effects on output. Interestingly, Eq. (57) implies that the autoregressive coefficient in the worst-case shock process for technology is greater than one. However, stability of the worst-case equilibrium is preserved through feedback among the worstcase shock processes, the real wage, and the shadow prices. Relative to the technology Inflation, 1% Technology Shock 0.02

0.40

−0.00 −0.02

0.30

Percent

Probability

Detection–Error Probability 0.50

0.20

−0.04 −0.06

0.10

RE Worst Approx. Eqm.

−0.08 −0.10

0.00

0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

2

4

6

8

10 12 14 16 18 20

log (θ)

Time

Output, 1% Technology Shock

Interest Rate, 1% Technology Shock

0.009 −0.02

Percentage points

Percent

0.007 0.005 0.003 0.001

−0.05 −0.08 −0.11 −0.14

−0.001 0

2

4

6

8

10 12 14 16 18 20

Time

0

2

4

6

8

10 12 14 16 18 20

Time

Fig. 1. Robust policy in the sticky price/sticky wage model (Section 3): (A) detection-error probability, (B) inflation, 1% technology shock, (C) output, 1% technology shock, and (D) interest rate, % technology shock.

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shock, the central bank is largely unconcerned about distortions to either the leisure shock or the consumption preference shock. Fig. 1 (Panel A) plots the relationship between (log) y and the probability of making a ‘detection error,’ given a sample of 200 observations. A detection-error probability is the probability that an econometrician observing equilibrium outcomes would infer incorrectly whether the approximating equilibrium or the worst-case equilibrium generated the data. Among other factors, detection-error probabilities depend on the sample size. A method for calculating detection-error probabilities is presented in Appendix B.14 As y increases (Z falls), the distortions become smaller and the probability of making a detection error rises. In fact, in the limit as y " 1 (Z # 0), the central bank’s rising confidence in its model means that the specification errors are so severely constrained that the approximating equilibrium and the worst-case equilibrium each converge to the rational expectations equilibrium, and the detection-error probability converges to 0:5. With y ¼ 4:75, Fig. 1 (Panels B–D) also displays the responses of inflation, output, and the interest rate to a positive 1% technology shock for the rational expectations equilibrium, the worst-case equilibrium, and the approximating equilibrium. Focusing first on the rational expectations equilibrium, Fig. 1 reveals that inflation and the interest rate fall while the output gap rises in response to a 1% technological innovation. The positive technology shock lowers the cost of producing a unit of output (real marginal costs), which, through competitive forces and a relatively large weight on wage stabilization in the policy objective function, causes prices and inflation (due to sticky prices) to fall. With inflation falling, the central bank cuts interest rates, stimulating demand and opening up a positive output gap. As the effects of the shock begin to dissipate, and as the demand stimulus exerts upward pressure on inflation, both inflation and interest rates begin to rise, causing the output gap to fall. The robust policy in the approximating equilibrium initially cuts interest rates by more than the rational expectations policy to guard against a sustained and persistent decline in inflation, but then raises interest rates quickly as inflationary pressures begin to build. The central bank’s fear of misspecification reveals itself most prominently in how the central bank responds initially to the shock. In terms of Fig. 1, the greatest differences between the rational expectations equilibrium and the approximating equilibrium can be found in the behavior of output, the variable that receives the least weight in the policy objective function. Formulating the robust Stackelberg problem according to Section 7, I set the detection-error probability to 0:1, which implies y ¼ 3:077 (logðyÞ ¼ 1:124), and with this parameterization the specification errors that the central bank fears are15 vxtþ1 ¼ 0:122xt þ 0:001qt  0:003zt  0:319wt1  0:005pgt ,

14

(60)

See also Hansen et al. (2002), Dennis et al. (2006a), or Hansen and Sargent (2007, chapter 9). Although the central bank’s promises about future policy have no direct effect on the evil agent, through their influence on households and firms, the shadow prices still enter the worst-case distortions. 15

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vqtþ1 ¼ 0:001xt þ 0:001wt1 ,

(61)

vztþ1 ¼ 0:002xt  0:004wt1 .

(62)

These specification errors are qualitatively similar to those in Eqs. (57)–(59) insomuch as the central bank primarily fears distortions to the technology shock, Eq. (60), and is less concerned about distortions to the leisure shock and the consumption preference shock. Eq. (60) is also qualitatively similar to Eq. (57) in that the worst-case distortion serves to make the technology shock more persistent and (unconditionally) correlated with the real wage. Fig. 2 (Panel A) maps out the relationship between the robustness parameter, y, and the probability of making a detection error. Panels B–D display the responses to a positive 1% technology shock in the rational expectations equilibrium, the approximating equilibrium, and the worst-case equilibrium. It is clear from Fig. 2 that robustness primarily affects the behavior of output. In this model, output is a non-predetermined variable whose behavior is governed Inflation, 1% Technology Shock 0.02

0.40

−0.00

0.30

Percent

Probability

Detection–Error Probability 0.50

0.20

−0.02 −0.04 RE Worst Approx. Eqm.

−0.06 0.10

−0.08

0.00 0

1

2

3

4

5

−0.10

6

0

2

4

6

8

10 12 14 16 18 20

log (θ)

Time

Output, 1% Technology Shock

Interest Rate, 1% Technology Shock

0.009 −0.02

Percentage points

Percent

0.007 0.005 0.003 0.001 −0.001

−0.05 −0.08 −0.11 −0.14

0

2

4

6

8

10 12 14 16 18 20

Time

0

2

4

6

8

10 12 14 16 18 20

Time

Fig. 2. Robust policy in the sticky price/sticky wage model (Section 7): (A) detection-error probability, (B) inflation, 1% technology shock, (C) output, 1% technology shock, and (D) interest rate, % technology shock.

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partly by central bank promises about future policy. Indeed, comparing the behavior of the robust and the nonrobust economies, shows that the main differences reside in the laws of motion for the shadow prices (not shown), particularly those of wage growth and inflation. Although robustness has little effect on the central bank’s interest rate response (Panel D) or on the behavior of inflation (Panel B), the central bank’s desire for robustness is manifest in the promises it makes about future policy, and these promises assert themselves in the behavior of output (Panel C). In this respect, Figs. 1 and 2 are qualitatively similar.

9. Conclusion In this paper I analyze decisionmaking in economies where a Stackelberg leader formulates policy under commitment while seeking robustness to model misspecification. My first contribution is to show that the Hansen and Sargent (2003) solution to this class of robust control problems is generally incorrect. As is by now well known, shadow prices enter the solution of control problems in which the leader commits. These shadow prices encode the history dependence of the optimal policy and direct private agents toward the equilibrium the Stackelberg leader desires. I show that the approximating equilibrium that Hansen and Sargent (2003) develop is generally incorrect because their method for obtaining it fails to recognize the role these shadow prices play in determining equilibrium. I then show how the approximating equilibrium can be constructed correctly, establishing as a general principle that the law of motion for the shadow prices in the approximating equilibrium should be the same as it is in the worst-case equilibrium. Next I consider the treatment of robust Stackelberg problems given in Hansen and Sargent (2007), which, due to the issues raised in this paper, differs from Hansen and Sargent (2003). I show that in the solution to this alternative formulation of the robust Stackelberg problem, conditional on the leader’s robust decision rule, the law of motion for the shadow prices is unaffected by the worst-case distortions. As a consequence, the method Hansen and Sargent (2007) use to construct the approximating equilibrium coincides with the method developed in this paper because it satisfies the principle that the law of motion for the shadow prices must be the same in the approximating equilibrium as it is in the worst-case equilibrium. In addition, I show that the first-order condition determining the worstcase distortions does not depend on the non-predetermined variables, implying that the worst-case distortions are unaffected by promises about future policy and are not subject to a time-consistency problem. This result also implies that the solution to the problem in which the Stackelberg leader and the fictitious evil agent both make decisions once-and-for-all is also the solution to the problem in which the Stackelberg leader makes decisions once-and-for-all, but the fictitious evil agent makes decisions sequentially. Having studied two formulations of the robust Stackelberg problem and shown how to construct the approximating equilibrium, I use a sticky wage/sticky price new Keynesian business cycle model to analyze robust monetary policy. An important

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result that emerges from this analysis is that a robust central bank should fear that its approximating model understates the persistence of technology shocks, which are difficult for the central bank to counteract. With the robust Stackelberg problem formulated according to Hansen and Sargent (2007), I find that the robust policy relies heavily on promises about future policy, with the central bank attempting to manipulate private-agent expectations in an effort to counteract the specification errors that it fears. Throughout this paper I have focused on robust Stackelberg games, games in which the Stackelberg leader commits to its policy. An alternative would be to have the leader conduct policy under discretion while seeking robustness. Giordani and So¨derlind (2004) consider this possibility, showing how the time-consistent solution procedure developed by Backus and Driffill (1986) can be applied to this setting. According to the Giordani and So¨derlind (2004) formulation, the leader revisits its optimization problem period-by-period, and each time it does so it takes into account its fear that its model may be misspecified. In other words, the evil agent optimizes only when the leader does and the robust policy is identified by imposing Markov perfection on the worst case equilibrium. An alternative approach might have the evil agent commit to its worst case distortions, but have the leader conduct policy period-by-period. For this alternative formulation, the relationship between the evil agent and the leader might more naturally be described in terms of a leadership game. I leave these important issues for future work.

Acknowledgments I would like to thank the editor and two anonymous referees for comments. I would also like to thank Tom Sargent for discussions on robust control, Eric Swanson and Carl Walsh for comments on an earlier draft, and Anita Todd for editorial suggestions. The views expressed in this paper do not necessarily reflect those of the Federal Reserve Bank of San Francisco or the Federal Reserve System.

Appendix A. Solving for the worst-case equilibrium With the optimization problem written as min max E0 fut g fvtþ1 g

1 X t¼0

e ut þ e e ut , bt ½z0t Wzt þ 2z0t Ue u0t Re

(A.1)

subject to e tþ1 , e ut þ Ce ztþ1 ¼ Azt þ Be the policy problem conforms to the standard linear-quadratic framework studied and solved by Backus and Driffill (1986). To obtain the optimal policy, Backus and Driffill (1986) first take z0 as given and write the optimization problem recursively as

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e ut þ e e ut þ bE t ðz0 Vztþ1 þ dÞ, z0t Vzt þ d  min max½z0t Wzt þ 2z0t Ue u0t Re tþ1 ut

vtþ1

2081

(A.2)

subject to e tþ1 . e ut þ Ce ztþ1 ¼ Azt þ Be

(A.3)

Using dynamic programming they obtain the standard solution e tþ1 e t þ Ce ztþ1 ¼ ðA  BFÞz

(A.4)

e ut ¼ Fzt ,

(A.5)

where e 0 VB e þ RÞ e 1 ðbB e 0 VA þ U e 0 Þ, F ¼ ðbB

(A.6) 0

e þ W  UF e  F0 U e 0 VðA  BFÞ e þ F0 RF, e V ¼ bðA  BFÞ

(A.7)

0

e VCEðe e tþ1 e0 Þ btr½C tþ1 . d¼ 1b

(A.8)

Since y0 is not predetermined and the expectation errors are not exogenous, the Stackelberg leader also chooses y0 and feyt g1 1 to minimize the value function " #" # x0 V11 V12 z00 Vz0 þ d ¼ ½x00 y00  þ d, (A.9) y0 V21 V22 obtaining y0 ¼ V1 22 V21 x0 ,

(A.10)

eyt ¼ V1 22 V21 C1 ext .

(A.11)

In the equilibrium that the leader desires, in the initial period the nonpredetermined variables relate to the predetermined variables according to Eq. (A.10), and in each period the expectational errors relate to the innovations according to Eq. (A.11). To achieve this equilibrium, the leader represents its policy as a function of a vector of shadow prices of the non-predetermined variables with the shadow prices inducing private agents to form expectations that are consistent with the desired equilibrium. Thus, introducing a vector of shadow prices of the nonpredetermined variables " # xt pt  ½V21 V22  (A.12) yt

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and noting that pt is predetermined (implied by Eq. (A.11) with initial value p0 ¼ 0 (implied by Eq. (A.10), the dynamics of xt and pt are given by " # " # xtþ1 xt 1 e tþ1 , e ðA:13Þ ¼ TðA  BFÞT þ TCe ptþ1 pt " # " # xt C1 1 e ¼ TðA  BFÞT ðA:14Þ þ extþ1 , pt 0 where " T¼

# 0 . V22

I V21

ut are determined according to Finally, yt and e " # xt 1 1 yt ¼ ½V22 V21 V22  pt

(A.15)

(A.16)

and " e ut ¼ FT

1

xt pt

# ,

(A.17)

respectively. Eqs. (A.14), (A.16), and (A.17) describe the state-contingent behavior of the predetermined variables, xt and pt , the non-predetermined variables, yt , and the decision variables, e ut , in the unique stable worst-case equilibrium in which the leader sets policy with commitment.

Appendix B. Detection-error probability Let A denote the approximating model and B denote the worst-case model; then, assigning equal prior weight to each model and assuming that model selection is based on the likelihood ratio principle, Hansen et al. (2002) show that detectionerror probabilities are calculated according to probðAjBÞ þ probðBjAÞ , (B.1) 2 where probðAjBÞ (probðBjAÞ) represents the probability that the econometrician erroneously chooses A (B) when B (A) generated the data. Let fzBt gT1 denote a finite sequence of economic outcomes (the shocks, the shadow prices, the endogenous variables, and the followers’ and leader’s decision variables) generated by the worst-case equilibrium, and let LAB and LBB denote the likelihood associated with models A and B, respectively; then the econometrician chooses A over B if logðLBB =LAB Þo0. Generating M independent sequences fzBt gT1 , probðAjBÞ pðyÞ ¼

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can be calculated according to   m   M 1 X LBB I log m probðAjBÞ  o0 , M m¼1 LAB

2083

(B.2)

m where I½logðLm BB =LAB Þo0 is the indicator function that equals one when its argument is satisfied and equals zero otherwise; probðBjAÞ is calculated analogously using data generated from the approximating model. Let

ztþ1 ¼ HA zt þ Gtþ1 ,

(B.3)

ztþ1 ¼ HB zt þ Gtþ1

(B.4)

govern equilibrium outcomes under the approximating equilibrium and the worstcase equilibrium, respectively. Using the Moore-Penrose inverse, ijj betþ1 ¼ ðG0 GÞ1 G0 ðzjtþ1  Hi zjt Þ;

fi; jg 2 fA; Bg

(B.5)

are the inferred innovations in period t þ 1 when model i is fitted to data fzjt gT1 ijj b generated from model j, and let S be the associated estimates of the innovation variance-covariance matrices. Note that the Moore–Penrose inverse picks out the shock process from among the variables in zt . Assuming that the innovations are normally distributed, it is easy to show that   LAA 1 b BjA b AjA log  R Þ, (B.6) ¼ trðR 2 LBA 

LBB log LAB

 ¼

1 b AjB b BjB trðR  R Þ. 2

(B.7)

Given Eqs. (B.6) and (B.7), Eq. (B.2) is used to estimate probðAjBÞ and (similarly) probðBjAÞ, which are needed to construct the detection-error probability, as per Eq. (B.1). The robustness parameter, y, is then determined by selecting a detectionerror probability and inverting Eq. (B.1). This inversion can be performed numerically by constructing the mapping between y and the detection-error probability for a given sample size. Note that any sequence of specification errors that satisfies Eq. (7) will be at least as difficult to distinguish from the approximating model as is a sequence that satisfies Eq. (7) with equality; as such, pðyÞ represents a lower bound on the probability of making a detection error. References Backus, D., Driffill, J., 1986. The consistency of optimal policy in stochastic rational expectations models. Centre for Economic Policy Research Discussion Paper #124. Cagetti, M., Hansen, L., Sargent, T., Williams, N., 2002. Robustness and pricing with uncertain growth. The Review of Financial Studies 15 (2), 363–404. Clarida, R., Galı´ , J., Gertler, M., 1999. The science of monetary policy: a New Keynesian perspective. Journal of Economic Literature 37 (4), 1661–1707.

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