MonetaryPolicyShocks: WhatHaveWeLearnedand


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1 MonetaryPolicyShocks: WhatHaveWeLearnedand towhatend? LawrenceJ.Christiano,MartinEichenbaum y andcharlesl.evans z August31, 1998 Abstract This paper reviews recentresearchthat grapples withthe question: What happens after an exogenous shock to monetary policy? We argue that this question is interesting because it lies at the center of a particular approach to assessing the empirical plausibility of structural economic models that can be used to think about systematic changes in monetary policy institutions and rules. The literature has not yet converged on a particular set of assumptions for identifying the e ects of an exogenous shock to monetary policy. Nevertheless, there is considerable agreement about the qualitative e ects of a monetary policy shock in the sense that inference is robust across a large subset of the identi cation schemes that have been considered in the literature. We document the nature of this agreement as it pertains to key economic aggregates. Contents 1 Introduction MonetaryPolicyShocks: SomePossibleInterpretations Vector Autoregressions and Identi cation The E ects of a Monetary Policy Shock: ARecursiveness Assumption TheRecursivenessAssumptionandVARs Three Benchmark Identi cation Schemes TheBenchmarkPolicyShocks Displayed WhatHappensAfteraBenchmarkPolicyShock? Results for other Economic Aggregates U.S.DomesticAggregates Exchange Rates and Monetary Policy Shocks Robustness ofthebenchmarkanalysis ExcludingCurrentOutputandPricesFrom t Northwestern University, NBER and the Federal Reserve Bank of Chicago y Northwestern University, NBER and the Federal Reserve Bank of Chicago z Federal Reserve Bank of Chicago
2 4.4.2 ExcludingCommodityPricesfrom t : ThePricePuzzle EquatingthePolicyInstrument,S t ;WithM0,M1orM Using Information FromtheFederalFunds Futures Market Sample Period Sensitivity Discriminating Between the Benchmark Identi cation Schemes The Coleman,Gilles and Labadie Identi cation Scheme TheBernankeMihovCritique Monetary Policy Shocks and Volatility The E ects of Monetary Policy Shocks: Abandoning the Recursiveness Approach AFullySimultaneous System SimsZha: Model Speci cation and Identi cation EmpiricalResults SomePitfalls in Interpreting Estimated MonetaryPolicyRules The E ects of a Monetary Policy Shock: The Narrative Approach Conclusion
3 1.Introduction In the past decade there has been a resurgence of interest in developing quantitative,monetary generalequilibriummodels ofthebusiness cycle. In part,this re ects theimportance of ongoing debates that center on monetary policy issues. What caused the increased in ation experienced by many countries in the1970s? What sorts of monetary policies and institutions would reduce the likelihood of it happening again? How should the FederalReserve respond to shocks that impact the economy? What are the welfare costs and bene ts of moving to a common currency area in Europe? To make fundamental progress on these types of questions requires that we address them within the con nes of quantitative general equilibriummodels. Assessing the e ect of a change in monetary policy institutions or rules could be accomplished using purely statistical methods. But only if we had data drawn from otherwise identical economies operating under the monetary institutions or rules we areinterested in evaluating. We don't. So purely statistical approaches to these sorts of questions aren't feasible. And,real world experimentation is not an option. The only place we can perform experiments is in structuralmodels. Butwenowhaveatourdisposalahostofcompetingmodels,eachofwhichemphasizes di erent frictions and embodies di erent policy implications. Which model should we use for conducting policy experiments? This paper discusses a literature that pursues one approach toansweringthisquestion. ItisinthespiritofasuggestionmadebyR.E.Lucas(1980). He argues that economists \...need to test them(models) as useful imitations of reality by subjecting them to shocks for which we are fairly certain how actual economies or parts of economies would react. The more dimensions on which the model mimics the answers actual economies give to simple questions, the morewe trust its answers to harder questions." The literature we review applies the Lucas program using monetary policy shocks. These shocks are good candidates for use in this program because di erent models respond very differentlytomonetarypolicyshocks(seechristiano,eichenbaumandevans(cee)(1997a)). 1 The program is operationalized in three steps: ² First, one isolates monetary policy shocks in actual economies and characterizes the nature of the corresponding monetary experiments. 1 Other applications of the Lucas program include the work of Gali (1997) who studies the dynamic e ects of technology shocks, and Rotemberg and Woodford (1992) and Ramey and Shapiro (1997), who study the dynamic e ects of shocks to government purchases. 3
4 ² Second, one characterizes theactual economy's response to these monetary experiments. ² Third, one performs the same experiments in the model economies to be evaluated and compares the outcomes with actual economies' responses to the corresponding experiments. These steps are designed to assist in the selection of a model that convincingly answers the question,`how does theeconomy respond to an exogenous monetary policy shock?' Granted, thefactthatamodelpassesthistestis notsu±cientto giveuscompletecon denceinits answerstothetypesofquestionsweareinterestedin. Howeverthistestdoeshelpnarrow ourchoicesandgivesguidanceinthedevelopmentofexistingtheory. A central feature of the program is the analysis of monetary policy shocks. Why not simply focus on the actions of monetary policy makers? Because monetary policy actions re ect,in part,policy makers'responses to nonmonetary developments in theeconomy. A givenpolicyactionandtheeconomiceventsthatfollowitre ectthee ectsofalltheshocks to theeconomy. Our application ofthelucas programfocuses on thee ects ofa monetary policy shock per se. An important practical reason for focusing on this type of shock is that di erent models respond very di erently to the experiment of a monetary policy shock. Inorder to usethis informationweneed to know whathappens inresponse totheanalog experiment in the actual economy. There is no point in comparing a model's response to one experiment with the outcome of a di erent experiment in the actualeconomy. So, to proceedwithourprogram,wemustknowwhathappensintheactualeconomyafterashock to monetarypolicy. The literature explores threegeneralstrategies for isolating monetary policyshocks. The rst is theprimaryfocus of our analysis. It involves making enough identifying assumptions to allow the analyst to estimate the parameters of the Federal Reserve's feedback rule,i.e., the rule which relates policymakers'actions to the state of the economy. The necessary identifying assumptions include functionalform assumptions,assumptions about which variables thefedlooksatwhensettingitsoperatinginstrumentandanassumptionaboutwhatthe operating instrument is. In addition, assumptions must be made about the nature of the interaction of thepolicyshockwith thevariables in the feedbackrule. One assumption is that thepolicy shock is orthogonal to these variables. Throughout,we refer to this as the recursiveness assumption. Along with linearity of thefed's feedbackrule,this assumption justi es estimating policy shocks by the tted residuals in theordinaryleast squares regression of the Fed's policyinstrument on the variables in the Fed's information set. The economic content of the recursiveness assumption is that the time t variables in the Fed's 4
5 information set do not respond to time trealizations of the monetary policy shock. As an example, CEE(1996a) assume that the Fed looks at current prices and output, among other things,when setting the time tvalue of its policy instrument. In that application,the recursivenessassumptionimpliesthatoutputandpricesrespondonlywithalag toamonetary policyshock. While there are models that are consistent with the previous recursiveness assumption, it is nevertheless controversial. 2 This is why authors like Bernanke (1986), Sims (1986), SimsandZha(1995)andLeeper,SimsandZha(1996)adoptanalternativeapproach. No doubt there are some advantages to abandoning the recursiveness assumption. But there is also a substantial cost: a broader set of economic relations must be identi ed. And the assumptions involved can also becontroversial. For example, Sims and Zha(1995) assume, among other things,that the Fed does not look at the contemporaneous price level or output when setting its policy instrument and that contemporaneous movements in the interest rate do not directly a ect aggregate output. Both assumptions are clearly debatable. Finally, it should be noted that abandoning the recursiveness assumption doesn't require one to adopt an identi cation scheme in which a policy shock has a contemporaneous impact on all nonpolicy variables. For example, Leeper and Gordon(1992) and Leeper, Sims and Zha (1996) assume that aggregate real output and the price level are not a ected in the impact period ofa monetarypolicyshock. The second and third strategies for identifying monetary policy shocks do not involve explicitly modelling the monetary authority's feedback rule. The second strategy involves looking at data that purportedly signal exogenous monetary policy actions. For example, Romer and Romer(1989) examine records of the Fed's policy deliberations to identify times in which they claim there were exogenous monetary policy shocks. Other authors like Rudebusch(1995a) assume that,in certain sample periods, exogenous changes in monetary policyare wellmeasured by changes in thefederalfunds rate. Finally,authors like Cooley and Hansen (1989, 1997), King (1991), Christiano (1991) and Christiano and Eichenbaum (1995) assume that all movements in money re ect exogenous movements in monetary policy. The third strategy identi es monetarypolicyshocks bythe assumption that theydo not a ect economicactivityin thelong run. 3 Wewillnot discuss this approach in detail. We referthereadertofaustandleeper(1997)andpaganandrobertson(1995)fordiscussions and critiques of this literature. Thepreviousoverviewmakesclearthattheliteraturehasnotyetconvergedonapartic 2 See Christiano, Eichenbaum and Evans (1997b) and Rotemberg and Woodford (1997) for models that are consistent with the assumption that contemporaneous output and the price level do not respond to a monetary policy shock. 3 For an early example of this approach see Gali (1992). 5
6 ular set of assumptions for identifying the e ects of an exogenous shock to monetary policy. Nevertheless,as we show,there is considerable agreement about the qualitative e ects of a monetary policy shock in thesense that inferenceis robust across a largesubset of theidenti cation schemes that have been considered in the literature. The nature of this agreement is as follows: after a contractionary monetary policy shock, short term interest rates rise, aggregateoutput,employment,pro ts and various monetaryaggregates fall,the aggregate price level responds very slowly, and various measures of wages fall, albeit by very modest amounts. In addition, there is agreement that monetary policy shocks account for only a very modest age of the volatility of aggregate output; they account for even less of themovementsintheaggregatepricelevel. 4 Theliteraturehasgonebeyondthistoprovide a richer more detailed picture of the economy's response to a monetary policy shock (see section 4.6). But even this small list of ndings has proven to be usefulin evaluating the empiricalplausibilityofalternativemonetary business cycle models(seecee (1997a)). In this sense thelucas program, as applied to monetary policy shocks, is already proving to bea fruitfulone. Identi cation schemes do exist which lead to di erent inferences about the e ects of a monetary policy shock than the consensus view just discussed. How should we select between competing identifying assumptions? We suggest one selection scheme:eliminate a policyshock measureif it implies a set of impulseresponse functions that is inconsistent with everyelement in theset of monetary models that we wish to discriminatebetween. This is equivalenttoannouncingthatifnoneofthemodelsthatweareinterestedincanaccountfor thequalitativefeatures ofa set of impulses responsefunctions,wereject thecorresponding identifying assumptions,not the entireset of models. In practice,this amounts to a set of sign and shape restrictions on impulse response functions(see Uhlig(1997) for a particular formalization of this argument). Since we have been explicit about the restrictions we impose, readerscanmaketheirowndecisionsaboutwhethertorejecttheidentifyingassumptionsin question. In the end, the key contribution of the monetary policy shock literature may be this: it has clari ed the mapping from identi cation assumptions to inference about the e ects of monetary policy shocks. This substantially eases the task of readers and model builders in evaluating potentially con icting claims about what actually happens after a monetary policyshock. The remainder of the paper is organized as follows: ² Section 2: We discuss possible interpretations of monetary policy shocks. 4 These latter two ndings say nothing about the impact of the systematic component of monetary policy on aggregate output and the price level. The literature that we review is silent on this point. 6
7 ² Section 3: Wediscuss the main statisticaltoolused in the analysis,namelythe Vector Autoregression (VAR). In addition we present a reasonably selfcontained discussion of the identi cation issues involved in estimating the economic e ects of a monetary policyshock. ² Section 4: Wediscuss inference about the e ects of a monetary policy shock using therecursiveness assumption. First,wediscuss thelink between therecursiveness assumption and identi ed VAR's. Second, we display thedynamic responseofvarious economic aggregates to a monetary policy shock under three benchmark identi cation schemes,each of which satis es the recursiveness assumption. In addition,we discuss related ndings in theliterature concerning other aggregates not explicitlyanalyzed here. Third,we discuss the robustness of inference to various perturbations including: alternative identi cation schemes which also impose the recursiveness assumption,incorporating information fromthe federalfunds futures market into theanalysis and varying the subsample over which the analysis is conducted. Fourth,we consider some critiques of the benchmark identi cation schemes. Fifth,we consider the implications of the benchmark identi cation schemes for thevolatility of various economic aggregates. ² Section 5: We consider other approaches which focus on the monetaryauthority's feedback rule,but which do not impose the recursiveness assumption. ² Section 6: We discuss the di±culty of directly interpreting estimated monetary policy rules. ² Section 7: We consider the narrative approach to assessing the e ects of a monetary policyshock. ² Section 8: We concludewith a brief discussion of various approaches to implementing the third step of the Lucas program as applied to monetary policy shocks. In particular we review a particular approach to performing monetary experiments in model economies,theoutcomesofwhichcanbecomparedtotheestimatede ectsofapolicy shock in actual economies. In addition we provide some summary remarks. 2.MonetaryPolicyShocks: SomePossibleInterpretations Many economists think that a signi cant fraction of the variation in central bank policy actions re ects policy makers'systematic responses to variations in the state of the economy. As noted in the introduction, this systematic component is typically formalized with the 7
8 conceptofafeedbackrule,orreactionfunction. Asapracticalmatter,itisrecognizedthat not allvariationsin centralbankpolicycan be accounted for as a reaction to thestate of the economy. The unaccounted variation is formalized with thenotion of a monetary policy shock. Given the large role that the concepts of a feedback rule and a policy shock play in the literature,we begin by discussing several sources of exogenous variation in monetary policy. Throughout this paper we identify a monetary policy shock with the disturbance term inanequationoftheform S t =f( t )+¾ s " s t: (2.1) Here S t is the instrument of the monetary authority, say the federal funds rate or some monetary aggregate, and f is a linear function that relates S t to the information set t : Therandomvariable,¾ s " s t ;isamonetarypolicyshock. Here,"s t isnormalizedtohaveunit variance,andwereferto¾ s asthestandarddeviationofthemonetarypolicyshock. Oneinterpretationoff and t isthattheyrepresentthemonetaryauthority'sfeedback rule and information set,respectively. As we indicate in section 6,there are other ways to thinkaboutf and t whichpreservetheinterpretationof" s t asashocktomonetarypolicy. What is the economic interpretation of these policy shocks? We o er three interpretations. The rst is that " s t re ects exogenous shocks to the preferences of the monetary authority,perhaps due to stochastic shifts in the relative weight given to unemployment and in ation. These shifts could re ect shocks to the preferences of the members of the Federal Open Market Committee(FOMC),or to the weights by which their views are aggregated. A change in weights mayre ect shifts in thepoliticalpower of individualcommitteemembers or in the factions that they represent. A second source of exogenous variation in policy can arise because of the strategic considerations developed in Ball(1995) and Chari, Christiano and Eichenbaum(1997). These authors argue that the Fed's desire to avoid the social costs of disappointing private agents'expectations can give rise to an exogenous source of variation inpolicylikethatcapturedby" s t:speci cally,shockstoprivateagents'expectationsabout Fedpolicycanbeselfful llingandleadtoexogenousvariationsinmonetarypolicy. Athird source of exogenous variation in Fed policy could re ect various technical factors. For one set of possibilities,see Hamilton(1997). Another set of possibilities,stressed by Bernanke and Mihov (1995), focuses on the measurement error in the preliminary data available to thefomcatthetimeitmakesitsdecision. We nd it useful to elaborate on Bernanke and Mihov's suggestion for three reasons. First,their suggestion is ofindependent interest. Second,we use it in section 6 to illustrate some of the di±culties involved in trying to interpret the parameters of f:third,we use a version of their argument to illustrate how the interpretation of monetary policy shocks can 8
9 interactwiththeplausibilityofalternativeassumptionsforidentifying" s t : Suppose the monetary authority sets the policy variable, S t ; as an exact function of currentandlaggedobservationsonasetofvariables,x t. Wedenotethetimetobservations onx t andx t 1 byx t (0)andx t 1 (1)where: x t (0)=x t +v t ; andx t 1 (1)=x t 1 +u t 1 : (2.2) So, v t represents the contemporaneous measurement error in x t ; while u t represents the measurementerrorinx t fromthestandpointofperiodt+1:ifx t isobservedperfectlywith aoneperioddelay,thenu t 0forallt:SupposethatthepolicymakersetsS t asfollows: S t = 0S t 1 + 1x t (0)+ 2x t 1 (1): (2.3) Expressed in terms ofcorrectly measured variables,this policy rule reduces to equation(2.1) with: f( t )= 0S t 1 + 1x t + 2x t 1 ; ¾ s " s t = 1v t + 2u t 1 : (2.4) This illustrates how noise in the data collection process can be a source of exogenous variation in monetary policy actions. This example can be used to illustrate how one's interpretation of the error term can a ect the plausibility of alternative assumptions used to identify " s t. Recall the recursiveness assumption, according to which " s t is orthogonal to the elements of t : Under what circumstances would this assumption be correct under the measurement error interpretation of" s t? To answer this, suppose that v t and u t are classical measurement errors, i.e. they are uncorrelated with x t at all leads and lags. If 0 = 0; then the recursiveness assumption is satis ed. Now suppose that 0 6= 0: If u t 0; then this assumption is still satis ed. However, in the more plausible case where 2 6= 0; u t 6= 0 and u t and v t are correlated with each other,then therecursiveness condition fails. This last case provides an important caveat to measurement error as an interpretation of the monetary policy shocks estimated byanalysts who make use of the recursiveness assumption. We suspect that this mayalso be true for analysts who do not use therecursiveness assumption (see Section 5 below), because in developing identifying restrictions,they typically abstract from the possibility of measurement error. 3.Vector Autoregressions and Identi cation A fundamentaltool in the literature that we review is the vector autoregression (VAR). A VAR is a convenient device for summarizing the rst and second moment properties 9
10 of the data. We begin by de ning moreprecisely what a VAR is. Wethen discuss the identi cation probleminvolved in measuring the dynamic response of economic aggregates to a fundamental economic shock. The basic problem is that a given set of second moments is consistent with many such dynamic response functions. Solving this problem amounts to making explicit assumptions that justify focusing on a particular dynamic response function. AVARforakdimensionalvectorofvariables,Z t ;isgivenby: Z t =B 1 Z t 1 +:::+B q Z t q +u t ; Eu t u 0 t =V: (3.1) Here, q is a nonnegative integer and u t is uncorrelated with all variables dated t 1 and earlier. 5 ConsistentestimatesoftheB i 'scanbeobtainedbyrunningordinaryleastsquares equation by equation on(3.1). One can then estimate V from the tted residuals. SupposethatweknewtheB i 's,theu t 'sandv. Itstillwouldnotbepossibletocompute thedynamicresponsefunctionofz t tothefundamentalshocksintheeconomy. Thebasic reason is that u t is the one step ahead forecast error in Z t :In general, each element of u t re ects thee ects ofallthefundamentaleconomicshocks. Thereis no reason to presume thatanyelementofu t correspondstoaparticulareconomicshock,sayforexample,ashock to monetarypolicy. To proceed, we assume that the relationship between the VAR disturbances and the fundamental economic shocks; " t, is given by A 0 u t = " t : Here, A 0 is an invertible, square matrixande" t " 0 t =D;whereDisapositivede nitematrix.6 Premultiplying(3.1)byA 0, weobtain: A 0 Z t =A 1 Z t 1 +:::+A q Z t q +" t : (3.2) HereA i isakxkmatrixofconstants,i=1;:::qand B i =A 1 0 A i;i=1;:::;q; andv =A 1 0 D ³ 0: A 1 0 (3.3) The response of Z t+h to a unitshockin " t ; h ; can be computed as follows. Let ~ h be the solution to the following di erence equation: ~ h =B 1 ~ h 1 +:::+B q ~ h q ; h=1;2;::: (3.4) with initial conditions ~ 0 =I; ~ 1 =~ 2 =::::=~ q =0: (3.5) 5 For a discussion of the class of processes that VAR's summarize, see Sargent (1987). The absence of a constant term in (3.1) is without loss of generality, since we are free to set one of the elements of Z t to be identically equal to unity. 6 This corresponds to the assumption that the economic shocks are recoverable from a nite list of current and past Z t 's. For our analysis, we only require that a subset of the " t 's be recoverable from current and past Z t 's. 10
11 Then, h =~ h A 1 0 ; h=0;1;::: (3.6) Here,the(j;l)elementof h representstheresponseofthej th componentofz t+h toaunit shockinthel th componentof" t :The h 'scharacterizethe`impulseresponsefunction'ofthe elementsofz t totheelementsof" t : Relation (3.6) implies we need to know A 0 as well as the B i 's in order to compute theimpulseresponsefunction. WhiletheB i 's canbeestimated viaordinaryleastsquares regressions,gettinga 0 isnotsoeasy. TheonlyinformationinthedataaboutA 0 isthatit solvestheequationsin(3.3). AbsentrestrictionsonA 0 thereareingeneralmanysolutions to theseequations. The traditional simultaneous equations literature places no assumptions ond;sothattheequationsrepresentedbyv =A 1 0 D ³ 0providenoinformationabout A 1 0 A 0 :Instead,thatliteraturedevelopsrestrictionsonA i ;i=0;:::;q thatguaranteeaunique solutiontoa 0 B i =A i,i=1;:::;q: In contrast,the literature we survey always imposes the restriction that the fundamental economic shocks are uncorrelated(i.e. D is a diagonal matrix),and places no restrictions on A i ;i=1;:::;q: 7 AbsentadditionalrestrictionsonA 0 wecanset D=I: (3.7) AlsonotethatwithoutanyrestrictionsontheA i 's;theequationsrepresentedbya 0 B i =A i, i = 1;:::;q provide no information about A 0 : All of the information about this matrix is ³ containedintherelationship,v =A 1 0 A 1 0 0:De nethesetofsolutionstothisequation by Q V = ½ A 0 :A 1 0 ³ ¾ A 1 0=V 0 : (3.8) Ingeneral,thissetcontainsmanyelements. ThisisbecauseA 0 hask 2 parameterswhilethe symmetricmatrix,v;hasatmostk(k+1)=2distinctnumbers. So,Q V isthesetofsolutions to k(k+1)=2 equations in k 2 unknowns. As long as k > 1, therewillin general bemany solutions to this set of equations,i.e.,there is an identi cation problem. To solvethis problem we must nd and defend restrictions on A 0 so that thereis only one element in Q V satisfying them. In practice, the literature works with two types of restrictions: a set of linear restrictions on the elements of A 0 and a requirement that the diagonalelementsofa 0 bepositive. Supposethattheanalysthasinmindllinearrestrictions on A 0. Thesecan berepresentedas therequirement vec(a 0 )=0;where is amatrixof dimensionl k 2 andvec(a 0 )isthek 2 1vectorcomposedofthekcolumnsofA 0. Eachof 7 See Leeper, Sims and Zha (1996) for a discussion of (3.7). 11
12 thelrowsof representsadi erentrestrictionontheelementsofa 0 :Wedenotethesetof A 0 satisfyingtheserestrictionsby: Q = fa 0 : vec(a 0 )=0g: (3.9) In the literature that we survey,the restrictions summarized by are either zero restrictions on the elements of A 0 or restrictions across the elements of individual rows of A 0 : Cross equation restrictions, i.e., restrictions across the elements of di erent rows of A 0 ; are not considered. NextwemotivatethesignrestrictionsthatthediagonalelementsofA 0 mustbestrictly positive. 8 IfQ \Q V isnonempty,itcanneverbecomposedofjustasinglematrix. Thisis becauseifa 0 liesinq V \Q ;then ~ A 0 obtainedfroma 0 bychangingthesignofallelements ofanarbitrarysubsetofrowsofa 0 alsoliesinq \Q V :Toseethis,letW beadiagonal matrixwith an arbitrary pattern of ones and minus ones along the diagonal. It is obvious thatwa 0 2Q :Also,becauseW isorthonormal(i.e.,w 0 W =I),WA 0 2Q V aswell. SupposeweimposetherestrictionthatthediagonalelementsofA 0 bestrictlypositive. Thisrulesoutmatrices ~A 0 thatareobtainedfromana 0 2Q \Q V bychangingthesigns ofalltheelementsofa 0 :InwhatfollowsweonlyconsiderA 0 matricesthat obeythesign restrictions. Thatis,weinsistthatA 0 2Q S ;where Q S = fa 0 :A 0 hasstrictlypositivediagonalelementsg: (3.10) >From(3.2)weseethatthei th diagonalofa 0 beingpositivecorrespondstothenormalization thatapositiveshockto thei th elementof" t representsapositiveshocktothei th element ofz t whentheotherelementsofz t areheld xed. WhenthereismorethanoneelementinthesetQ V \Q \Q S wesaythatthesystem is `underidenti ed', or, `not identi ed'. When Q V \Q \Q S has one element, we say it is `identi ed'. So, in these terms, solving the identi cation problem requires selecting a which causes the system to be identi ed. NotethatQ V \Q isthesetofsolutionstok(k+1)=2+lequationsinthek 2 unknowns ofa 0 :Inpractice,theliteratureseekstoachieveidenti cationbyselectingafullrowrank satisfying the order condition, l k(k 1)=2:However, the order and sign conditions are not su±cient for identi cation. For example, when l = k(k 1)=2 underidenti cation could occur for two reasons. First, a neighborhood of a given A 0 2 Q V \Q \Q S could contain other matrices belonging to Q V \Q \Q S. This possibility can be ruled out by 8 The following discussion ignores the possibility that Q \ Q V contains a matrix with one or more diagonal elements that are exactly zero. A suitable modi cation of the argument below can accommodate this possibility. 12
13 verifying a simple rank condition, namely that the matrix derivative with respect to A 0 of the equations de ning (3.8) is of full rank. 9 In this case, we say we have established local identi cation. A second possibility is that there may be other matrices belonging to Q V \Q \Q S but which are not in a small neighborhood of A 0 : 10 In general, no known simple conditions rule out this possibility. Ifwe do manage to rule it out,we say thesystem is globally identi ed. 11 In practice, we use the rank and order conditions to verify local identi cation. Global identi cation must be established on a case bycasebasis. Sometimes, as in our discussion of Bernankeand Mihov (1995), this can be done analytically. More typically,one is limited to building con dencein globalidenti cation byconducting an ad hocnumerical search through theparameter spaceto determineif there are other elements inq V \Q \Q S : The di±culty of establishing global identi cation in the literature we survey stands in contrast to thesituation in the traditional simultaneous equations context. There,theidenti cation problem only involves systems of linear equations. Under these circumstances, local identi cation obtains if and only if global identi cation obtains. The traditional simultaneous equations literature provides a simple set of rank and order conditions that are necessary and su±cient for identi cation. These conditions are only su±cient to characterize localidenti cationforthesystemsthatweconsider. 12 Moreover,theyareneithernecessary nor su±cient for global identi cation. We now describetwo examples which illustratethediscussion above. In the rst case,the order and sign conditions are su±cient to guarantee global identi cation. In the second,the 9 Here we de ne a particular rank condition and establish that the rank and order conditions are su±cient for local identi cation. Let be the k(k + 1)=2 dimensional column vector of parameters in A 0 that remain free after imposing (3.9), so that A 0 ( ) 2 Q for all : Let f( ) denote the k(k + 1)=2 dimensional row vector composed of the upper triangular part of A 0 ( ) 1 A 0 ( ) 1 0 V: Let F( ) denote the k(k +1)=2 by k(k+1)=2 derivative matrix of f( ) with respect to : Let satisfy f( ) = 0. Consider the following rank condition: F( ) has full rank for all 2 D( ); where D( ) is some neighborhood of : We assume that f iscontinuous and that F iswellde ned. Astraightforward application of themean valuetheorem (seebartle (1976), p.196) establishes that this rank condition guarantees f( ) 6= 0 for all 2 D( ) and 6= : Let g : [" ;" ]! R k(k+1)=2 be de ned by g (") = f( + "); where is an arbitrary nonzero k(k +1)=2 column vector, and " and " are the smallest and largest values, respectively, of " such that ( + ") 2 D( ). Note that g (") 0 = 0 F( + ") and " < 0 < " : By the mean value theorem, g (") = g (0) + g ( )" 0 for some between 0 and ": This can be written g (") = 0 F( + ")": The rank condition implies that the expression to the right of the equality is nonzero, as long as " 6= 0: Since the choice of 6= 0 was arbitrary, the result is established. 10 A simple example is (x a)(x b) = 0; which is one equation with two isolated solutions, x = a and x = b: 11 We can also di erentiate other concepts of identi cation. For example, asymptotic and small sample identi cation correspond to the cases where V is the population and nite sample value of the variance covariance matrix of the VAR disturbances, respectively. Obviously, asymptotic identi cation could hold while nite sample identi cation fails, as well as the converse. 12 To show that the rank condition is not necessary for local identi cation, consider f(x) = (x a) 2 : For this function there is a globally unique zero at x = a; yet f 0 (a) = 0: 13
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