Robust Monetary Policy with Misspecified Models: Does Model Uncertainty Always Call for Attenuated Policy?

Robust Monetary Policy with Misspecified Models: Does Model Uncertainty Always Call for Attenuated Policy? Robert J. Tetlow and Peter von zur Muehlen (cid:3) Abstract ThispaperexploresKnightianmodeluncertainty asapossibleexplanation oftheconsiderable difference between estimated interest rate rules and optimal feedback descriptions of monetary policy. We focus on two types of uncertainty: (i) unstructured model uncertainty reflected in additive shock error processes that result from omitted-variable misspecifications, and (ii) structured model uncertainty, where one or more parameters are identified as the source of misspecification. For an estimated forward-looking model of the U.S. economy, we find that rules that are robust against uncertainty, the nature of whichisunspecifiable, oragainstone-timeparametricshifts,aremoreaggressivethanthe optimal linear quadratic rule. However, policies designed to protect the economy against theworst-case consequences ofmisspecified dynamics areless aggressive and turn outto begoodapproximationsoftheestimatedrule. Apossibledrawbackofsuchpoliciesisthat the losses incurred from protecting against worst-case scenarios are concentrated among thesamebusiness cyclefrequencies thatnormallyoccupy theattention ofpolicymakers. JEL Classification: C6, E5 Keywords: modeluncertainty,robustcontrol,monetarypolicy,Stackelberg games. (cid:3) Federal Reserve Board, Washington, DC 20551. Tetlow: rtetlow@frb.govor by mail to stop 61; von zur Muehlen:pmuehlen@frb.govorbymailtostop76. Forthcoming,theJournalofEconomicDynamics & Control. WethankDavidReifschneiderandthreeanonymousrefereesforusefulguidanceandespeciallyAlexeiOnatskifor generouslyprovidingtwoprototypicalMatlabprogramsandsubsequenthelpinunderstandingthemethodology. Thanksare also due to FredericoS. Finan for his usual, excellentresearch assistance. All remainingerrorsare ours. Any opinions expressed herein do not necessarily reflect those of the staff or members of the Board of Governors.

1 Introduction Recent articles have uncovered a puzzle in monetary policy: Interest-rate reaction functions derived from solving optimization problems call for much more aggressive responsiveness of policyinstrumentstooutputandinflationthandorulesestimatedwithUSdata.1 Whatexplains theobservedlack ofaggressiveness–theattenuation–ofpolicy? Three distinct arguments have been advanced to explain the observed reluctance to act aggressively. Thefirstisthatitissimplyamatteroftaste: Policyisslowand adjustssmoothly in response to shocks because central bankers prefer it that way, either as an inherent taste or as a device to avoid public scrutiny and criticism (see, e.g., Drazen (2000), Chapter 10). The second argues that partial adjustment in interest rates aids policy by exploiting private agents’ expectations of future short-term rates to move long-term interest rates in a way that is conduciveto monetary control (see, e.g., Goodfriend (1991), Woodford (1999), Tetlow and vonzurMuehlen(2000)). Thethirdcontentionisthatattenuatedpolicyistheoptimalresponseofpolicymakersfacing uncertaintyinmodelparameters,inthenatureofstochasticdisturbances,inthedatathemselves given statistical revisions, and in the measurement of latent state variables such as potential output,theNAIRU,andthesteady-staterealinterestrate. Blinder(1998),EstrellaandMishkin (1998), Orphanides (1998), Rudebusch (1998), Sack (1998a), Smets (1999), Orphanides et.al. (2000), Sack and Wieland (2000), Wieland (1998), and Tetlow (2000) all support this general argument, followingthe lineof research that began with Brainard (1967).2. The present paper is concerned with this third explanation for policy attentuation. There is no unanamity on this third line of argument, however. Chow (1975) and Craine (1979) demonstrated long ago that uncertainty can lead to the opposite result of more aggressive policy than in the certainty equivalencecase—orwhat wemightdubasanti-attenuation. So¨derstro¨m(1999a)providesan empirical example of such a case. Moreover, possible deficiencies in the Brainard-style story are hinted at in the range of uncertainties required in papers by Sack (1998a) and Rudebusch (1998) to come even close to explaining observed policy behavior.3 Lastly, time-variation in uncertainty can, in some circumstances, lead to anti-attentuation of policy as shown by Mercado and Kendrick(1999). The concept ofmodel uncertainty underlyingthepapers cited aboveis Bayesian in nature: 1ThelistofpapersincludesRudebusch(1998),Sack(1998b),So¨derstro¨m(1999b),Tetlow,vonzurMuehlen, andFinan(1999)andTetlowandvonzurMuehlen(2000). 2OtherimportantearlyreferencesincludeAoki(1967),Johansen(1973),Johansen(1978),andCraine(1979) 3Rudebusch(1998)findsthatdatauncertaintyhasonlyaslightattenuatingeffectonoptimalpolicy.Similarly, Onatski and Stock (2000) find that data uncertainty has a minimal effect on reactions formed when monetary policyisrobustinthesenseofthetermdescribedbelow. 2

A researcher faces a well-defined range of possibilities for the true economy over which he or she must formulate a probability distribution function (see Easley and Kiefer (1988)). All of the usual laws from probability theory can be brought to bear on such questions. These are problems of risk, and risks can be priced. More realistically, however, central bankers see themselves as facing far more profound uncertainties. They seem to view the world as so complex and time varying that the assignment of probability distributions to parameters or models is impossible. Perhaps for this reason, no central bank is currently committed to a policyrule(otherthananexchangeratepeg). Inacknowledgmentofthisview,thispaperarises from the conception of uncertainty, in the sense of Knight, wherein probability distributions forparameters ormodelscannotbearticulated. We consider two approaches to model uncertainty that differ in the nature of the specification errors envisioned and in the robustness criterion applied to the problem. One approach treats errors as manifested in arbitrarily serially correlated shock processes, in addition to the model’s normal stochastic disturbances. This formulation, called unstructured model uncertainty, followsin the tradition of Caravani and Papavassilopoulos(1990) and Hansen, Sargent and Tallarini (1999), among others. A second approach puts specific structure on misspecification errors in selected parameters in a model. It is possible, for example, to analyze the effect on policy of the worst possible one-time shift in one or more parameter. Alternatively, misspecification in model lag structures could be examined. Such unmodeled dynamics, will affect robustpolicy. Theseminaleconomicspaperin thisarea ofstructuredmodeluncertainty isOnatskiand Stock (2000). The inability to characterize risk in probability terms compels the monetary authority to protectlossesagainstworst-caseoutcomes,toplayamentalgameagainstnature,asitwere. In thecaseofunstructureduncertainty,thesolutiontothegameisan H 1 problemor,inarelated special case, a problem that minimizes absolute deviations of targets. In the case of structured uncertainty, the monetary authority ends up choosing a reaction function that minimizes the chance of model instability. In both cases, the authority adopts a bounded “worst-case” strategy, planning against nature’s “conspiring” to produce the most disadvantageous parameterizationofthetruemodel. With the exception of Hansen and Sargent (1999b), Kasa (2000), and Giannoni (2000), robust decision theory has been applied solely to backward looking models. Hansen and Sargent(1999b)andKasa(2000)derivepoliciesundertheassumptionofunstructureduncertainty, while Giannoni (2000) solves a problem with structured uncertainty, wherein policies are derived subject to uncertainty bounds on selected parameters of the model. In this paper, we break new ground in that we consider a number of cases of unstructured as well as structured 3

uncertainty,doingsoforanestimatedforward-lookingmodel,andwithaparticularreal-world policy issue in mind. Also, unlike Hansen and Sargent (1999b) and Kasa (2000), but like Giannoni (2000) and Onatski and Stock (2000), we derive robust simple policy rules, similar in form to the well-known Taylor (1993) rule. Our analysis differs from Giannoni (2000) in that it is less parametric, relies on numerical techniques, and is amenable to treatment of larger modelsandcoversunstructuredas well as structureduncertainty. Therestofthispaperunfoldsasfollows. Insection2,weintroducestructuredandunstructured perturbations as a way of modeling specification errors to a reference model considered to be the authority’s best approximation to true but unknown model. We define a number of Stackelberg games that differ according to the central bank’s assessment of the bounds on uncertaintyanditslossfunction. Toanalyzethespecificquestionsathand,inthethirdsection,we estimate a small forward-looking macro model with Keynesian features. The model is a form ofcontractingmodel,in thespiritofTaylor(1980)andCalvo (1983),and isbroadlysimilarto that of Fuhrer and Moore (1995a). Section 4 provides our results leaving Section 5 to sum up and conclude. Topresagetheresults,althoughweareabletoproducearobustrulethatisnearlyidentical to the estimated rule, it is not clear how much of the issue this resolves. Robustness, per se, cannotexplainattenuatedpolicy. Asothershavefound,whenpolicyisrobustagainstacombinationofshockandmisspecificationerrors, monetarypolicybecomesevenmorereactivethan the linear quadratic optimal rule. Indeed, we observe a seeming inverse relationship between reactiveness and the degreeof structureimposed on uncertainty. At one extreme, unstructured uncertainty justifies the most reactive set of rules. At the other extreme, heavily attenuated policiesaregenerated incases withalotofstructureonmodeluncertainty. Inparticular, ifthe monetary authority chooses a policy that is robust only to misspecificationof thelag structure ofthemodel,theoptimalinterestratereactionrulebecomesverysimilartotheestimatedrule. When the only criterion is robustness to misspecifications of the lagged output coefficients in the aggregate demand equation, the robust rule and the estimated rule are practically identical. It is tempting to infer from this result that Federal Reserve policy in the last twenty years has been influenced bya specialconcern aboutill-understoodfutureeffects ofcurrent actions. However,oursisnotsufficientevidencetoestablishthattheUSmonetarypolicyauthorityhas orhas notinfact been a robustdecisionmaker. 4

2 Model Uncertainty in the Sense of Knight Few observers of the U.S. macroeconomic scene in 1995 would have forecast the happy coincidence of very strong output growth coupled with low inflation that was observed over the second half of the decade. Such forecast errors tend to bring out the humble side of model builders and forecasters alike and reinforce the lesson that considerable uncertainty surrounds our understanding of the real world, a fact that should be taken account of in planning. In the presence of such pervasive ambiguity, the notion of Knightian uncertainty has an obvious appeal. Whateverthecircumstances,it is easy to imaginethat thebest guessof thetruemodel a policymaker can bring to the issue of monetary control is flawed in a serious but unspecifiableway. Weconsideranapproach,foundedonrecentdevelopmentsinrobustdecisiontheory, in which the authority contemplates approximation errors of unspecified nature but of a size contained within a bounded neighborhood of its reference model. As will become apparent shortly, this approach has the advantage of fitting into a linear quadratic framework, but without its usual association with certainty equivalence. The absence of a probability distribution forpossiblemisspecificationsleads,inturn,toadesiretominimizeworst-caseoutcomes. Previousresearch has shownthatthiscan beusefullyformulatedasatwo-persongameplayed,in thisinstance,between themonetaryauthorityand amalevolentnature.4 Itiseasytoconceptualizehowmodelmisspecificationcanberepresentedasagame. Imagine amonetary authorityattemptingto control a misspecified model. As it does, themisspecification will manifest itself, out of sample, as residuals whose time-series pattern differs from those derived by estimating the model. The nature of these ex post residuals will depend on thenatureofthemisspecification,theboundsonitssize, andthespecifics ofthefeedback rule employed by the authority. However, the uncertainty involved—uncertainty in the sense of Knight—isnotamenabletostandardoptimizationtechniques,reliantas theyareon themeans and variancesofstatevariables. Itis reasonableforapolicymakerthatis unableto distinguish the likelihood of good and bad events to seek protection from the worst outcomes. This lends a non-cooperative gaming aspect to the problem, with the authority planning to protect itself against the possibility that nature will throw the worst possible set of ex post shocks allowed by theboundsoftheproblem. Different assumptionsabout uncertaintybounds alterthenature ofeach gamein ways that 4The policy environment used in this paper is similar to one used in von zur Muehlen (1982), which examined several two-person Stackelberg games with Knightian uncertainty modeled as uniform distributions over the stochastic processes of the parameters. Early treatments of control as two-person games include Sworder (1964)and Ferguson(1967). More recentpapersinclude Glover and Doyle (1988), Caravani and Papavassilopoulos (1990), Caravani (1995), and Basar and Bernhard (1991), Hansen et al. (1999), Hansen and Sargent(1996,1998,1999a). 5

will be explored below. In addition, the precise formulation of each game will be determined by the amount of structure placed on approximation errors assumed to arise from misspecification. In this regard, we first assume that specification errors, viewed as perturbations to the reference model, are unstructured in the sense of being completely reflected in the additive stochastic error process driving the reference model. This uncertainty about the model is parameterized by a bound on extreme rather than average values of the loss function, which rise as unspecifiable uncertainty increases, as we will show.5 With these restrictions, the decision maker is compelled to act cautiously by assuring a minimum level of performance under the worst possible conditions. Whether or not such caution leads to less or more intensified feedback, is aquestionwewillanswerin thecontextofourempiricalmodel.6 Asecondsetofexercises,alsowithintheframeworkofKnightianuncertainty,examinesrobustdecisionsundermodeluncertaintyalone. There,weshalldistinguishbetweenuncertainty in parameters and uncertainty in dynamics. Parameter uncertainty is a fairly clear concept, exceptthatwithKnightianuncertaintywemakenodistributionalassumptions. Inaddition,we shall distinguish between one-time shifts and time varying approximation errors that may be either linear or nonlinear. Dynamicmisspecification, includingomitted or unmodeled dynamics,iscommonandimpliesthaterrorprocessesarenotwhitenoise. Forsuchcases,weareable to use a recently developed technique, called (cid:22) analysis, to determine interest-rate rules that are robust to worst-case misspecifications of the lag structure in a model. A drawback of the solutions for robust control rules under structured uncertainty is that computational methods forobtaininguniqueminimaofriskfunctionsaregenerallynotavailable. However,itispossibleto generate robustrules resultingin outputwithfinitenorms that guaranteerobust stability butnotnecessarily robustperformanceoflosses.7 2.1 A generic linear rational expectations model Let x t denote an n (cid:2) 1 vector of endogenous variables in the model. A number, n 1 , of these variablesisassumedtobeexpectational(nonpredetermined,forwardlooking),andtheremainder, n 2 = n (cid:0) n 1 , are predetermined (backward looking). The economy is assumed to evolve 5The literatureon robustcontrolhas its genesisin the seminalpaperby Zames(1981). The firstsystematic treatmentofrobustcontrolintermsofexplicitstatefeedbackrules, basedon H 1 normsona system’stransfer functions,isbyDoyle,Glover,KhargonekarandFrancis(1989). 6Anotherwayofaddressingthisissueistoconstruct,byexperimentation,rulesthatarerobusttoavarietyof models. Anfine exampleof thisis Levin, Wielandand Williams(1999)which tests interest-raterulesfortheir performance in four structural models. While these results are quite instructive, they are limited in generality bythesmallsetofmodelsconsidered. ThepremiseofKnightianuncertaintyisthatsuchspecific knowledgeis lacking. 7The relevant methodology is described in Zhou, Doyle and Glover (1996) and Dahleh and Diaz-Bobillo (1995). 6

according tothefollowinglawofmotion, x t+ 1 = A x t + B u t + v t+ 1 ; (1) where u t isa k -dimensionalvectorofpolicyinstruments,and v t+ 1 isavectorofrandomshocks, the properties of which we detail below.8 Throughout, we assume that the authority uses only one instrument and commits to the stationary rule, u t = K x t , where K is an 1 (cid:2) n vector of parameters to be chosen.9 Let T t denote a target vector, for example inflation, output, and possiblythepolicymaker’s controlvariable. T is thusrepresented as themapping, T t = M x x t + M u u t : (2) For u t = K x t ,thetargetis, T t = M x t , where M = M x + K M u is m (cid:2) n . Asisusuallydone in this literature, the periodic loss function is a quadratic form involving the target vector, T t : L t = T 0 Q t T t , where Q is an m (cid:2) m (diagonal) weighting matrix of fixed scalars assigned by theauthority. Next,itisconvenienttodefine theoutputvector, z t = Q 1 2 T t = H x t ; (3) where H = Q 1 2 M is an m (cid:2) n matrix. Withthis,theperiodiclossfunctionbecomes, L t = z 0 t z t = x 0 t H 0 H x t : (4) Theauthority’sobjectivefunctionisthediscountedsumofperiodiclosses, V 0 = 1 Xt= 0 (cid:12) t L t = 1 Xt= 0 (cid:12) t z 0 t z t ; (5) where 0 < (cid:12) (cid:20) 1 is a time discount factor.10 Note that, contrary to the usual practice, we do notexpressfuturelossesin stochasticterms, giventhedefinitionofuncertaintyas Knightian. Given a stabilizing vector, K , the unique saddlepath, with exactly n 1 roots within the unit 8Thevectorofresiduals, v t+ 1 maycontainzeros. Equation(1)maybethoughtofthesolvedcompanionform ofastructuralmodel,inwhichisues,suchassingularitieshavebeenresolvedbyappropriatesolutiontechniques. 9The feedbackrule may be a “synthesized”optimalor robustcontrolrule or it may be a restricted (simple) optimalorrobustfeedbackrule,whensomeelementsof K arerestrictedtobezero. 10In the calculations, we set (cid:12) = 1 , which we can do withoutloss of generality, providedwe are willing to assumetheexistenceofacommitmenttechnology. Valueslessthanonecanbetriviallyaccommodatedbywell knowntransformations. 7

circle, and n 2 outside,impliestherestrictedreduced-form 11 x t+ 1 = (cid:5) x t + C v t+ 1 ; (6) where (cid:5) and C dependnonlinearlyontheparametersinthepolicyruleaswellasonthestructural parameters of the model. 12 This non-linear dependence of the reduced-form parameters onstructuralandpolicyparametersimpliesthatspecificationerrorsinthestructural(reference) modelarealsoreflected intheparameters ofthereducedform. Asiscommonpractice, weexpressmodeluncertaintybyaugmentingthetransitionmatrix—inourcase,thereducedformof thereferencemodel—withamatrixofperturbations, (cid:1) (cid:5) ,representingmisspecificationerrors, x t+ 1 = ( (cid:5) + (cid:1) (cid:5) ) x t + C v t+ 1 : (7) In the next few pages we shall tackle two cases: (1) unstructured model uncertainty, represented by combined model and shock uncertainty, in which we do not distinguish between (cid:1) (cid:5) x t and C v t , and (2) structuredmodel uncertainty,in which uncertainty is solely associated withthemodel’sparameters. 2.2 Unstructured uncertainty We treat the general case of unstructured model uncertainty first, combining errors arising from independentdisturbances with thosethat arise from misspecification. If misspecification errors are due to omitted variables, the implied additive shocks will be heteroskedastic, with additional dependence on the decision rule. Further, as noted by Hansen and Sargent (1998), by feeding back on endogenous variables, misspecified shock processes capture misspecified endogenous dynamics. If these errors in specification manifest themselves through the same dynamicsasthemodel’sadditiveshocks—astheywouldinthecaseofomittedvariables—they may be reflected in size and autocorrelation of the residuals, possibly at frequencies that may damage the policy maker’s stabilization goals. Under the assumption that the distribution of shocksisunknown,theauthoritychooseselementsin K topreventorminimizeworstoutcomes totheperformancemetric. Setting w t+ 1 = (cid:1) (cid:5) x t + C v t+ 1 , and combining (7) and (3), the system may be compactly 11Asolutionmaybeobtainedusinganyofavarietyoftechniquesforsolvinglineardynamicperfectforesight models. We use the Anderson-Moorealgorithm, which computes saddle-pointsolutions using the QR decomposition to obviate problems with possible non-singularities. See Anderson and Moore (1985) and Anderson (2000). 12Notethatineveryinstance,thecomputationsproperlyaccountforthisdependenceaswellasthedependence onthefeedbackparameters, K . 8

represented by, 2 4 x z t+ t 1 3 5 = 2 4 (cid:5) H I 0 3 5 2 4 x w t t+ 1 3 5 : (8) This describes a state-space system with output, z t , and input, w t+ 1 , where, as noted earlier, w t+ 1 should be viewed as the nature’s control variable. With w t+ 1 as the instrument of one of our players, we henceforth treat it not as a stochastic process but as a deterministic sequence ofboundedapproximationerrors. We nowintroducethemappingfrom shocks, w t , tothetarget vector, z t , called thetransfer function, G . It isobtainedby solving(8)for z t as afunctionof w t : z t = H ( I (cid:0) (cid:5) L ) (cid:0) 1 w t (cid:17) G w t : (9) We are interested in the size of G , because it measures how disturbances, including perturbationsto themodel, affect target performance. However G ismeasured, i.e., whatevernorm we adopt,smallervaluesare alwaysto bepreferred tolarger ones. As first proposed by Basar and Bernhard (1991), the problem of the monetary authority facingKnightianuncertaintymaybecastasatwo-persongamewithaNashequilibrium: nature choosingthesequence f w t+ 1 g , taking theauthority’sfeedback ruleas given,and theauthority choosingthevector, K ,whichfeedsbackonthestatevariables. UsingtheHansenandSargent (1999a)formulation,thedecisionmakerchooses K tominimizewelfarelosses,andmalevolent naturechooses w t to maximizewelfare losses: m K i n m w a t x 1 Xt= 0 z 0 t z t ; (10) subjectto (8)and 1 Xt= 0 w 0 t w t (cid:20) (cid:17) 2 + w 0 0 w 0 ; (11) x 0 = w 0 : The above formulation is a very general and powerful representation of a class of games. In the special case where (cid:17) = 0 , the policymaker is solely concerned with additiveshock disturbances. In the more general case where (cid:17) > 0 , the game is determined by the initial value of theshockprocess, w 0 , which,represents theopponent’scommitmenttoits hostilestrategy. Severalresultsaswellasthecomputationsinsection4requireuseofthefrequencydomain. 9

Accordingly,weintroducethefollowingone-sidedFouriertransform, X ( (cid:24) ) (cid:17) 1 Xt= 0 x t (cid:24) t ; where (cid:24) = e i! , and ! is a pointin the frequency range (cid:0) (cid:25) ; (cid:25) . Applyingthistransformationto (8)and(9),yieldsthefollowingreformulationoftheoutputvector, z t ,andthetransferfunction, G , analogoustothetransferfunction, G , defined inthetimedomain, Z G ( ( (cid:24) (cid:24) ) ) = = G H ( (cid:24) ) ( I W (cid:0) ( (cid:24) (cid:24) (cid:5) ) ; ) (cid:0) 1 : ApplyingParseval’sequality,thegamein(10)is equivalentlyexpressed as thefinding ofa pair, [ K ; W ] , thatsolves i n K f s u W p 1 Xt= 0 z 0 t z t = i n K f s u W p 2 1 (cid:25) Z (cid:0) (cid:25) (cid:25) W ( (cid:24) ) 0 G ( (cid:24) ) 0 G ( (cid:24) ) W ( (cid:24) ) d ! (12) Z (cid:0) (cid:25) (cid:25) W ( (cid:24) ) 0 W ( (cid:24) ) d ! (cid:20) (cid:17) 2 + w 0 0 w 0 ; subjectto (8). Noticethattheintegralisdefined on theunitdisk. This formulation allows us to describe several possible games, depending on particular assumptionsmadeabouttheformofthelossfunctionand theboundsplaced onKnightianuncertainty. Theauthoritymay,forvariousreasons,beinterestedinminimizingasquaremeasure of loss, such as a quadratic form, or it may want to minimize the largest absolute deviation of its losses. In the absense of distributional assumptions, the authority must also decide how largenature’sshockscan be. Typicalassumptionsarethatshocksaresquare-summableorthat their largest absolute value is less than some finite number. Such assumptions are typically expressed as norms. In our case, we shall be mainly interested in two such norms, denoted ‘ 2 and ‘ 1 , respectively,13 13Let n lp ( Z ) be the space of all vector-valued real sequences on integers of dimension n , where ( (cid:1) (cid:1) (cid:1) ; x ( (cid:0) 1 ) ; x ( 0 ) ; x ( 1 ) ; (cid:1) (cid:1) (cid:1)) x = with x ( k ) 2 < n ,suchthat jjx jjp = 0 @ k =X 1 (cid:0) 1 jX n = 1 jx j ( k ) p j 1 A 1 = p < 1 : If p = 1 , jjx jj1 belongsin ‘ 1 andisthesumofabsolutevalues. For p = 2 ,thenorm, jjx jj2 belongsin ‘ 2 ,andits squareistheamountof“energy”inasignal,which,instatisticsoreconomics,isakintoacovarianceorquadratic loss. For p = 1 , jjx jj1 belongsin ‘ 1 andisthemaximum“magnitude”or“amplitude”asignalcanattainover alltime. 10

1. ‘ 2 ( X ) = ( P 1 0 j X ( j ) j 2 ) 1 = 2 : square-root of thesum of squares of X; when X = z , this is thelinearquadraticregulator(LQR) lossfunctionin (10). The constraintshownin (11), given (cid:17) = 6 0 , isan ‘ 2 norm,where X = w . 2. ‘ 1 ( X ) = s u p j j X ( j ) j : the largest absolute size of X . For X = w , it measures the maximumamplitudeofdisturbances. Wenowpresentthreeversionsoftheprecedinggame,eachdistinguishedfromtheotherby an assumptionmade about (cid:17) and the manner in which nature is assumed to have“committed” to a strategy, W , as manifested by its choice (or lack thereof) of the initial shock, w 0 . Given nature’s commitment to a strategy, W , the authority can solve the opponent’s maximization problem to eliminate W and reduce the problem to one involving an indirect loss function. In the language of linear operator theory, the resulting loss function, expressed as a norm on G , is said to be “induced”.14 Of course, the central bank does not know nature’s strategy other thanthatitisboundedinsomesense. Wediscussthreeversionsofthegamein(10)-(11),each determinedby particularassumptionsconcerning theform ofthelossfunctionandthebounds placeduponKnightianuncertainty. Asisshownnext,byspecifyingwhichoftheabovenorms appliestothelossfunctionandwhichtotheshockprocess,theauthoritydeterminesthekindof one-player, ‘induced” loss function it seeks to minimize. An important feature of that indirect lossfunctionisthat isindependentoftheopponent’sstrategy, f w t+ 1 g . Table3anticipatestheindirectlossfunctionswewillderiveinthenextthreesections. The top row specifies the norm assumed to bound uncertainty, and the leftmost column shows the assumed norm for the loss function. The cells in the table display the indirect loss functions that result when one combines assumptions from the top row with loss functions at the left. The ‘ 1 normis defined infootnote13 and insection2.2.3. Table1: Loss Functions Induced by TypeofUnstructured Model Uncertainty ModelUncertainty: (cid:17) > 0 CertaintyEquivalence: (cid:17) = 0 UnderlyingLoss ‘ 2 ‘ 1 ‘ 2 Quadratic Loss: ‘ 2 H 1 (cid:0) H 2 Maximum Loss: ‘ 1 (cid:0) ‘ 1 (cid:0) 14LetG( G = ( g ij ) 2 < m (cid:2) n )beatransferfunctionfrom w to z : z = G w , thenthe ‘ p inducednormof G is anoperatorfrom ( < n ; j:jp ) to( < m ), jjG jjp (cid:0) in d u c e d = s w u 6= p 0 jjG jjw w jjp jjp ; which is the amountof amplification the operator G exerts on the space, Z . Further, G is a boundedoperator normfrom W to Z ,ifitsinducednormisfinite, jjG jjp < 1 . 11

2.2.1 ‘ 2 losswithout model uncertainty: the linearquadratic regulator(LQR) Assume that model uncertainty is not an issue, that is, let (cid:17) = 0 and allow W ( 0 ) = w 0 to be arbitrary. In this case, (11) is satisfied only if w t = 0 for all t > 0 , so that W ( (cid:24) ) = w 0 , a constant for all (cid:24) . Equivalently,theshocks, w , have no spectral density, so that I W ( (cid:24) ) 0 W ( (cid:24) ) = . Thegame(12)reduces tothelinear-quadraticproblemwithcertainty equivalence, i n K f s u W p 1 Xt= 0 z 0 z t t (cid:17) = i i n K n K f f s w u W p 0 0 j j j G j z j j j j 2 2 2 2 w = 0 : i n K f s u W p w 0 0 (cid:20) 2 1 (cid:25) Z (cid:0) (cid:25) (cid:25) G ( (cid:24) ) 0 G ( (cid:24) ) d ! (cid:21) w 0 (13) Minimizationoftheloss functionistherefore equivalentto minimizationofthenorm j j G j j 2 in H 2 ,(theHardy spaceofsquare-summableanalyticfunctionsontheunitdisk),where j j G j j 2 = 2 1 (cid:25) Z (cid:0) (cid:25) (cid:25) t r a c e [ G 0 ( (cid:24) ) G ( (cid:24) ) ] d ! ; is a function of the authority’s decision rule, K , but not of w . Each point on a plot of t r a c e [ G ( (cid:24) ) 0 G ( (cid:24) ) ] represents the contribution of the shock process at frequency point ! to the total loss. Notice that j j G j j 2 is related to a generalized variance, defined by integrating over spectral frequencies withequal weightingacross frequencies. As weshall showbelow,therobustauthoritydoesnotassignequalweightstoallfrequencies; rather,itassignslargerweights tofrequenciestowhichtheeconomyismostsusceptibletodamagefromwellchosenshocks,in accordancewithitsstrategytoavoidworst-caseoutcomes. Sinceallpolicies,robustornot,implyavariancemeasureofloss,differentpoliciescanbecomparedbyplotting t r a c e [ G ( (cid:24) ) 0 G ( (cid:24) ) ] , measuringtheirrelativestrengthsand vulnerabilitiesatvariousfrequencies. 2.2.2 ‘ 2 losswith ‘ 2 bounded model uncertainty: the H 1 problem As noted before, the LQR, with its implication of certainty equivalence, may distort policy if riskisarealconcern. Iftheauthoritybelievesrisktobeasignificantfeatureoftheenvironment it faces, a different approach is required. In terms of the game (12), model uncertainty is equivalenttoletting (cid:17) > 0 bearbitrary. Forthepresentcase,alsoassumethat W ( 0 ) = w 0 = 0 . The initial setting, w 0 therefore disappears from the constraint, making the problem the same as if nature made no commitment to an initial w 0 , at all.15 As before, the authority’s nominal lossfunctionisassumedtobequadratic. InHansenandSargent(1999a)andZhouetal.(1996) 15Absenceofcommitmentto w 0 is a hallmarkof H 1 control. Asshownin subsection2.2.4,gamesthatare intermediateto H 2 and H 1 ,assumethatnaturedoescommittosome w 0 6= 0 . 12

itis shownthat for(12),thisproblemimpliesasingle-agentminimizationproblemin H 1 , i n K f s u W p 2 1 (cid:25) Z (cid:0) (cid:25) (cid:25) W ( (cid:24) ) 0 G ( (cid:24) ) 0 G ( (cid:24) ) W ( (cid:24) ) d ! (cid:20) (cid:20) i i n K n K f f s s u W u W p p ! (cid:22)(cid:27) s u 2 [(cid:0) 2 (cid:17) p (cid:25) 2 ;(cid:25) = ] (cid:22)(cid:27) i 2 n K 2 f 1 (cid:25) j j G Z (cid:25) (cid:0) (cid:25) 2 1 j j W (cid:17) 2 ( ; (cid:24) ) 0 W ( (cid:24) ) d ! (14) where (cid:22)(cid:27) denotes the singular value of G , defined as the square root of the largest eigenvalue of G 0 G .16 The idea behind this approach is to spread the consequences of unknown serial correlations across frequencies by designing a rule that works well over a range of values of W ( (cid:24) ) 0 W ( (cid:24) ) ,takingtheviewthat j j w j j 2 istheworstthatnaturecando. Thesaddlepointsolution of(14)isequivalentto theinfinumofthe H 1 norm, j j G j j 1 , i n K f j j G ( K ) j j 1 = i n K f s u W p j j G j ( w j K j ) 2 j j j 2 (cid:17) (cid:22) (cid:18) : (15) Denotingtheminimizingfeedback of K by K ^ , (cid:22)(cid:18) satisfiestheinequality, j j G ( ^K ) j j 2 (cid:20) (cid:22)(cid:18) j j w j j 2 ; f o r a l l w 2 ‘ 2 ; (16) demonstratingthatrobustpolicycanlimittheratioofthetwonorms. Confiningthisratiotoacceptablelevelsiscalleddisturbanceattenuationintheengineeringcontrolliterature. Whilean increasein (cid:22)(cid:18) alwaysimpliesanincreaseinthelevelofuncertainty,HansenandSargent(1999a) identify declining values of (cid:18) with rising levels of preference for robustness. Conversely, as (cid:18) ! 1 , thepreceding criterionconverges onthestandard LQRpolicy. 2.2.3 ‘ 1 losswith ‘ 1 bounded model uncertainty: theMAD criterion Insteadofminimizingdeviationsfromtargetpathsagainstshockswithsquare-summablebounds, the policy maker is now assumed to avoid worst-case scenarios by minimizing the maximum amplitudeoftargetdeviations, j j z j j 1 ,againstthelargestpossibleshocksatisfying j j w j j 1 < (cid:17) 2 . Thisassumptionrepresentspoliciesofanauthoritythatfeelsespeciallysusceptibletounfortunate shocks. This combination of loss function and size of uncertainty induces an ‘ 1 indirect loss on G , defined as the weighted sum of absolute deviations of targets from their desired 16Arefereehaspointedoutthedependenceof w t+ 1 onthefeedback, K ,asapotentialproblemintheanalysis of H 1 problems. Sincenaturemaychoosea w t+ 1 inresponseto K thatishugeincomparisontoothershocks, evenif G ( K ) amplifiesthisshockaslittleaspossible,thedamagecanbeverylarge.Framedintermsofthegame, thevariancerestrictionontheshocksdependson K ,sothat P w 0 ( K ) w ( K ) < (cid:17) 2 + w 0 0 w 0 . Note,however,the assumptionthat (cid:17) isarbitrarymeansthattheimpliedrobustpolicywillguaranteeadequateperformanceagainst the worst that naturecan do, includingreacting adverselyto K . In any case, this featureis notconfinedto the forward-lookingmodelanalyzedhere;thenatureofthegamesolutionmakesitgenerictoalldynamicmodels. 13

levels, j j z j j 1 .17 Formally,theminimumabsolutedeviation(MAD)problemsolves, m s i K : t n s j j u w w p j j j 1 j G (cid:20) w j j (cid:17) 1 2 ; whichis equivalentto m K i n w s :jjw u p 1 jj (cid:20) (cid:17) 2 j G j j j w w j j j 1 j 1 = m K i n j j G j j 1 = m K i n m 1 (cid:20) a i(cid:20) x m n Xi= 1 j G ij j : 2.2.4 Between H 2 and H 1 : Minimum Entropy Adigressionthatisusefulforcastingrobustcontrolinlanguagethatisfamiliartoeconomists, is the case of minimum entropy. Observe that in the H 2 control problem, misspecification becomes irrelevant when (cid:17) = 0 and w 0 is allowed to be arbitrary. By contrast, in the H 1 robust control problem, (cid:17) > 0 allows for model misspecification, while w 0 is assumed to be zero,or,equivalently,free. Anintermediatecasecommitsnaturetospecifyaninitialcondition, w 0 = 6 0 , leadingto theLagrangian multipliergame, i n K f w :jjw s u 2 jj (cid:20) p 2 (cid:17) + w 0 0 w 0 j j z j j 2 : (17) An interesting result due to Whittle (1990) is that the preceding game reduces to a singleplayer control problem with the authority minimizingentropyor, equivalently,a risk-sensitive functioncloselyrelated to riskaversioninutilitytheory. i n K f (cid:20) 2 1 (cid:25) Z (cid:0) (cid:25) (cid:25) l o g d e t [ ( G ( (cid:24) ) 0 G ( (cid:24) ) (cid:0) (cid:18) I ) ] d ! (cid:21) = i n K f (cid:0) 2 (cid:11) l o g E ( e (cid:0) (cid:11) 2 jjG w 2 jj ) ; (18) which is defined only for (cid:18) > (cid:22)(cid:18) , where (cid:18) is the Lagrangian multiplier in (17), and (cid:22)(cid:18) is the smallest positive scalar for which the integrand is negative semidefinite.18 The relationship between the entropy criterion and the H 1 game is that (cid:22)(cid:18) is the infinum of the H 1 norm in 17In estimation theory, the ‘ 1 -normestimator is knownas the least absolute deviationestimator (LAD), proposedbyPowell(1981). 18Representing model uncertainty as a game may seem like a stretch to some readers since it gives to what is normallyan exogenousprocessthe fiction ofstrategic non-cooperativebehavior. There is, however,roomto modulatethisbehaviorbyadjustingpreferencesforrobustnesstofittheprobleminaspecificway. Inthegame (17),itisthemultipliertotheconstraintwhichbecomestheinstrumentfordiscipliningtheperceivedbehaviorof nature,earningittheinterpretationas(theinverseof)ameasureofpreferenceforrobustness.Accordingly,nature, seekingtomaximizelosses,ispenalizedif w 0 w (cid:21) (cid:17) 2 + w 0 0 w 0 . Theaddedcriterionofrobustnessisequivalentto theintroductonofapessimisticattitude, w 0 w ,theimportanceofwhichisgovernedby (cid:18) : smallvaluesof (cid:18) imply a w thatcanbelarge,whilelargevaluesof (cid:18) makenature’sthreatlessimportant. 14

(15). Theparameter, (cid:11) = (cid:0) 1 = (cid:18) ,hasaninterestinginterpretationasaKnightianrisksensitivity parameter. The relationship between risk sensitivity and robustness is further illuminated by expandingthelastterm aboveinpowersof (cid:11) < 0 , l o g E ( e (cid:0) (cid:11) 2 jjG w 2 jj ) (cid:25) E K ( j j G w j j 2 ) + (cid:11) 4 v a r K ( j j G w j j 2 ) + O ( (cid:11) 2 ) ; where E K generates a mathematical expectation, and v a r K is the variance operator. As (cid:11) approaches (cid:0) 1 = (cid:22)(cid:18) from above, (becoming larger in absolute value), dislike of increasing values of j j G w j j 2 rises. Conversely, as (cid:11) approaches 0 , the problemincreasingly reduces to thetraditional linearquadraticcontrol problem, as thevariance term disappears. Ofcourse, a decrease in (cid:22)(cid:11) ofthe H 1 bounditselfrepresents adecrease in actualmodeluncertainty. Finally,HansenandSargent(1995)haveshownthat(18)canbesolvedusingtherecursion, V t = z 0 t z t (cid:0) 2 (cid:11) l o g E u e (cid:0) 1 2 (cid:11) V +t 1 ; which demonstrates that preference for robustness is like a discount factor, suggesting that with (cid:11) = 6 0 , the authority has an incentive to forestall future consequences of current model uncertaintyby actingaggressively. 2.3 Structured model uncertainty 2.3.1 Defining thegame We now turn to structured model uncertainty, where uncertainties are assigned to parameters of the model instead of being consigned to additive shock processes. For convenience we repeatequation(7)butshowthelagoperator, L ,asanexplicitargumentof (cid:5) toemphasizethe dynamicnatureofthemodel, x t+ 1 = ( (cid:5) ( L ) + (cid:1) (cid:5) ) x t + C v t+ 1 : ( 7 ) Uncertainty about parameters or about unmodeled dynamics is formally treated as perturbations, (cid:1) (cid:5) 2 (cid:1) , where (cid:1) is a perturbation block spanning all approximation errors. These can include one-time jumps in individual parameters, misspecifications in contemporaneous channels frompolicytothestatevariables,and omissionofcriticallag structures affectingthe dynamicbehavioroftheeconomy. Analytically,thestructuredperturbations, (cid:1) (cid:5) ,areoperators defined independentlyofthestatevector, x t . If theauthority believes approximationerrors to lie within somesmall neighborhood,then 15

individual elements, (cid:1) (cid:5) ij , corresponding to the elements of (cid:5) ( L ) , will have small norms less than r Æ ij ,where r istheradiusofallowableperturbations,and Æ ij isascalingconstantassigned to the i j -th parameter in (cid:5) ( L ) .19 In this case the perturbation block may be defined as the diagonal matrix, (cid:1) = d i a g f (cid:1) (cid:5) ij = Æ ij g . Theadmissibleset ofperturbations is formally denoted D r = f (cid:1) : j j (cid:1) (cid:5) ij = Æ ij j j < r g , where r (cid:20) (cid:22)r < 1 . Accordingly, the set, D r , is widened as the radiusofallowableperturbationsisincreased. Agamesimilarto(12)istheStackelberg game, m K i n (cid:1) s (cid:5) u 2 p D r j j z j j 2 2 ; subject to (7), where K is the vector of parameters chosen by the authority, and (cid:1) (cid:5) is a diagonal matrix of the perturbations (controlled by nature).20 Naturally, the authority is interested in protecting itselfagainst thewidest set of perturbations, but no policy may exist that accomplishes this task. Conversely, as one narrows the range of permissible misspecifications, the menu of possible rules that achieve stability widens and may become unmanageable. As a result, the authority may seek a unique rule, that, at least for benchmark purposes, guarantees stabilityoftheeconomyforthemaximumrangeofmisspecificationsofparameters orlag structures. 2.3.2 A frameworkforanalyzing structured perturbations Tocharacterizetheperturbationblock, (cid:1) ,wemayconsidertime-varyingversustime-invariant perturbationsandlinearversusnonlinearperturbations. Amonglinearperturbations,thosewith the greatest amount of imposed structure are time-invariant scalar (LTI-scalar) perturbations, representingsucheventsasone-timeshiftsinparametersandstructuralbreaks. Iftheauthority is concerned with misspecification in the lag structure of the model, then the case of infinite movingaverage (LTI-MA)perturbationsmightbe themostappropriateto consider, sincethey involve perturbations with long memory. An authority that fears dynamic misspecification to be in the form of LTI lag polynomials seeks out policies that protect against the worst possible lag misspecifications. For LTI perturbations, the literature has developed techniques in the frequency domain, which help conserve on the number of parameters. The last in the 19Intheexercises,the Æ ij arefunctionsofthestandarderrorsofestimatesofthemodel. 20Theperturbationoperator, (cid:1) ,isdefinedindependentlyofthestate, x t . Asaconsequence,weanalyzerobust system stability, in contrast with the procedures for unstructured uncertainty, which yield policies that assure robustperformance. Forabackwardlookingmodel,OnatskiandStock(2000)areabletosolveforperformance robustnessin a special case where the two criteria coincide. Onatski(1999)derivesoptimalminimaxrules for parametricaswellaslag-structureuncertaintyinabackwardlookingmodel.Inthepresentmodel,givenforwardlooking agents and the need to work with the reduced form representation of the model, this is not possible. 16

groupoflinearperturbationsaretime-varying(LTV)perturbations. Withintheclassofnonlinear perturbations, we are able to consider both nonlinear time-invariant (NTI), and nonlinear time-varying(NTV) perturbations, where thelatter allow thewidest latitudeand greatest nonparametric generality of model uncertainty. The last three types of perturbations to the model (LTV, NTI, and NTV) can be treated as one, and we shall refer to them collectively as NTV, because, as itturnsout, thestabilityconditionsforeach areidentical. Turning to the task of specifying the perturbation block, denote the space of all causal perturbationsby ‘ (cid:1) . Asbefore, (cid:1) , denotestheclassofallowableperturbations,i.e.,thosethat carry with them the structure information of the perturbations. In our case, (cid:1) is the set of all diagonalperturbationsoftheform, (cid:1) = f d i a g ( (cid:1) (cid:5) ij = Æ ij ) j (cid:1) (cid:5) ij 2 ‘ (cid:1) g wherethe (cid:1) (cid:5) ij ,whichcanbeanyofthetypesofperturbationsdiscussedbefore,areassumedto be ‘ p -stable, and p = 2 or 1 . The subsetof (cid:1) containingelementswith ‘ p norm smallerthan than r isdenoted B (cid:1) ;p , so that B (cid:1) ;p = f (cid:1) (cid:5) 2 (cid:1) j j j (cid:1) (cid:5) j j ‘ p (cid:0) in d < r g : For reasons that will become clear presently, we represent the perturbed model as an interconnected system of equations linking the state vector, x t , and a vector of perturbations. Let us supposethat a subset of the elements in (cid:5) are misspecified, as represented by bounded perturbations, and suppose this involves k elements of the state vector, x t . Accordingly, let (cid:0) and U betheappropriatelydimensionedselectormatrices,filledwithzeros andones,thatpick thecorrectelementsof (cid:5) tobeperturbedandelementsof x t thatbecomeinvolved,respectively. Theperturbedand truemodelis, x t+ 1 = ( (cid:5) + (cid:0) (cid:1) (cid:5) U ) x t + C v t+ 1 : By defining an augmented output vector, p t = U x t and a corresponding input vector, (cid:1) (cid:5) x t h t = , thepreceding equationisequivalenttotheaugmentedfeedback loop, 2 4 x t+ p t 1 3 5 = 2 4 (cid:5) U C 0 (cid:0) 0 3 5 2 6 6 6 4 v x t t+ h t 1 3 7 7 7 5 ; (19) h t = (cid:1) (cid:5) p t ; (20) 17

whichis inaform amenabletothetechniquestobeused below. Once again, we want to analyze the size of a transfer function, although this time around, thetransferfunctiondoesnotinvolvethetarget vector,perse. Wesolve(19)for x t and p t as a mappingfrom v t and h t . Thisyieldsthetransfermatrix, G , writteninpartitionedform, 2 4 x p t t 3 5 = (cid:17) 2 4 2 4 I U G G n 1 2 3 5 1 1 ( I G G (cid:0) 1 2 2 2 (cid:5) 3 5 L 2 4 ) (cid:0) v h 1 t t h 3 5 C : (cid:0) i 2 4 v h t t 3 5 ; (21) Noticethattheinterconnectionbetween h t and p t isrepresentedbytwochannels: feedforward p t = G 2 2 h t and feedback h t = (cid:1) (cid:5) p t . Why is G 2 2 interesting? For an answer, appeal is made to the “small gain theorem” (see Dahleh-Diaz-Bobillo (1995) and Zhou, Doyle, and Glover (1996)), which states that for all (cid:1) (cid:5) 2 < H p , j j G 2 2 j j p < 1 = r if and only if (cid:1) (cid:5) (cid:20) r . In words, if the policy rule, K , stabilizes the nominal model (7), then the augmented model (19)-(20) is stable if and only if the feedback interconnection between h t and p t in (21) is stable. As a consequence, only the stability of G 2 2 need be examined for any desired norm to assure stability of the full model, under the same criterion. For p = 2 and p = 1 , the mathematics forevaluatingstabilityunderstructuredperturbationsinvolvesthestructurednorm, S N (cid:1) ;p ( G 2 2 ) = i n f (cid:1) (cid:5) [ j j (cid:1) (cid:5) j j ‘ p (cid:0) in d u c e d j ( I (cid:0) 1 G 2 2 (cid:1) (cid:5) ) (cid:0) 1 i s n o t ‘ p (cid:0) s t a b l e ] ; (22) suchthatif ( I (cid:0) G 2 2 (cid:1) (cid:5) ) (cid:0) 1 is ‘ p -stableforevery (cid:1) (cid:5) 2 D r ,then S N (cid:1) ;p ( G 2 2 ) = 0 . Importantly, themaximalallowableradiusofperturbationsisgivenby r = 1 = S N (cid:1) ;p ( G 2 2 ) . Oneimplication ofthesmallgaintheorem istheresultthatthestructurednormis alowerboundon G 2 2 , S N (cid:1) ;p ( G 2 2 ) (cid:20) j j G 2 2 j j p ; (23) because if j j (cid:1) (cid:5) j j p < 1 = j j G 2 2 j j p , then ( I (cid:0) G 2 2 (cid:1) (cid:5) ) (cid:0) 1 is ‘ p -stable. Therefore, if S N (cid:1) ;p is ‘ p -normbounded,thenso is G 2 2 . 2.3.3 Stability forLTV, NTV, and NTI perturbations The monetary authority is assumed to choose the elements of K , that minimizethe structured norm, S N (cid:1) ;p , p = 2 or p = 1 . The elements of the transfer matrix, G 2 2 , are linear, timeinvariant, and hence ‘ 1 -stable. This allows us to define a n (cid:2) n matrix, N ^ , of ‘ 1 norms of the 18

elementsof G 2 2 , N ^ = 0 B B B B B B B B B @ G j j G j j 1 2 : : : n 2 1 2 1 2 j j j j 1 1 (cid:1) (cid:1) (cid:1) : : : (cid:1) (cid:1) (cid:1) G j j G j j 1 n 2 2 : : : n n 2 2 j j j j 1 1 1 C C C C C C C C C A ; whichmustbeapproximatednumerically. Dahleh and Diaz-Bobillo (1995) show that for LTV, NTV, and NTI perturbations and bounded ‘ 1 -induced norms, ( p = 1 ), it is necessary and sufficient that the spectral radius of N ^ be smaller than the inverse of the radius of allowable perturbations, (cid:26) ( N ^ ) (cid:20) 1 = r , where (cid:26) ( N ^ ) is defined as the largest stable root of N ^ . With bounded ‘ 2 -induced norms, ( p = 2 ), the condition (cid:26) ( N ^ ) (cid:20) 1 = r is sufficient. Further, since the stability conditions are the same for all three types of perturbations, the computations are greatly simplified, involving minimization of (cid:26) ( N ^ ) overtheelementsof K inallcases;andthemaximalacceptableradiusofperturbation becomes 1 = ^(cid:26) ( N ^ ) . These three cases seem to be the most relevant ones for a monetary authority forming robust policy, given that G 2 2 is a function of the reduced-form solution of the parameters of a forward-looking structural model. If the authority is concerned about the about worst-case consequencesofmisspecificationinsomeorallstructuralparameters,theeffectonthereduced form will be nonlinear and will in general, though not always, involve most of its elements. In addition, by the Lucas critique, the elements of (cid:5) may be time varying, so that, generally, parameterperturbationsofanykindinthestructuralmodelmayalsotranslateintotime-varying perturbationsofthereduced form. 2.3.4 Stability withLTI perturbations Lineartimeinvariantperturbationscaninvolveone-timeparametricshiftsorchangesinvolving lags. The analysis is best carried out in the complex plane, which allows for an economical treatment ofperturbations with infinitemovingaverages. The ideais the following.21 Assume we can model the i j -th diagonal component of the worst-case perturbation, (cid:1) (cid:5) ij ( L ) , obeying j j (cid:1) (cid:5) ij = Æ ij j j 1 = r ij < r , with a very general infinite-order moving average, expressed as a 21WewouldliketothankAlexeiOnatskiforclarifyingforusanumberofpointsinthefollowingdiscussion. 19

polynomialon L , (cid:1) (cid:5) ij = Æ ij = = (cid:6) (cid:6) r r ij ij ( L (cid:0) h (cid:0) z ij z ij + ) = ( ( 1 1 (cid:0) (cid:0) z z 2 ij L ij ) L ) + ( 1 (cid:0) z 2 ij ) z ij L 2 + (cid:1) (cid:1) (cid:1) i ; (24) where z ij 2 ( (cid:0) 1 ; 1 ) . Notice that for z ij =0, the effect of a perturbation is via the (cid:6) r ij L operator, so that the worst-case effect of a dynamic misspecification involves only one lag. Conversely, if j z ij j is close to 1, all terms except the first are vanishingly small. Hence, this situationrepresentsthecasewhentheeffect oftheoperatorisnotdynamicbutscalar: (cid:6) r ij z ij , such as a structural break. For values of z ij between 0 and 1, (cid:1) (cid:5) ij ( L ) represents a complex perturbation equivalent to an infinite-order polynomial in L with rate of decay determined by z ij .22 In Zhouet al. (1996)itisshownthatthelineartimeinvariantperturbationin(24)satisfies, (cid:1) (cid:5) ij ( ! 0 ) = Æ ij r ij e i! ij ; (25) where i = p (cid:0) 1 , and ! 0 2 [ 0 ; (cid:25) ) . The following correspondence can be established between (24) and (25): the sign in (24) is positive if ! ij 2 [ 0 ; (cid:25) ) , and it is negative if ! ij 2 ( (cid:0) (cid:25) ; 0 ) . Hence, in both instances, z ij = ( e i! 0 (cid:0) e i! = ( 1 (cid:0) e i( ! 0 + ! ij ) ) . Therefore, by identifying ! 0 and r ij ,it ispossibleto unravelthestructureof (cid:1) (cid:5) ij . The next step is to determine a feedback vector K for which the important condition, j j (cid:1) (cid:5) ij = Æ ij j j 1 = r ij < r ,underwhich thepreceding resultshold,is,indeed, satisfied. Thesmall gaintheoremthen guarantees thatthesystemwillbestableundertheworst-casescenario. Equation(25)canbeevaluatednumerically,againbyexploitingthesmallgaintheorem,this timeinthefrequencydomain,todefineatransferfunction G 2 2 eqivalentto G 2 2 above. Consider theaugmentedinputoutputsystem(19)-(20). Thetransferfunctionfrombothdisturbances, v t , and perturbations, h t , to the state, x t , and the augmented state, p t , is denoted G to distinguish itfrom G : 2 4 x p t t 3 5 = (cid:17) 2 4 2 4 I U G G n 1 2 1 1 3 5 (cid:16) G G I 1 2 e 2 2 (cid:24) (cid:0) 3 5 2 4 (cid:5) (cid:17) v h (cid:0) t t 1 3 5 h : C (cid:0) i 2 4 v h t t 3 5 ; 22Forthenumericalexercises,wefoundthatthe z ij ’s,whichwedonotreport,impliedanaveragepersistence withmeanlagofabout3quarters. 20

TodevelopanotionofstabilityunderLTIperturbations,wedefineyetanothervariationon structured norms, the structured singular value function. The feedback loop between p t and h t ,whichlinkstheperturbations, (cid:1) (cid:5) ,tothesystemvariables, x t ,is p t = G 2 2 h t . For p = 2 and p = 1 , thestructuredsingularvaluefunctionis thendefined as, (cid:22) (cid:1) 2 L T I p [ G 2 2 ] = i n f (cid:1) (cid:5) 2 L T I p f (cid:22)(cid:27) [ ^(cid:1) (cid:5) ] j d 1 e t ( I (cid:0) G 2 2 (cid:1) (cid:5) ) = 0 g = (cid:1) (cid:5) i n 2 L f T I p (cid:26) ( G 2 2 (cid:1) (cid:5) ) ; and ifthereisno (cid:1) (cid:5) 2 D r suchthat d e t ( I (cid:0) G 2 2 (cid:1) (cid:5) ) = 0 ,then (cid:22) (cid:1) (cid:5) 2 L T I ( G 2 2 ) = 0 . The structured singular value function may be thought of as the frequency domain parallel to the structured norm function (22). The authority now chooses K to minimize the structured singularvalue, (cid:22) = (cid:22) i n K f S N (cid:1) L T V ;p ( G 2 2 ) = i n K f ! s 2 u [0 p ;2 (cid:25) ] (cid:22) (cid:1) L T I p [ G 2 2 ] (cid:20) i n K f s u (cid:24) p (cid:22)(cid:27) ( G 2 2 ) ; where (cid:27) ( G 2 2 ) isthesingularvalueof ( G 2 2 ) . Themaximumradius ofallowableLTIperturbationsis theinverseof (cid:22) , s u K p r = m i n K s u p ! 2 [0 1 ;2 (cid:25) ] (cid:22) (cid:1) L T I p [ G 2 2 ] : 3 The model We seek a framework for policy that is simple, empirical, and realistic from the point of view of a monetary authority. Towards this objective, we construct a simpleNew Keynesian model alongthelinesofFuhrerandMoore(1995b). Thekeytothismodel,asinanyKeynesianmodel, is the price equation or Phillips curve. Our formulation is very much in the same style as the realwagecontractingmodelofFuhrerandMoore(1995a). BymakinguseoftheFuhrer-Moore formulation, we ‘slip the derivative’in the price equation, making inflation stickyand not just the price level, thereby ruling out the possibility of costless disinflation. However, instead of thefixed-termcontractspecificationofFuhrer-Moore,weadoptthestochasticcontractduration formulation of Calvo. In doing this, we significantly reduce the state space of the model, thereby accelerating thenumericalexercisesthatfollow. Equations (26) and (27) together comprise a forward-looking Phillips curve, with (cid:25) and c measuring aggregate and core inflation, respectively, and y is the output gap, a measure of excessdemand. Equation(26)givesinflationasaweightedaverageofinheritedinflation, (cid:25) t(cid:0) 1 , and expected core inflation, E t(cid:0) 1 c t+ 1 . Following Calvo (1983), the expiration of contracts 21

is given by an exponential distribution with hazard rate, Æ . Assuming that terminations of contracts are independent of one another, the proportion of contracts negotiated s periods ago thatare stillin forcetoday is ( 1 (cid:0) Æ ) Æ t(cid:0) s . (cid:25) t = Æ (cid:25) t(cid:0) 1 + ( 1 (cid:0) Æ ) E t(cid:0) 1 c t + v (cid:25) t (26) c t = ( 1 (cid:0) Æ ) E t(cid:0) 1 ( (cid:25) t + (cid:13) y t ) + Æ E t(cid:0) 1 c t+ 1 (27) y t = (cid:30) 1 y t(cid:0) 1 + (cid:30) 2 y t(cid:0) 2 + (cid:30) 3 R t(cid:0) 1 + v y t (28) R t = 1 + 1 D E t(cid:0) 1 1 Xi= 0 ( 1 D (cid:0) D ) i ( r t+ i (cid:0) (cid:25) t+ i ) (29) r t = g r r t(cid:0) 1 + ( 1 (cid:0) g r ) ( (cid:22)(cid:25) t + r (cid:3) ) + g (cid:25) ( (cid:22)(cid:25) t (cid:0) (cid:25) (cid:3) ) + g y y t ; (30) where it is assumed that the central bank reacts to the behavior of the average rate of inflation over the past year, where (cid:22)(cid:25) = 1 4 P 4 1 (cid:25) t(cid:0) i . In equation (27), core inflation is seen to be a weighted average of future core inflation and a markup of excess demand overinherited inflation. Equations (26)and (27) differ from the standard Calvo model only in that the dependent variables are rates of changes rather than levels. Equation (28) is a very simple aggregate demand equation with output being a function of two lags of output as well as the lagged ex antelong-termrealinterestrate, R . Equation(29)followsFuhrerandMoore(1995b)inusinga constantapproximationtodurationformulabyMacaulay(1938)definingtheexantelong-term realinterestrateasageometricallydecliningweightedaverageofcurrentandfutureshort-term realinterestrates. Finally,equation(30)isagenericinterestratereactionfunction,writtenhere simplyto completethe model. We may assume that it is the empirical manifestationof an optimal decision rule by the Federal Reserve, which manipulates the nominal federal funds rate, r , and implicitly deviations of the real rate from its equilibrium level, r (cid:3) (cid:0) (cid:25) (cid:3) , with the aim of movingaverage annual inflation to its target level, (cid:25) (cid:3) , reducing excess demand to zero, and penalizingmovementsin theinstrumentitself. Themodelisstylized,butitdoescapturewhatwewouldtaketobethefundamentalaspects of models that are useful for the analysis of monetary policy. Among these, stickiness of inflation is foremost. Other integral features of the model include that policy acts on demand and prices with a lag. This rules out monetary policy that can instantaneously offset shocks as they occur. The model also assumes that disturbances to aggregate demand have persistent effects, as are the effects of demand itself on inflation. These features imply that in order to be effective, monetary policy must look ahead, setting the federal funds rate today to achieve objectivesinthefuture. However,thestochasticnatureoftheeconomyimpliesthattheseplans will not be achieved on a period-by-period basis. Rather, the contingent plan set out by the authority in any one period will have to be updated as new information is revealed regarding 22

theshocksthathavebeen borneby theeconomy. 3.1 The estimated model WeestimatedthekeyequationsofthemodelonU.S.datafrom1972Q1to1996Q4.23 Sincethe precise empirical estimates of the model are not fundamental to the issues examined here, we will keep our discussion of them concise. A couple of important points should be mentioned however. We measure goods-price inflation, (cid:25) , with the quarterly change in the chain-weight GDP price index, a producer’s price. However, we proxy E t(cid:0) 1 c t+ 1 with the median of the Michigan survey of expected future inflation. The survey has some good features as a proxy. First, it is an unbiased predictor of future inflation. At the same time, it is not efficient: Other variables do help in predicting movements in future inflation. Second, it measures consumer priceinflationexpectations,preciselytheratesthatwouldtheoreticallygointowagebargaining decisions, and thereby into unit labor costs. GDP price inflation can then be thought of as a pseudo-mark-up over these expected future costs. The disadvantage is that the survey is for inflation over the next twelve months, which does not match the quarterly frequency of our model. However, most of the predictive power of the survey to predict inflation over the next twelvemonthscomesfromitsabilitytopredictinflationintheveryshorttermratherthanlater on,so thisproblemisnottoo serious. Equation(27)can besubstitutedintoequation(26)toyielda restrictedPhillipscurve. The estimatesofthisequationalongwithtwoothersarepresentedinTable1below. Unemployment gaps–defined as the deviation of the demographically adjusted unemployment rate less the NAIRU–performed better in estimation than did output gaps, and so the former appears in equation (A) of the table. We then supplemented the empirical model with a simple Okun’s Law relationship, equation (C), and then substituted it in order to arrive at the appropriate estimatesfortheequations(26)through(30). The equation of primary interest is our Phillips curve. As equation (A) in the table shows, wesupplementedthebasicformulationwithasmallnumberofexogenoussupplyshockterms, includingoilprices,avariabletocapturetheeffectsoftheNixonwage-and-pricecontrols,and aconstantterm. Thesearetraditionalanduncontroversialinclusions. Roberts(1995)hasfound oilprices tobeimportantforexplaininginflationinestimation,usingMichigansurveydata. Thekeyparametersarethe‘contractduration’parameter, ^ Æ ,andtheexcessdemandparameter, ^(cid:13) . If this were a level contracts model, Æ = : 4 1 would be an unreasonably low number since it implies a very short average contract length. For the present model, this interpretation 23Theestimatedinterestreactionfunctionisbasedondatarangingfrom1980Q1to1998Q4toreflectthenow relevantpolicyregimespannedbytheVolcker-GreenspanchairmanshipsoftheFederalReserveBoard. 23

Table2: Estimates oftheBasicContract Model (1972Q1-1996Q4) A. (cid:25) = [1 (cid:0) ( 1 (cid:0) Æ ) 2 (cid:0) ] 1 [Æ (cid:25) (cid:0)t 1 + ( 1 (cid:0) Æ ) 2 (cid:13) u (cid:0)t 1 + ( 1 (cid:0) Æ ) c t;t+ 1 ] + (cid:0) Z ab Nixonpricecontrols ^Z 1 -5.35 (2.20) R 2 =.97 Change in oil prices ^Z 2 0.0019 (.70) SEE=1.02 Unemployment ^(cid:13) (cid:0) 0.23 (1.49) B-G(1)=.01 Contract duration ^Æ 0.41 (4.65) ConstrainedlinearIV B. y = (cid:30) 0 + (cid:30) 1 y (cid:0)t 1 + (cid:30) 2 y (cid:0)t 2 + (cid:30) 3 r l (cid:0)t 1 Firstlagoutput (cid:30) 1 1.22 (12.16) R 2 =.88 SEE=1.21 Second lagoutput (cid:30) 2 (cid:0) 0.36 (4.02) B-G(4)=0.05 Real5-yearrealrate (cid:30) 3 (cid:0) 0.26 (2.41) OLS C. u = (cid:13) 0 + (cid:13) 1 y t + (cid:13) 2 T + (cid:13) 3 ( p o i lt = p t ) Outputgap ^(cid:13) 1 (cid:0) 0.34 (9.01) R 2 =0.80SEE=0.60 Timetrend ^(cid:13) 2 0.008 (1.81) B-G(4)=0.00 Relativeoilprice ^(cid:13) 3 0.57 (3.07) 2SLS D. r = ( 1 (cid:0) g r ) ( r (cid:3) (cid:0) (cid:25) (cid:3) ) + g r r (cid:0)t 1 + ( 1 (cid:0) g r + g (cid:25) ) (cid:22)(cid:25) t + g y y t Inflation ( 1 (cid:0) g r + g (cid:25) ) 0.324 (2.37) R 2 =0.90SEE=1.23 OutputGap g y 0.148 (2.22 B-G(4)=0.00 InterestRate g r 0.803 (13.8) RestrictedOLS aData: changeinoilpricesisafour-quartermovingaverageofthepriceofoilimportedintothe U.S.;isthequarterlychangeatannualratesofthechain-weightGDPpriceindex;isthedemographicallycorrectedunemploymentrate,lessthenaturalrateofunemploymentfromtheFRB/USmodel database;isproxiedbythemedianoftheMichigansurveyofexpectedinflation,12monthsahead; is the output gap for the U.S. from the FRB/US model database; is the real interest rate defined as the quarterlyaverageof the federalfundsrate less a four-quartermovingaverage of the chainweightGDPpriceindex;isthepriceofimportedoilrelativetheGDPpriceindex;andNixonprice controlsequalsunity in 1971Q4and-0.6 in 1972Q1. All regressionsalso includedan unreported constantterm. Constantswereneverstatisticallysignificant. B-G(1)istheprobabilityvalueofthe Breusch-Godfreytestoffirst-orderserialcorrelation. bNotes: Equation(A)isestimatedwithinstruments:constant,timetrend,laggedunemployment gap, fourlags of the changein importedoilprices; two lags of inflation, lagged realinterest rate, lagged Nixon wage-price control dummy, and the lagged relative price of imported oil. Standard errorsforallthreeequationswerecorrectedforautocorrelatedresidualsofunspecifiedformusing theNewey-West(1987)method. 24

is not warranted, however. An estimate of Æ = : 4 1 implies substantial persistence in inflation, much more so than any nominal wage contracting model could furnish. In fact, when equation (A) is solved, its reduced-form coefficient on lagged inflation is seen to be 0.846. This is substantialinflationstickinessbyanymeasure. Turningtotheaggregatedemandfunction,it isconventionallybelievedthatdemand inthe U.S. responds to movements in the long-term real interest rate.24 Accordingly, we define the ex ante real interest rate, R t , as the five-year government bond rate less the average inflation rate that is expected over the next five years, and compute the latter using a small-scale vector autoregression.25. Five years is about the time period for which consumer durables and automobiles are typically financed. The duration, D, in equation (29), is set at 20 quarters in conformationwiththedefinitionofR. The estimates of the aggregate demand function show the humped shape pattern of responses to output to demand shocks; that is, an exogenous disturbance to demand tends to overshoot initially—as determined by ^(cid:30) 1 = 1 : 2 2 > 1 —and then drop back, as indicated by ^ (cid:30) 2 = (cid:0) : 3 6 < 0 . Theinterestelasticityofaggregatedemandis largeand negativeas expected. Aftersubstitutingequation(C)intoequation(A)anddroppingthoseargumentsthatarenot of interestto us, wearrivewhere we began: withequations (26)through (30). The parameters of the estimated model are broadly similar to estimates of other models, and are reasonable. Impulse responses of the model to exogenous shocks to the price equation and the aggregate demandfunctionareconsistentwiththehistoricalexperienceintheU.S.asmeasuredbysimple vectorautoregressions. Theestimatedequationsdoshowsomeremainingresidualcorrelation. This is a common finding in structural price equations—Roberts (1995) uncovered the same phenomenon—so as noted above, we have corrected the variance-covariance matrix for this autocorrelation using the Newey and West (1987) technique. We conclude that our model is appropriateforthequestionswewishtoaddress. 4 Robust Policy Exercises The monetary policy authority’s task is to keep inflation, (cid:25) , close to its target level, taken without loss of generality to be zero, and to do so at minimum cost in terms of output losses, y , defined as thepercent deviation of GDP from trend. We also includea third target variable, 24See FuhrerandMoore(1995a)foranextensivediscussionofthe linkagebetweenmonetarypolicyandthe long-terminterestrate. 25This is the same methodology as employed in the FRB/US macroeconomic model of the US, built and maintainedbythe FederalReserveBoard. Formoreinformation,see BraytonandTinsley(1996)andBrayton, Mauskopf,Reifschneider,TinsleyandWilliams(1997) 25

the change in the short-term interest rate, (cid:1) r t , because in the present model, as in most other models, the parameters in the interest rate reaction function are absurdly large in the absence ofsomepenaltyon instrumentvariability. Themodelisimplicitlyderivedfromfirst-orderconditionsdeterminedbyoptimizingagents whoformrationalexpectationsbysolvingthemodel,whichtheyassumetobetrue. Incontrast, the central bank is assumed to have doubts about that model as a true representation of the economy.26 In terms of the generic model outlined in Section 2.1, the state vector of the preceding model is defined by, x t+ 1 = [ y t ; (cid:25) t ; r t ; y t(cid:0) 1 ; (cid:25) t(cid:0) 1 ; r t(cid:0) 1 ; (cid:25) t(cid:0) 2 ; (cid:25) t(cid:0) 3 ; c t+ 1 ; R t+ 1 ] 0 . The instrument is r t , and the target vector is, T t = [ y t ; (cid:25) t ; (cid:1) r t ] 0 . The selector matrices for the mapping from statesto targetsin (2)and theweightingmatrix Q , are M x = 2 6 6 6 4 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 (cid:0) 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 7 7 7 5 ; M u = 2 6 6 6 4 0 0 1 3 7 7 7 5 ; Q = 2 6 6 6 4 p (cid:13) 0 0 y p 0 (cid:13) 0 (cid:25) p 0 0 (cid:13) (cid:1) r 3 7 7 7 5 We will distinguish two cases of preference orderings over inflation and output performance: (1)acentralbankwith“strong”inflationpreferences,i.e.,onethatputsrelativelymore emphasis on controlling inflation, and (2) a central bank with “weak” inflation preferences, i.e., one that puts relatively more emphasis on controlling output. Formally, the preference parameters are, Strong: (cid:13) y = : 2 , (cid:13) (cid:25) = : 8 ,and (cid:13) (cid:1) r = : 0 1 ; Weak: (cid:13) y = : 8 , (cid:13) (cid:25) = : 2 , and (cid:13) (cid:1) r = : 0 1 . In each case, a miniscule weight is placed on interest rate changes. The optimal control literatureaswellasmostpapersonrobustcontroltraditionallyplaceemphasisonsynthesizing fully parameterized feedback rules. Given the strong interest in simple, Taylor-style interest raterulesandtheimpracticalityofoptimalcontrolsolutionswithlargemodels,weshallrestrict our attention to simple robust rules that respond to the previous period’s movements in only three state variables, the output gap, y , the inflation rate, (cid:25) , and the federal funds rate, r . Obviously, this restriction imposes an additional burden of choosing a suitable subset of the state variables, an issue that is still under debate as the growing literature on simple monetary policy rules demonstrates. Experience suggests that the cost of such parsimony appears to be negligible. For example, Tetlow and von zur Muehlen (1996) show that for a model very similarto theone used here, the differences in welfare loss resulting from optimalcontrol and 26We assumethat,whilethemonetaryauthorityisrisk-sensitive,agentsintheeconomybelievethereference modeltobetrue. HansenandSargent(1999b)assumethatagentsandtheauthoritysharethesamepreferencefor robustness,indexedbyasingleparmater, (cid:18) . 26

optimalsimplerules aresmall. The parameter vector, K , of the policy rule introduced in section 2.1 becomes [ g r ; g (cid:25) ; g y ] K = , multiplying the lagged values of the federal funds rate, the inflation rate, and the outputgap,respectively. Recall that in section 2, we defined five optimization problems, three associated with unstructureduncertainty,andtwowithstructureduncertainty. Table3servesasareminderofthe methods utilized in what follows. Under the heading “Unstructured Uncertainty,” the authority chooses a feedback rule, u t = K x t that minimizes the three loss functions H 2 , H 1 , and ‘ 1 . Under the heading “Structured Uncertainty,” the authority chooses K that maximizes the radius, ^r , ofallowableperturbations,as previouslydefined. Table3: Criteria forRobustPolicies Criteriafor Criteriafor UnstructuredUncertainty StructuredUncertainty NTV LTI m in H 2 ; m in H 1 ; m in ‘ 1 ^r = m a x ^(cid:26) 1 ^ ( N ) ^r = m a x ^(cid:22) ( 1 G 2 2 ) Weconsidertwowaysoflookingattheconsequencesofapplyingthevariousrobustpolicy rules derived here to the estimated model. In each case, we compare the derived rule with the estimatedruleortheoptimal(restricted)linear-quadraticrule. The first and most obvious measures of performance are the rules themselves, their implicationsforthesteadystate,andtheimpliedlosses. Inthecaseofstructuredmodeluncertainty, the allowable radius of perturbations (the inverse of the minimum structured norm) gives a measure of the size of maximum uncertainty the authority is capable of tolerating. Since in most cases, this may be too strict a criterion, a preferred interpretation of the “maximum of allowable perturbations” as an upper bound. A wider range of policies becomes available if the central bank considers the opponent to be less malevolent, that is, if uncertainty can be boundedina smallerset. Asecondwaytoviewtheimplicationsofvariouspoliciesistoevaluate H 2 welfarelossesat variousfrequencies,makinguseofthedefinitionin(14). Forexample,whiletheoptimallinearquadraticruleperformswellforseriallyuncorrelatedshocksthatgiverisetocyclesatquarterly to2-yearfrequencies,itturnsoutthatitleavespolicyvulnerabletosmallmispecificationsofthe temporal and feedback properties ofshocks affecting businesscycle frequencies. A surprising 27

finding is that this is particularly true of historic policy as represented by our estimated rule and filtered throughthemodel. 4.1 Unstructured model uncertainty The sets of rules that stabilize performance of the target variables y , (cid:25) , and in a minor way, (cid:1) r , were calculated for H 2 , H 1 , ‘ 1 losses. We omit minimumentropy results since it is clear thattheyconstituteintermediateoutcomesbetween H 2 and H 1 policies. Tables4and5showthedetailedresultsfor“strong”and“weak”anti-inflationpreferences, respectively. The left-hand column indicates the norms imposed on the loss function and uncertainty; the next column to the right shows the induced norm, i.e., the indirect loss function impliedbythefirstcolumn;thenextthreecolumnspresenttheimpactcoefficientsofthecalculatedoptimalrule, andthefollowingtwocolumnspresenttheimpliedequilibriumreactionsto inflationand output;thelargest stableroots arelistedunder“spectral radius.” Thelastcolumn givestheimplied H 2 losses, which measurethe welfare loss under theoriginal LQR criterion. Thismeasureshowmuchperformancetheauthoritygivesupifitistrulyparanoidandtheonly shocks the economy experiences are those of the estimated variance covariance matrix. This columnshowsthatrobustnessagainstworst-caseoutcomescanonlybeachievedattheexpense ofaverageperformance. Interestingly,theestimatedrulegenerates thehighest H 2 loss. The H 2 LQR rule, Both tables show that the LQR rule, while less autoregressive than the estimated rule, has significantly larger reaction coefficients for inflation and output than the estimated rule. This nowfamiliarresulthas been thespark for muchofrecent writingon policytimidity,including thepresentpaper. The H 1 robust rule Theresultsfor H 1 policiesareabitmoresubtleanddependontheassumedinflationfighting attitude of the authority. A “strong” inflation fighter practices a policy of anti-attenuation, reacting to outputfluctuationsand especially discrepancies ofinflationfrom target withgusto. The “weak” inflation targeting authority is more circumspect: Its policy rule is much more comparable to the certainty equivalent rule. In addition, the “weak” authority tends to show a substantially greater degree of persistence—or instrument smoothing— in its setting of the fundsrate,asshownbythelargecoefficientonthelaggedfundsratetermintherule. Thistoo, ismuchlikethecertainty-equivalent(LQR)rule. Thekeytounderstandingthisresultisrecognizing that outputcontrol appears earlier in the monetarypolicy transmissionmechanism,and thatit isgovernedby an AR(2)process: Controlof outputoperates more-or-less directly—but 28

withpersistence—whilecontrolofinflation,toafirstapproximation,mustbecarriedoutindirectlythroughmanipulatingoutput. Anauthoritythatcaresrelativelyagreatdealaboutoutput will, in the presence of large and persistent (perceived) shocks tend to match that persistence, using the persistence in output itself, along with persistent movements in the funds rate. That this may produce poor outcomes in terms of inflation is less of a worry since inflation carries a low weight in the loss function. A “strong” inflation fighting central bank, by contrast, will tend to aggressively adjust the funds rate to move output, and work against the persistence in output,to attack inflationanytimeitdeviatesfrom target. It is worth noting that the results reported here for robust control with unstructured uncertaintyechotheresultsinpreviousresearch. Sargent(1999),forexample,findsthatrobustrules appliedtothebackward-lookingopen-economymodelofBall(1998)aremoreaggressivethan the optimal LQR rule, and become increasingly aggressive as risk sensitivity approachs that implied by an optimal H 1 solution. Similarly, Onatski and Stock (2000), find that ‘ 1 and H 1 criteria produce more aggressive feedback rules in the Rudebusch and Svensson (1999) backward-lookingclosed-economymodeloftheU.S. economy. Table4: Rules under Shock and Model Uncertainty:“Strong” InflationPreference Bounds Induced ParametersintheRule Spectral Implied Loss/Perturbation Norm Impact Equilibrium SpectralRadius H 2 Loss r (cid:22)(cid:25) y (cid:22)(cid:25) y LQR H 2 0.73 4.24 1.81 16.7 6.7 0.82 3.8 ‘ 2 l o s s = ‘ 2 s h o c k ; H 1 0.25 16.6 8.5 23.1 11.3 0.65 5.0 ‘ 1 l o s s = ‘ 1 s h o c k ‘ 1 0.53 9.60 5.23 21.4 11.2 0.73 4.2 EstimatedRule 0.80 0.32 0.15 2.6 0.8 0.88 5.8 Table5: Rules under Shock and Model Uncertainty: “Weak”InflationPreference Bounds Induced ParametersintheRule Spectral Implied Loss/Perturbation Norm Impact Equilibrium SpectralRadius H 2 Loss r (cid:22)(cid:25) y (cid:22)(cid:25) y LQR H 2 0.66 1.89 3.51 6.6 10.3 0.79 3.8 ‘ 2 l o s s = ‘ 2 s h o c k H 1 0.70 2.10 2.35 8.0 7.8 0.76 5.0 ‘ 1 l o s s = ‘ 1 s h o c k ‘ 1 0.25 5.88 16.4 8.8 21.9 0.84 6.6 EstimatedRule 0.80 0.32 0.15 2.6 0.8 0.88 7.6 The ‘ 1 (MAD)robust rule Under the ‘ 1 loss criterion, the authority minimizes absolute target deviations. As stated earlier, this criterion is the induced loss function when the authority minimizes the maximum absolute value of target deviations, j j z j j 1 , against the largest possible shock satisfying j j w j j 1 < 1 . 29

The tables show that MAD policies are universally more reactive than the LQR rules and less aggressive than H 1 policies under strong inflation preferences than under weak inflation preferences,bothwithrespecttoautoregressivityoftheinterest-rateruleandthesizeofimpact reactionparameters. Incomparingthesetwopolicies,notethatunder H 1 control,thedecision maker protects the average square metric of performance against an average square metric of misspecificationerrors, whilein aMADrule, theattemptisto guardagainstoutlieroutcomes. Giventhecharacteroftheestimatedpolicychannelfrominterestratemovementstooutputand inflation in this model, it appears that a strong preference for preventing economic booms or busts produces faster and stronger responses to signals from the economy than a concern for smootheconomicbehavior. Frequency decompositions oflosses To illuminate how preference for robustness manifests itself in the potential performance oftheauthority’sgoals,itisusefultoplotthe H 2 lossesimpliedbythefouralternativerules.27 Figure1showsthisforthestronginflationtargetingauthority. Thecurvescapturehowdifferent attitudestowardrobustnessaffectaverageperformance,undertheassumptionthatthereference model is correct. A relatively flat curve suggests that the authority has managed to insulate itself against shocks in a broad range of frequencies, while a curve that has a peak—power concentrated within a narrow frequency band—indicates that the authority is vulnerable to shocks affecting those frequencies. By assumption, nature could choose to concentrate its choiceofshockstoattack thatrange TheLQRrule,which,bydefinitionhasthesmallestareabeneathit,hasapronouncedpeak atafour-yearfrequency. Bycontrast,the H 1 andMADpoliciestendtomoreevenlydispersed overallfrequencies. Asmightbeexpected,theMADrule,beingthemostrisksensitive,isthe flattest, followed by the H 1 rule. Under these criteria, the authority surrenders some average performancebyincreasingtheareaunderthecurveinordertominimizethemaximumloss—in otherwords,thepeak. In contrast to the risk sensitive policies just discussed, which tend to immunize losses againstadverseoutcomesconcentratedatparticularfrequencies,theestimatedruleimplies H 2 lossesthatarestronglyclustered aroundeightyears, leavingthepolicymakermostvulnerable at typicalbusinesscyclefrequencies. 27Technically,theplotsshow t r a c e [G 0 ( (cid:24) ) G ( (cid:24) ) ] ,whichmeasuresthe H 2 lossatfrequency ! . G ( (cid:24) ) measuresthe amplificationofshocksasmeasuredbytheperformanceofthelossfunctioninfrequencyspace. Thehorizontal axisdisplaysquarters,andtheverticalaxisshowsthevalueoftheloss. 30

4.2 Structured model uncertainty In this section,we offeranswers to questionslike: “what shouldrobustpolicybe ifthecentral bankfearsmisspecificationofaparticularparameter,suchastheslopeofthePhillipscurve?”28 or,“whatisthebestpolicyagainstlossesarisingfromworst-caseconsequencesofmisspecified dynamics?” Misspecified or unmodelled dynamics are perhaps the most common sources of misspecification, given the difficulty of fully capturing all lagged effects in typical economic models. AsindicatedinSection2.2.2,weworkwiththereducedformbasedonthestructuralmodel, givenaspecificationofthepolicyrule. Inourcase,itturnsoutthatthetoprowofthereducedform matrix, (cid:5) , reproduces the structural IS equation with zero elements corresponding to all other state variables not specified in (28). This means that the reduced-form parameters for the IS curve are independent of the parameters in the remainder of the model. Any presumed parameteruncertaintyintheremainderofthemodelhas noeffect ontheuncertaintyinvolving the IS curve parameters. The remaining parameters in (cid:5) are nonlinear functions of all other parameters in the structural model, so that perturbations to any parameter in the structural model will affect parameters in all but the first row of the reduced form, with the exception of the own parameters in the IS curve: a perturbation to the slope of the IS curve, for example, affects only that parameter in the IS curve and all the elements in the second to eleventh rows of (cid:5) . The exercises, while not specifically directed at perturbing structural parameters (as they would be in a backward looking model), can be interpreted as such in ways to be spelled out presently. Four interesting cases can be distinguished: (1) perturbations to all parameters, (2) perturbations to lagged output parameters, (3) perturbations to the slope of the IS curve, and (4) perturbations to the slope of the Phillips curve. As outlined in the theoretical sections, we distinguishthreetypesofmodelperturbations. TheLTI-scalarcaseismostrestrictive,limiting uncertaintytoverysimpleevents,suchasone-timestructuralbreaksinoneormoreparameters. Nonlinear time-varying (NTV) perturbations reflect the most general case of model structure uncertainty. An intermediatecase is LTI-MA, which assumes that the approximationerrors of selected parameters can be modeled as infinite movingaverages representing misspecified lag structuresin themodel. Theresultsforthesecases aresummarizedin Table6.29 28ThesourceofmisspecificationoftheslopeofthePhillipscurve, ( 1 (cid:0) Æ ) 2 (cid:13) ,maybemismeasurementofthe outputgap.Wemaydecompose y as y = lo g Y (cid:0) lo g ^Y (cid:0) (cid:15) y ,where Y and ^Y arethelevelsofactualandcapacity output,respectively,and (cid:15) y isthemeasurementerrorofcapacityoutput. Thus,ifonecannotdistinguishbetween ashockto y andashockto ( 1 (cid:0) Æ ) 2 (cid:13) ,thenuncertaintyaboutmeasurementisreflectedinuncertaintyaboutthe slope. Asnotedintheintroduction,Orphanides,Porter,Reifschneider,TetlowandFinan(2000)findthatoutput gapuncertaintyleadstoattenuationofinterestratereactionsifthemonetaryauthorityminimizesBayesianrisk. 29Therelevantaugmentedoutputvector(seeequation(19))is p t = [(cid:25) (cid:0)t 1 ; r (cid:0)t 1 ; y (cid:0)t 1 ; (cid:25) (cid:0)t 2 ; (cid:25) (cid:0)t 3 ; r r lt 0 ] ,and U 31

Table6: Interest RateRules withStructured Model Uncertainty ParametersintheRule (1) (2) (3) (4) (5) (6) (7) (8) Typeof Impact Equilibrium SpectralRadius Radiusof Implied Perturbation r (cid:22)(cid:25) y (cid:22)(cid:25) y (cid:22)(cid:26) Perturbationsa H 2 L o s s b PerturbationstoAllModelParameters (1) LTI scalar 0.05 1.05 -1.30 2.1 -1.4 0.89 0.24 17.3 (2) LTI MA 0.92 0.08 0.39 1.1 4.8 0.88 0.21 13.6 (3) NTV 0.65 0.74 0.78 3.1 2.2 0.76 0.15 6.6 Perturbationstolaggedoutputparameters( (cid:30) 1 ; (cid:30) 2 ) (4) LTI scalar 0.84 0.16 0.12 2.0 0.8 0.90 0.31 10.8 (5) LTI MA 0.93 0.07 0.38 2.0 5.4 0.89 0.22 14.9 (6) NTV 0.65 0.78 0.82 3.2 2.3 0.76 0.16 6.4 PerturbationstoslopeofIScurve( (cid:30) 3 ) (7) LTI scalar 0.44 1.24 0.65 3.2 1.1 0.77 0.22 4.9 (8) LTI MA -0.08 1.08 -0.48 2.0 -0.4 0.86 0.23 11.4 (9) NTV 0.64 0.73 0.71 3.2 2.0 0.76 0.19 4.8 PerturbationstoslopeofPhillipscurve( Æ ; (cid:13) ) (10) LTI scalar -.97 1.97 -2.29 2.0 -1.2 0.97 0.39 13.8 (11) LTI MA -0.98 1.98 -0.91 2.0 -0.5 0.98 0.24 13.0 (12) NTV 0.58 0.83 0.70 3.0 1.7 0.75 0.20 4.9 (13) L Q R b 0.66 1.89 3.51 6.6 10.3 0.79 3.8 (14) EstimatedRule 0.80 0.32 0.15 2.6 0.8 0.88 7.6 aTheradiusofallowableperturbationsistheinverseoftheimputed ‘ 1 loss. bEvaluatedforweakinflationtargetingpreference. ischosenappropriately.Thecalculationsforthissectionassumethatperturbationstothestructuralparameterslie approximatelywithintwostandarddeviationsoftheestimates. 32

To jump to the punchline, as shown in Table 6, in contrast to robust policy against combined model and shock uncertainty, rules that are robust to structured model uncertainty are less aggressive than optimal LQR rules, in some cases approaching the estimated rule to a remarkable degree. A convenient measure of the relative aggressiveness of policy is the set of equilibrium response coefficients exhibited in columns 4 and 5. Compared with the LQR response to inflation of 6.6, shown in row 13, the inflation responses of the robust rules are considerably weaker, ranging from 1.1 to 3.2. A similar conclusion holds for responses to the output gap. By contrast, the equilibrium responses of the robust rules are much more closely aligned with the estimated rule (row 14), especially in the case of the scalar LTI perturbations to the lagged output coefficients of the aggregate demand function in row 4. In general, there appears to be a rough hierarchy from more to less aggressive, depending on how much structure is imposed on model uncertainty. Policies based on the greatest amount of structure, the LTI rules, are the most attenuated, while policies based on the least amount of structure, the NTV rules, are the most aggressive.30 This finding allows us to put into context our earlier findings on attenuation with unstructured uncertainty with the Bayesian approaches surveyed above: it appears that the denser the fog of Knightian uncertainty, the greater the tendency for decisiveaction. Smallerand morelocalizeduncertaintiesbringaboutcautiousbehavior, while moregeneralized ambiguitieselicitstrongerresponses. Structural breaks The rows marked “LTI scalar,” show robust rules when the authority attempts to avoid worst-case outcomes resulting from one-time permanent structural breaks in either the entire model, or in selected parameters. In each of thesecases, policies become substantiallyattenuated;thatis,thecoefficientsareuniformlylowerthantheLQRcoefficientsdisplayedinrow13 of the table. In particular, robust rules that protect against breaks in the parameters of the aggregatedemand function—rows4 and 7—giveequilibriumruleparameters that are veryclose totheestimatedrule. Thesameresultappliesforbreaks intheslopeofthePhillipscurve(row 10), and in all model parameters simultaneously (row 1)—but only for inflation coefficients: One-timeperturbationstotheslopeofthePhillipscurveshowninrow10leadto“perverse”responsestooutputowingtothepossiblebreakdownofthenormallead-lagrelationshipbetween outputandfutureinflation. Misspecified orunmodeled dynamics Policiesthataresensitivetomisspecifiedorunmodeleddynamicsarerepresentedbylinear 30This conformswith Onatski and Stock (2000), who also find optimalpolicies becomingless aggressive as morestructureisplacedonuncertainty. 33

time-invariant moving-average (LTI-MA) perturbations to any or all parameters in the model (rows 2, 5, 8 and 11). In general, theLTI-MA rules produceequilibriumresponses to inflation that are close to that of the estimated rule, but with wide variations in equilibrium output gap coefficients. Considering, once again, the case of perturbations to the persistence parameters of the aggregatedemand function, (cid:30) 1 and (cid:30) 2 , shown in row 5, we see that the resulting policy alsocomesclosetotheestimatedrule,althoughitproducestoomuchpersistenceandtoomuch aggressivenessinresponsetotheoutputgap. Nonlineartime-varying perturbations The most noteworthy thing about the NTV rules shown in Table 6 (rows 3, 6, 9 and 12) is how similarthey are notwithstandingthedifferent origins of theperturbations considered. All the rules have impact coefficients on output and inflation of about 0.7 or so, and a coefficient on the lagged federal funds rate of about 0.6. Moreover, the implied equilibrium coefficients are all quite close to those of the estimated rule, particularly for inflation. However, the computed impact coefficients are too large, and persistence parameters are too small, to match the estimatedrule. Both the similarity of the rules across cases and the lack of persistence are manifestations of two aspects of the model uncertainty that are assumed: The perturbations have very little temporal stability given their non-linear and time-varying nature, and no persistence. The formerimpliesthattheoriginsoftheshocksareoflittleuseforthedesignofpolicy—resulting inahomogeneityofpolicydesign. Thelattersuggeststhatwithoutpersistenceindisturbances, thereislittlebenefit to persistenceinpolicy. Robustness andits costs Before leaving this topic, it is worth reflecting on the extent to which the policies in Table 6 protect against large shocks, and the costs of this protection. The degree of protection is measured by the radius of perturbations shown in column 7. The higher the number in the table,thelargertheperturbationsthattheapplicablerulecanwithstand. Notsurprisingly,there is a tendency that the broader the scope of shocks the authority wants to protect against— such as in the NTV cases—the smaller the shock that can be protected against: Coverage against LTI-scalar shocks is shown to be greater than coverage against NTV shocks. Column 8,however,showsthecostofthisprotectioninthoseinstanceswherethereferencemodelturns outtobetrue;thatis,whentheeconomyactuallyfacesonlytheestimatedshocks. Thiscolumn of the table demonstrates that in most cases, the protection against one-time structural shifts comes at a substantial price. This contrasts with the NTV cases where losses are uniformly lower. Evidently, a tinge of apprehension about generalized misspecification comes at a low 34

costrelativeto stronganxiety aboutuncertaintyofaspecifictype. Frequency decompositionoflosses Finally, we turn to the decomposition of losses, analogous to those of Figure 1, discussed previously. Figures 3 and 5 show the susceptibility of the rules discussed in Table 6 to perturbations to all parameters simultaneously, and to the lagged output parameters. These are the cases that span the shocks to model persistence. Notice that the LTI-scalar and especially the LTI-MA rules have patterns that closely match those of estimated rule. This observation confirms the impression left by Table 6 that the estimated rule may have been the outcome of efforts to protect against misspecified output persistence. Broadly similar, if less pronounced, resultsare obtainedfortheothersources ofperturbationsshowninFigures 7 and9. The most dramatic conclusion for robust policy under structured model uncertainty is that almost as a general rule, the estimated rule and the policies that best mimic it, are the most vulnerable to disturbances that produce phenomena at business cycle frequencies. Arguably, thesearejustthesortofshocksthatcentralbankerswouldmostoftenworryabout. Theexception are the NTV perturbations shown as the dotted lines in the figures. Taken at face value, theseobservationssuggestthatifcentralbankscanovercomeestimationproblemswithrespect to such things as one-time breaks in trends, the remaining small but exotic perturbations can be handled well with rules that are robust, plausible, and protect against phenomena at business cycle frequencies. Moreover such rules are not costly to implement in terms of foregone performancein worldswherethereference modelistrue. 5 Concluding Remarks We began this paper by reflecting on a puzzle: if monetary policy seeks to minimize output and inflation fluctuations, how does one explain the fact that historical interest-rate responses tothesetwoindicatorshavebeen farmoremutedthansuggestedby optimalpolicyrules? Wehavefound,asothershave,thatoptimallinear-quadraticrulesderivedintheabsenceof modeluncertaintyare indeedmorereactivethanrules estimatedondatafortheUnitedStates. Stabilizingamonetaryeconomyisadifficultjob. Theauthorityhasbutoneinstrumentand usuallyat least twotargets. Theinstrumentworkswitha lag. Moreover,theauthorityfaces an economy that is constantly changing, resulting in profound uncertainties regarding estimated structural parameters. Can such uncertainties explain the observed attenuation of policy? Our resultssuggestthattheanswerisyes and no. We did find rules that protect against a class of specification errors, modeled as structured perturbations to a reference model, that resemble the estimated rule. However, we also found 35

that robust policy rules that seek to guard against very general forms of misspecifications are evenmorereactivethanthelinear-quadraticrule. Therobustrulethatcomesclosesttoapproximating the estimated rule is one that seeks to guarantee a minimum level of stability against worst-case specification errors in the dynamics of aggregate demand. It follows that one possible interpretation of Fed behavior of the last twenty years is that the observed attenuation in policy was motivatedby distrust regarding the estimated degree of output persistence. This motivationarisesinlargepartbecausetheaggregatedemandfunctiondeterminesthedominant root—andhencethestability—ofthemodel. Giventhattheaggregatedemandfunctionisspecifiedintermsofexcessdemand—meaning output relative to potential output—the literature on mismeasurment of potential output is of pertinence here. Work by Orphanides (1998), Smets (1999), Orphanides et al. (2000), and Tetlow (2000), among others, shows that potential output can be badly mismeasured and that correcting the measurement error can take a long time. Such errors could easily show up as mismeasuredpersistencein an aggregateddemandfunction. But if uncertainty of a particularstructure can explain observedFed behavior, what can be said about more generalized uncertainty? Our results suggest a hierarchy of policy responses measured in terms of attenuation or anti-attenuation indexed against the assumed degree of structureinKnightianuncertainty: Thegreaterthestructureontheuncertainty,themorelikely policyattenuationislikelytoarise. Atthesametimehowever,themorestructureisassumedin theperturbationsthe authorityfaces, thelarger thelosses that are borneifthe robustnessturns outto havebeen unnecessary. 36

References Anderson, G., Moore, G., 1985, A linear algebraic procedure for solving linear perfect foresightmodels,EconomicsLetters17, 247–252. Anderson, G. S., 2000, A reliable and computationally efficient algorithm for imposing the saddlepointpropertyindynamicmodels,Federal ReserveBoard FinanceandEconomics DiscussionSeries forthcoming. Aoki,M.1967, Optimizationofstochasticsystems,AcademicPress, New York. Ball, L. 1998, Policy rules for open economies, in: J. B. Taylor ed., Monetary policy rules, U(niversityofChicago Press, Chicago), 127–144. Basar, T., Bernhard, P. 1991, H 1 (cid:0) optimal control and related minimax design problems, (Birka¨user, Boston-Basel-Berlin). Blinder, A.1998, Central bankingintheoryand practice, (MIT Press, Cambridge). Brainard, W., 1967, Uncertainty and the effectiveness of policy, American Economic Review 58,411–425. Brayton, F., Mauskopf, E., Reifschneider, D., Tinsley, P., Williams, J., 1997, The role of expectationsintheFRB/US model,Federal Reserve Bulletin83,227–245. Brayton, F., Tinsley, P. A., 1996, A guide to FRB/US: a macroeconomic model of the United States, Federal Reserve Board Financeand EconomicsDiscussionSeries 1996-42. Calvo, G. A., 1983, Staggered prices in a utility-maximizingframework, Journal of Monetary Economics12, 383–398. Caravani,P.,1995,On H 1 criteriaformacroeconomicpolicyevaluation,JournalofEconomic DynamicsandControl 19,961–84. Caravani, P., Papavassilopoulos, G., 1990, A class of risk-sensitive noncooperative games, JournalofEconomicDynamicsandControl 14,117–149. Chow,G.C.1975,Analysisandcontrolofdynamiceconomicsystems,(JohnWileyandSons, NewYork). Craine, R., 1979, Optimal monetary policy with uncertainty, Journal of Economic Dynamics and Control1, 58–93. 37

Dahleh, M. A., Diaz-Bobillo, I. J. 1995, Control of uncertain systems: a linear programming approach, Prentice-Hall, EnglewoodCliffs. Doyle, J., Glover, K., Khargonekar, P., Francis, B., 1989, State space solutionsto standard H 2 and H 1 controlproblems,IEEE Transactionson AutomaticControl 34,831–847. Drazen, A. 2000, Political economy in macroeconomics, (Princeton University Press, Princeton). Easley, D., Kiefer, N., 1988, Controlling stochastic processes with unknown parameters, Econometrica56,1045–1064. Estrella, A., Mishkin, F. S. 1998, Rethinking the role of NAIRU in monetary policy: implicationsofmodelformulationanduncertainty. Unpublishedmanuscript. Ferguson, T.1967,Mathematicalstatistics,(AcademicPress, New York). Fuhrer, J., Moore, G., 1995a, Inflation persistence, Quarterly Journal of Economics125, 127– 159. Fuhrer, J., Moore, G., 1995b, Monetary policy trade-offs and the correlation between nominal interestrates and real output,American EconomicReview85, 219–239. Giannoni, M. P., 2000, Does model uncertainty justify caution? Robust optimal monetary policyin aforward-lookingmodel,MacroeconomicDynamics. Glover, K., Doyle, J., 1988, State-space formulae for all stabilizing controllers that satisfy an H 1 -norm bound and relations to risk sensitivity, Systems and Control Letters 11, 162– 172. Goodfriend, M., 1991, Interest rate smoothing in the conduct of monetary policy, Carnegie- RochesterConference Series onPublicPolicy34,7–30. Hansen, L. P., Sargent, T. J. 1999a, Five games and two objective functions that promote robustness. Unpublishedmanuscript,U. ofChicago and HooverInstitution. Hansen, L., Sargent, T. J., 1995, Discounted linear exponential quadratic gaussian control, IEEETransactionsonAutomaticControl40, 968–971. Hansen, L., Sargent, T. J. 1999b, Robustness and commitment: a monetary policy example. Unpublishedmanuscript,U. ofChicago and HooverInstitution. 38

Hansen,L.,Sargent,T.J.,Tallarini,T.D.,1999,Robustpermanentincomeandpricing,Review ofEconomicStudies66,873–907. Johansen, L. 1973, Targets and instruments under uncertainty, in: H. C. Boss, H. Linneman, P.deWolffeds.,Economicstructureanddevelopment: essaysinhonorofJanTinbergen, (North-Holland,Amsterdam), 3–20. Johansen,L. 1978,Lectures onmacroeconomicplanning,(North-Holland,Amsterdam). Kasa, K., 2000, Model uncertainty, robust policies, and the value of commitment, MacroeconomicDynamics. Levin, A., Wieland, V., Williams, J. 1999, Robustness of simple monetary policy rules under model uncertainty, in: J. B. Taylor ed., Monetary policy rules, (University of Chicago Press, Chicago), 263–309. Macaulay, F. R. 1938, Some theoretical problems suggested by the movements of interest rates, bond yields, and stock prices in the United States since 1956, (National Bureau of EconomicResearch, Cambridge). Mercado, R. P., Kendrick,D. A.,1999,Caution in macroeconomicpolicy: uncertaintyand the relativeintensityofpolicy,EconomicsLetters 68,37–41. Newey,W.K.,West,K.D.,1987,ASimplepositive-definiteheteroskedasticityandautocorrelationconsistentcovariancematrix,Econometrica55,703–708. Onatski, A. 1999, Minimax analysis of model uncertainty: comparison to Bayesian approach, worst possible economies, and optimal robust monetary policies. (Department of Economics,Harvard University). Onatski, A., Stock, J. H., 2000, Robust monetary policy under model uncertainty in a small modeloftheU.S. economy,NBER WorkingPaperSeries 7490. Orphanides, A., 1998, Monetary policy rules based on real time data, Federal Reserve Board Financeand EconomicsDiscussionSeries 1998-3. Orphanides,A., Porter, R. D., Reifschneider, D.,Tetlow,R. J.,Finan, F. S., 2000,Errors inthe measurementoftheoutputgap,andthedesignofmonetarypolicy,JournalofEconomics and Business52. Roberts, J. M., 1995, New Keynesian economics and the Phillips curve, Journal of Money Credit andBanking 27,975–984. 39

Rudebusch, G. D. 1998, Is theFed too timid: monetary policyin an uncertain world? Unpublishedmanuscript,Federal ReserveBank ofSan Francisco. Rudebusch,G.D.,Svensson,L.E.O.1999,Policyrulesforinflationtargeting,in: J.B.Taylor ed., Monetarypolicyrules, (UniversityofChicago Press, Chicago), 203–262. Sack, B., 1998a, Does the fed act gradually? A var analysis, Federal Reserve Board Finance and EconomicsDiscussionSeries 1998-17. Sack, B., 1998b, Uncertainty, learning, and gradual monetary policy, Federal Reserve Board Financeand EconomicsDiscussionSeries 1998-34. Sack, B., Wieland, V., 2000, Interest-rate smoothing and optimal monetary policy, Journal of Economicsand Business52. Sargent,T.J.1999,Discussionof’policyrulesforopeneconomies’byLaurenceBall,in: J.B. Taylored., Monetarypolicyrules,(UniversityofChicago Press, Chicago), 144–54. Smets,F.1999,Outputgapuncertainty: doesitmatterfortheTaylorrule?, in: B. Hunt,A.Orr eds., Monetary policy under uncertainty, (Reserve Bank of New Zealand, Wellington), 10–29. So¨derstro¨m, U., 1999a, Should central banks be more aggressive?, Stockholm University, WorkingPapers inEconomicsand Finance309. So¨derstro¨m, U., 1999b, Monetary policy with uncertain parameters, Stockholm University, WorkingPapers inEconomicsand Finance308. Sworder, D. D., 1964, Minmax control of discrete time stochastic systems, Society for IndustrialApliedMathematics,Series A, Control2 433–449. Taylor,J.B.,1980,Aggregatedynamicsandstaggeredcontracts,JournalofPoliticalEconomy 88,1–23. Taylor, J. B., 1993, Discretion versus policy rules in practice, Carnegie-Rochester Conference Series on PublicPolicy39, 195–214. Tetlow,R.J.2000,Uncertainpotentialoutputandmonetarypolicyinaforward-lookingmodel. Unpublishedmanuscript,Board ofGovernorsoftheFederal Reserve System. Whittle,P. 1990,Risk-sensitiveoptimalcontrol,Wiley,New York. 40

Wieland, V., 1998, Monetary policy and uncertainty about the natural unemployment rate, Federal Reserve Board Financeand EconomicsDiscussionSeries 1998-2. Woodford,M., 1999,Optimalmonetarypolicyinertia,ManchesterSchool67, 1–35. Zames, G., 1981, Feedback and optimal sensitivity: model reference transformations, multiplicativeseminorms,andapproximateinverses,IEEETransactionsonAutomaticControl AC-26, 301–320. Zhou, K., Doyle, J. C., Glover, K. 1996, Robust and optimal control, (Prentice-Hall, EnglewoodCliffs). 41

Figure1: FrequencyDecomposition ofExpectedLossesunderModelandShockUncertainty "STRONG" INFLATION PREFERENCE 8 trace[G(x )’G(x )] 7 Estimated 6 5 4 3 LQG L 2 1 H¥ 1 1 year 8 years 4 years 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure2: FrequencyDecomposition ofExpectedLossesunderModelandShockUncertainty "WEAK" INFLATION PREFERENCE 7 trace[G(x )’G(x )] 6 Estimated 5 4 3 LQG H¥ 2 L 1 1 1 year 8 years 4 years 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 42

Figure3: FrequencyDecomposition ofExpectedLosses: Perturbations toallParameters "STRONG" INFLATION PREFERENCE 12 trace[G(x )’G(x )] 10 LTI MA 8 LTI scalar 6 Estimated 4 NTV 2 LQG 1 year 8 years 4 years 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure4: FrequencyDecomposition ofExpectedLosses: Perturbations toallParameters "WEAK" INFLATION PREFERENCE 9 trace[G(x )’G(x )] 8 LTI scalar 7 Estimated 6 LTI MA 5 4 3 NTV LQG 2 1 1 year 8 years 4 years 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 43

Figure5: FrequencyDecomposition ofExpectedLosses: Perturbations toLaggedOutputCoefficients "STRONG INFLATION PREFERENCE" 14 trace[G(x )’G(x )] 12 LTI MA 10 LTI scalar 8 6 Estimated 4 NTV LQG 2 1 year 8 years 4 years 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure6: FrequencyDecomposition ofExpectedLosses: Perturbations toLaggedOutputCoefficients "WEAK INFLATION PREFERENCE" 8 trace[G(x )’G(x )] LTI scalar 7 Estimated 6 LTI MA 5 4 LQG 3 NTV 2 1 8 years 4 years 1 year 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 44

Figure7: FrequencyDecomposition ofExpectedLosses: Perturbations toSlopeofISCurve "STRONG" INFLATION PREFERENCE 9 trace[G(x )’G(x )] 8 LTI MA 7 Estimated 6 5 NTV 4 LTI scalar 3 LQG 2 1 1 year 8 years 4 years 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure8: FrequencyDecomposition ofExpectedLosses: Perturbations toSlopeofISCurve "WEAK" INFLATION PREFERENCE 7 trace[G(x )’G(x )] 6 LTI MA 5 Estimated 4 LTI scalar 3 NTV 2 LQG 1 1 year 8 years 4 years 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 45

Figure9: FrequencyDecompositionofExpectedLosses: PerturbationstoSlopeofPhillipsCurve "STRONG" INFLATION PREFERENCE 9 trace[G(x )’G(x )] 8 LTI scalar 7 LTI MA 6 Estimated 5 4 3 NTV 2 LQG 1 4 years 1 year 8 years 0 0 0.5 1 1.5 2 2.5 3 3.5 Figure 10: Frequency Decomposition of Expected Losses: Perturbations to Slope of Phillips Curve "WEAK" INFLATION PREFERENCE 8 trace[G(x )’G(x )] 7 LTI scalar Estimated 6 LTI MA 5 4 3 NTV LQG 2 1 1 year 8 years 4 years 0 0 0.5 1 1.5 2 2.5 3 3.5 46