MonetaryPolicyShocks: WhatHaveWeLearnedand


 Nicholas Owen
 1 years ago
 Views:
Transcription
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
14 order condition and sign conditions for identi cation hold,yet the system is not identi ed. In the rst example, we select so that all the elements above(alternatively, below) the diagonalofa 0 arezero. Ifinaddition,weimposethesignrestriction,thenitiswellknown thatthereisonlyoneelementinq V \Q \Q S ;i.e.,thesystemisgloballyidenti ed. This result is an implication of the uniqueness of the Cholesky factorization of a positive de nite symmetric matrix. This example plays a role in the section on identi cation of monetary policy shocks with a recursiveness assumption. Foroursecondexample,considerthecasek=3withthefollowingrestrictedA 0 matrix: A 0 = a 11 0 a 13 0 a 22 a 23 0 a 32 a 33 where a ii > 0 for i = 1;2;3: Since there are three zero restrictions, the order condition is satis ed. SupposethatA 0 2Q V ;sothata 0 2Q V \Q \Q S :LetW beablockdiagonal matrix with unity in the (1;1) element and an arbitrary 2 2 orthonormal matrix in the second diagonal block. Let W also have the property that WA 0 has positive elements on thediagonal. Then, WW 0 =I;and WA 0 2Q V \Q \Q S : 13 In this case we do not have identi cation, even though the order and sign conditions are satis ed. The reason for the failure of local identi cation is that the rank condition does not hold. If it did hold, then identi cation would have obtained. The failure of the rankcondition in this example re ects that the second and third equations in the system are indistinguishable. It is easy to show that every element in Q V \Q \Q S generates the same dynamic response function to the rst shock in thesystem. To see this, note from (3.5) that the rstcolumnofa 1 0 is what characterizes the response of all the variables to the rst shock. Similarly, the rst column of (WA 0 ) 1 controls theresponse ofthe transformed system to the rst shock. But, the result (WA 0 ) 1 = A 1 0 W 0 ; and our de nition of W imply that the rstcolumnsof(wa 0 ) 1 andofa 1 0 are thesame. So,if one is onlyinterested in the dynamic response of the system to the rst shock, then the choice of the second diagonal blockofw isirrelevant. Anextendedversionofthisobservationplaysanimportantrolein 3 7 5; our discussion of nonrecursive identi cation schemes below. 13 To see that this example is non empty, consider the case a 11 = 0:70; a 13 = 0:40; a 22 = 0:38; a 23 = 0:50; a 32 = 0:83; a 33 = 0:71 and let the 2 2 lower block in W be 0:4941 0:8694 : 0:8694 0:4941 It is easy to verify that WA 0 satis es the zero and sign restrictions on A 0. 14
15 4.The E ects of a Monetary Policy Shock: A Recursiveness Assumption In this section wediscuss one widelyused strategy for estimating the e ects of a monetary policyshock. Thestrategyis based on the recursiveness assumption, according to which monetary policy shocks are orthogonal to the information set of the monetary authority. Section 4.1 discusses therelationship between therecursiveness assumption and VARs. Section 4.2 describes three benchmark identi cation schemes which embody the recursiveness assumption. In addition,wedisplay estimates of the dynamic e ects of a monetary policy shock on various economic aggregates,obtained using the benchmark identi cation schemes. Section 4.3 reviews someresults in theliterature regarding the dynamic e ects of a monetary policyshock on other economicaggregates,obtained using close variants of the benchmark schemes. Section 4.4 considers robustness of theempiricalresults contained in section 4.2. Section 4.5 discusses various critiques of the benchmark identi cation schemes. Finally,section 4.6 investigates the implications of the benchmark schemes for the volatility of various economicaggregates. 4.1.The Recursiveness Assumption and VARs The recursiveness assumption justi es the following twostep procedure for estimating the dynamicresponse of a variable to a monetarypolicy shock. First,estimate thepolicy shocks bythe tted residuals intheordinaryleastsquaresregression ofs t on theelementsof t. Second,estimatethedynamicresponseofa variableto a monetarypolicy shockbyregressing the variable on the current and lagged values of the estimated policy shocks. In our analysis we nd it convenient to map the above twostep procedure into an asymptoticallyequivalent VARbased procedure. There are two reasons for this. First, the twostep approach implies that we lose a number of initial data points equal to the number ofdynamicresponsesthatwewishtoestimate,plusthenumberoflags,q;in t. Withthe VAR procedure we onlylosethelatter. Second,the VAR methodology provides a complete description of the data generating process for the elements of t. This allows us to use a straightforward bootstrap methodology for use in conducting hypothesis tests. We now indicate how the recursiveness assumption restricts A 0 in (3.2). Partition Z t intothreeblocks: thek 1 variables,x 1t ;whosecontemporaneousvaluesappearin t ;thek 2 variables, X 2t ; which only appear with a lag in t ; and S t itself. Then, k = k 1 +k 2 +1; wherekisthedimensionofz t :Thatis: Z t = 0 X 1t S t X 2t 1 C A: 15
16 Weconsiderk 1 ;k 2 0:Tomaketheanalysisinterestingweassumethatifk 1 =0;so that X 1t isabsentfromthede nition ofz t ;then k 2 >1:Similarly,ifk 2 =0;thenk 1 >1:The recursivenessassumptionplacesthefollowingzerorestrictionsona 0 : 2 A 0 = 6 4 a 11 (k 1 k 1 ) a 21 (1 k 1) a 31 (k 2 k 1) 0 (k 1 1) a 22 (1 1) a 32 (k 2 1) 0 (k 1 k 2 ) 0 (1 k 2 ) a 33 (k 2 k 2) 3 : (4.1) 7 5 Here,expressionsinparenthesesindicatethedimensionoftheassociatedmatrixanda 22 = 1=¾ s ;where¾ s >0: The zeros in the middle row of this matrix re ect the assumption that the policy maker does not see X 2t when S t is set. The two zero blocks in the rst row of A 0 re ect our assumption that the monetary policy shock is orthogonal to the elements in X 1t : These blocks correspond to the two distinct channels by which a monetary policy shock could in principlea ectthevariablesinx 1t :The rstoftheseblockscorrespondstothedirecte ect ofs t onx 1t. Thesecondblockcorrespondstotheindirecte ectthatoperatesviatheimpact ofamonetarypolicyshockonthevariablesinx 2t : We now show that the recursiveness assumption is not su±cient to identifyall the elements of A 0 : This is not surprising, in light of the fact that the rst k 1 equations are indistinguishablefrom each other, as arethelastk 2 equations. Signi cantly, however, the recursiveness assumption is su±cient to identify the object of interest: the dynamic response ofz t toamonetarypolicyshock. Speci cally,weestablishthreeresults. The rsttwoare as follows: (i) there is a nonempty familyofa 0 matrices, oneofwhich is lower triangular with positiveterms on the diagonal,which are consistent with the recursiveness assumption ³ (i.e.,satisfy(4.1))andsatisfya 1 0 A 1 0=V;and(ii)eachmemberofthisfamilygenerates 0 precisely the same dynamic response function of the elements of Z t to a monetary policy shock. The third result is that if we adopt the normalization of always selecting the lower triangulara 0 matrixidenti edin(i),thenthedynamicresponseofthevariablesinz t are invarianttotheorderingofvariablesinx 1t andx 2t : Toprove(i)(iii)itisusefultoestablishapreliminaryresult. Webeginbyde ningsome notation. Letthe((k 1 +1)k 2 +k 1 ) k 2 matrix summarizethezerorestrictionsona 0 in (4.1). So,Q isthesetofa 0 matricesconsistentwiththerecursivenessassumption. LetQ V be thesetofa 0 matrices de ned bythe propertythat A 1 0 (A 1 0 ) 0 (see (3.8)). Inaddition, let 2 3 W W = ; (4.2) 0 0 W 33 16
17 wherew ispartitionedconformablywitha 0 in(4.1)andw 11 andw 33 arearbitraryorthonormalmatrices. De ne Q¹A 0 = n A 0 :A 0 =W A ¹ 0 ; forsomew satisfying(4.2) o : Here ¹A 0 isamatrixconformablewithw. Wenow establish thefollowing result: Q¹ A0 =Q V \Q ; (4.3) where ¹ A 0 isanarbitraryelementofq V \Q :ItisstraightforwardtoestablishthatA 0 2Q¹A 0 impliesa 0 2Q V \Q. Theresult,A 0 2Q V followsfromorthonormalityofw andthefact, ¹A 0 2 Q V : The result, A 0 2 Q ; follows from the block diagonal structure of W in (4.2). NowconsideranarbitraryA 0 2Q V \Q :To showthata 0 2Q¹A 0 ;considerthecandidate orthonormal matrix W = A 0 ¹ A 1 0 ; where invertibility of ¹ A0 re ects ¹ A 0 2 Q V : Since W is theproduct of two blocklower triangular matrices, it too is blocklower triangular. Also, it is easy to verify that WW 0 = I: The orthonormality of W; together with blocklower triangularityimplythatw hastheform,(4.2). ThisestablishesA 0 2Q¹A 0 and,hence,(4.3). Wenowproveresult(i). ThefactthatQ V \Q isnotemptyfollowsfromthefactthat wecanalwaysseta 0 equaltotheinverseofthelowertriangularcholeskyfactorofv:the existenceandinvertabilityof this matrix is discussed in Hamilton (1994, p. 91). 14 To see thatthereismorethanoneelementinq V \Q ;usethecharacterizationresult,(4.3),with ¹ A 0 equal to the inverse of the Cholesky factor of V:Construct the orthonormal matrix W 6= Iby interchangingtwoofeitherthe rstk 1 rowsorthelastk 2 rowsofthekdimensionalidentity matrix. 15 Then,W ¹ A 0 6= ¹ A 0 :Result(i)isestablishedbecauseW ¹ A 0 2Q V \Q : Wenowproveresult(ii). Consideranytwomatrices,A 0 ; ~ A0 2Q V \Q :By(4.3)there existsaw satisfying(4.2)withtheproperty, ~A 0 =WA 0 ;sothat ~A 1 0 =A 1 0 W 0 : Inconjunctionwith(4.2),thisexpressionimpliesthatthe(k 1 +1) th columnof A ~ 1 0 anda 1 0 areidentical. But,by(3.6)theimplieddynamicresponsesofZ t+i,i=0;1;:::toamonetary policy shock are identical too. This establishes result(ii). 14 The Cholesky factor of a positive de nite, symmetric matrix, V; is a lower triangular matrix, C; with the properties (i) it has positive elements along the diagonal, and (ii) it satis es the property, CC 0 = V: 15 Recall, orthonormality of a matrix means that the inner product between two di erent columns is zero and the inner product of any column with itselfis unity. This property is obviously satis ed by theidentity matrix. Rearranging the rows of the identity matrix just changes the order of the terms being added in the inner products de ning orthonormality, and so does not alter the value of column inner products. Hence a matrix obtained from the identity matrix by arbitrarily rearranging the order of its rows is orthonormal. 17
18 We now prove (iii) using an argument essentially the same as the one used to prove (ii). Weaccomplish theproofbystarting with arepresentationofz t in whicha 0 is lower triangularwithpositivediagonalelements. Wethenarbitrarilyreorderthe rstk 1 andthe lastk 2 elementsofz t :TheanalogtoA 0 intheresultingsystemneednotbelowertriangular with positive elements. We then applya particular orthonormaltransformation which results in a lower triangular system with positive diagonal elements. The response of the variables inz t toamonetarypolicyshockisthesameinthissystemandintheoriginalsystem. Consider ~Z t = DZ t ; where D is the orthonormal matrix constructed by arbitrarily reordering the columns within the rst k 1 and the last k 2 columns of the identity matrix. 16 Then, ~ Zt correspondstoz t withthevariablesinx 1t andx 2t reorderedarbitrarily. LetB i ; i=1;:::;qandv characterizethevarofz t andleta 0 betheuniquelowertriangularmatrix ³ withpositivediagonaltermswiththepropertya 1 0 A 1 0=V:GiventheBi 0 's,a 0 characterizestheimpulseresponsefunctionofthez t 'sto" t (see(3.4)(3.6).) TheVARrepresentationof ~ Z t ;obtainedbysuitablyreorderingtheequationsin(3.1),ischaracterizedbydb i D 0, i=1;:::;q;anddvd 0 : 17 Also,itiseasilyveri edthat(a 0 D 0 ) 1 h (A 0 D 0 ) 1i 0 =DVD 0 ;and thatgiventhedb i D 0 's,a 0 D 0 characterizestheimpulseresponsefunctionofthe ~Z t 'sto" t : Moreover,theseresponses coincidewith theresponses ofthecorresponding variablesinz t to" t. NotethatA 0 D 0 isnotingenerallowertriangular. Let ~ A 0 =A 0 D 0 : ~A 0 = ~a ~a 21 ~a 22 0 ~a 31 ~a 32 ~a 33 where~a ii isfullrank,butnotnecessarilylowertriangular,fori=1;3:lettheqrdecompositionofthesematricesbe~a ii =Q i R i ;whereq i isasquare,orthonormalmatrix,andr i is lower triangular with positive elements along the diagonal. This decomposition exists as long as ~a ii ;i=1;3;is nonsingular, a property guaranteed by thefact A 0 2Q V \Q (see Strang(1976,p. 124)). 18 Let W = Note that WW 0 = I; ³ W ~A 0 1 ³W ~A Q Q ; 7 5: = DVD 0 ; and W ~A 0 is lower triangular with 16 The type of reasoning in the previous footnote indicates that permuting the columns of the identity matrix does not alter orthonormality. 17 To see this, simply premultiply (3.1) by D on both sides and note that B i Z t i = B i D 0 DZ t i ; because D 0 D = I: 18 Actually, it is customary to state the QR decomposition of the (n n) matrix A as A = QR; where R is upper triangular. We get it into lower triangular form by constructing the orthonormal matrix E with zeros everywhere and 1's in the (n + 1 i; i) th entries, i = 1;2;:::;n; and writing A = (QE)(E 0 R): The orthonormal matrix to which we refer in the text is actually QE: 18
19 positive elements along the diagonal. Since ³ W A ~ 0 1 = A ~ 1 0 W 0 ; the (k 1 +1) th columns of ~A 1 0 W 0 and ~A 1 0 coincide. We conclude that, under the normalization that A 0 is lower diagonal with positive diagonal terms, the response of the variables in Z t to a monetary policyshockisinvarianttotheorderingofvariablesinx 1t andx 2t :Thisestablishes(iii). Wenow summarizetheseresults in theformofa proposition. Proposition 4.1. ConsiderthesetsQ V andq : (i) ThesetQ V \Q isnonemptyandcontainsmorethanoneelement. (ii) The(k 1 +1) th columnof i ;i=0;1;:::in(3.6)isinvarianttothechoiceofa 0 2Q V \Q : (iii) Restricting A 0 2 Q V \Q to be lower triangular with positive diagonal terms, the (k 1 +1) th columnof i ;i=0;1;:::isinvarianttotheorderingoftheelementsinx 1t andx 2t : We now provide a brief discussion of (i)(iii). According to results (i) and (ii), under the recursiveness assumption the data are consistent with an entire family, Q V \Q ; of A 0 matrices. Itfollowsthattherecursiveness assumption is notsu±cientto pin downthe dynamic response functions of the variables in Z t to every element of " t : But, each A 0 2 Q V \Q doesgeneratethesameresponsetooneofthe" t 's,namelytheonecorrespondingto the monetary policyshock. In this sense,the recursiveness assumption identi es the dynamic responseofz t toamonetaryshock,butnottheresponsetoothershocks. In practice, computational convenience dictates the choice of some A 0 2 Q V \Q : A standard normalization adopted in the literature is that thea 0 matrix is lower triangular with nonnegative diagonal terms. This stillleaves open the question of how to order the variablesinx 1t andx 2t :But,accordingtoresult(iii),thedynamicresponseofthevariables inz t toamonetarypolicyshockisinvarianttothisordering. Atthesametime,thedynamic impactonz t ofthenonpolicyshocksissensitivetotheorderingofthevariablesinx 1t and X 2t :Therecursiveness assumptionhasnothing to sayaboutthis ordering. Absentfurther identifying restrictions,the nonpolicy shocks and the associated dynamic response functions simply re ect normalizations adopted for computationalconvenience. 4.2.Three BenchmarkIdenti cation Schemes We organize our empirical discussion around three benchmarkrecursiveidenti cation schemes. Thesecorrespondtodi erentspeci cationsofs t and t :Inour rstbenchmarksystem,we measure the policy instrument, S t, by the time t federal funds rate. This choice is motivated byinstitutional arguments in McCallum (1983), Bernankeand Blinder (1992) and 19
20 Sims(1986,1992). LetY t ;P t ;PCOM t ;FF t ;TR t ;NBR t ;andm t denotethetimetvaluesof the log of real GDP,the log of the implicit GDP de ator,the smoothed changein an index of sensitive commodity prices (a component in the Bureau of Economic Analysis'index of leading indicators),the federalfunds rate,the log of totalreserves,the log of nonborrowed reservesplus extendedcredit,andthe log ofeitherm1orm2;respectively. Herealldata arequarterly. Ourbenchmarkspeci cationof t includescurrentandfourlaggedvaluesof Y t ;P t ;andpcom t ;aswellasfourlaggedvaluesofff t ;NBR t ;TR t andm t. Wereferto the policy shock measure corresponding to this speci cation as an FF policy shock. InoursecondbenchmarksystemwemeasureS t bynbr t :Thischoiceismotivatedby arguments in Eichenbaum(1992) and Christiano and Eichenbaum(1992) that innovations to nonborrowed reserves primarily re ect exogenous shocks to monetary policy,while innovations to broader monetary aggregates primarily re ect shocks to money demand. We assume that t includescurrentandfourlaggedvaluesofy t ;P t ;andpcom t ;aswellasfourlagged valuesofff t ;NBR t ;TR t andm t. Werefertothepolicyshockmeasurecorrespondingto thisspeci cationasannbrpolicyshock. Notethatinbothbenchmarkspeci cations,themonetaryauthorityisassumedtoseey t ; P t andpcom t ;whenchoosings t : 19 Thisassumptioniscertainlyarguablebecausequarterly real GDP data and the GDP de ator are typically known only with a delay. Still,the Fed does haveat its disposal monthly data on aggregate employment,industrial output and other indicators of aggregate real economic activity. It also has substantial amounts of information regardingthepricelevel. InourviewtheassumptionthattheFedseesY t andp t whenthey chooses t seemsatleastasplausibleasassumingthattheydon't. 20 Belowwedocumentthe e ect of deviating from this benchmark assumption. Notice that under our assumptions, Y t ; P t and PCOM t do not change in the impact periodofeitheranff orannbrpolicyshock. CEE(1997b)presentadynamicstochastic general equilibrium model which is consistent with the notion that prices and output do not move appreciably in the impact period of a monetary policy shock. The assumption regarding PCOM t ismoredi±culttoassessontheoreticalgroundsabsentanexplicitmonetarygeneral equilibrium model that incorporates a market for commodity prices. In any event,weshow below that altering the benchmark speci cation to exclude the contemporaneous value of PCOM t from t hasvirtuallynoe ectonourresults. 21 In the following subsection we display the time series ofthetwo benchmarkpolicy shock 19 Examples of analyses which make this type of information assumption include Christiano and Eichenbaum (1992), CEE (1996a, 1997a), Eichenbaum and Evans (1995), Strongin (1995), Bernanke and Blinder (1992), Bernanke and Mihov (1995), and Gertler and Gilchrist (1994). 20 See for example the speci cations in Sims and Zha (1995) and Leeper, Sims and Zha (1996). 21 This does not mean that excluding lagged values from t has no e ect on our results. 20
Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem
Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become
More informationNormalization and Mixed Degrees of Integration in Cointegrated Time Series Systems
Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Robert J. Rossana Department of Economics, 04 F/AB, Wayne State University, Detroit MI 480 EMail: r.j.rossana@wayne.edu
More informationComments on \Do We Really Know that Oil Caused the Great Stag ation? A Monetary Alternative", by Robert Barsky and Lutz Kilian
Comments on \Do We Really Know that Oil Caused the Great Stag ation? A Monetary Alternative", by Robert Barsky and Lutz Kilian Olivier Blanchard July 2001 Revisionist history is always fun. But it is not
More informationy t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where
. Variance decomposition and innovation accounting Consider the VAR(p) model where (L)y t = t, (L) =I m L L p L p is the lag polynomial of order p with m m coe±cient matrices i, i =,...p. Provided that
More informationIDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011
IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION
More informationChapter 3: The Multiple Linear Regression Model
Chapter 3: The Multiple Linear Regression Model Advanced Econometrics  HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics
More informationSYSTEMS OF REGRESSION EQUATIONS
SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations
More informationWhy Does Consumption Lead the Business Cycle?
Why Does Consumption Lead the Business Cycle? Yi Wen Department of Economics Cornell University, Ithaca, N.Y. yw57@cornell.edu Abstract Consumption in the US leads output at the business cycle frequency.
More informationResearch Division Federal Reserve Bank of St. Louis Working Paper Series
Research Division Federal Reserve Bank of St. Louis Working Paper Series Comment on "Taylor Rule Exchange Rate Forecasting During the Financial Crisis" Michael W. McCracken Working Paper 2012030A http://research.stlouisfed.org/wp/2012/2012030.pdf
More informationCAPM, Arbitrage, and Linear Factor Models
CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose meanvariance e cient portfolios. By equating these investors
More informationChapter 2. Dynamic panel data models
Chapter 2. Dynamic panel data models Master of Science in Economics  University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans April 2010 Introduction De nition We now consider
More informationBank Loan Portfolios and the Canadian Monetary Transmission Mechanism
Bank Loan Portfolios and the Canadian Monetary Transmission Mechanism Wouter J. DEN HAAN, y Steven W. SUMNER, z Guy M. YAMASHIRO x May 29, 2008 Abstract Following a monetary tightening, bank loans to consumers
More informationMonetary Policy Surprises, Credit Costs. and. Economic Activity
Monetary Policy Surprises, Credit Costs and Economic Activity Mark Gertler and Peter Karadi NYU and ECB BIS, March 215 The views expressed are those of the authors and do not necessarily reflect the offi
More informationChapter 4: Vector Autoregressive Models
Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationFiscal and Monetary Policy in Australia: an SVAR Model
Fiscal and Monetary Policy in Australia: an SVAR Model Mardi Dungey and Renée Fry University of Tasmania, CFAP University of Cambridge, CAMA Australian National University September 21 ungey and Fry (University
More informationCan we rely upon fiscal policy estimates in countries with a tax evasion of 15% of GDP?
Can we rely upon fiscal policy estimates in countries with a tax evasion of 15% of GDP? Raffaella Basile, Ministry of Economy and Finance, Dept. of Treasury Bruno Chiarini, University of Naples Parthenope,
More informationIn ation Tax and In ation Subsidies: Working Capital in a Cashinadvance model
In ation Tax and In ation Subsidies: Working Capital in a Cashinadvance model George T. McCandless March 3, 006 Abstract This paper studies the nature of monetary policy with nancial intermediaries that
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationIntermediate Macroeconomics: The Real Business Cycle Model
Intermediate Macroeconomics: The Real Business Cycle Model Eric Sims University of Notre Dame Fall 2012 1 Introduction Having developed an operational model of the economy, we want to ask ourselves the
More informationPanel Data Econometrics
Panel Data Econometrics Master of Science in Economics  University of Geneva Christophe Hurlin, Université d Orléans University of Orléans January 2010 De nition A longitudinal, or panel, data set is
More informationDiscussion of Capital Injection, Monetary Policy, and Financial Accelerators
Discussion of Capital Injection, Monetary Policy, and Financial Accelerators Karl Walentin Sveriges Riksbank 1. Background This paper is part of the large literature that takes as its starting point the
More informationAn Introduction into the SVAR Methodology: Identification, Interpretation and Limitations of SVAR models
Kiel Institute of World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1072 An Introduction into the SVAR Methodology: Identification, Interpretation and Limitations of SVAR
More informationDEMB Working Paper Series N. 53. What Drives US Inflation and Unemployment in the Long Run? Antonio Ribba* May 2015
DEMB Working Paper Series N. 53 What Drives US Inflation and Unemployment in the Long Run? Antonio Ribba* May 2015 *University of Modena and Reggio Emilia RECent (Center for Economic Research) Address:
More informationBias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes
Bias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationTopic 5: Stochastic Growth and Real Business Cycles
Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45 Lag Operators The lag operator (L) is de ned as Similar
More informationImport Prices and Inflation
Import Prices and Inflation James D. Hamilton Department of Economics, University of California, San Diego Understanding the consequences of international developments for domestic inflation is an extremely
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulationbased method for estimating the parameters of economic models. Its
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationIndeterminacy, Aggregate Demand, and the Real Business Cycle
Indeterminacy, Aggregate Demand, and the Real Business Cycle Jess Benhabib Department of Economics New York University jess.benhabib@nyu.edu Yi Wen Department of Economics Cornell University Yw57@cornell.edu
More informationReal Business Cycles. Federal Reserve Bank of Minneapolis Research Department Staff Report 370. February 2006. Ellen R. McGrattan
Federal Reserve Bank of Minneapolis Research Department Staff Report 370 February 2006 Real Business Cycles Ellen R. McGrattan Federal Reserve Bank of Minneapolis and University of Minnesota Abstract:
More informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN13: 9780470860809 ISBN10: 0470860804 Editors Brian S Everitt & David
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 201112) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More informationChapter 11. Keynesianism: The Macroeconomics of Wage and Price Rigidity. 2008 Pearson AddisonWesley. All rights reserved
Chapter 11 Keynesianism: The Macroeconomics of Wage and Price Rigidity Chapter Outline RealWage Rigidity Price Stickiness Monetary and Fiscal Policy in the Keynesian Model The Keynesian Theory of Business
More informationExact Nonparametric Tests for Comparing Means  A Personal Summary
Exact Nonparametric Tests for Comparing Means  A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
More informationHas Monetary Policy Become Less Powerful?
Has Monetary Policy Become Less Powerful? Jean Boivin y Columbia University Marc Giannoni z Federal Reserve Bank First Draft: March 2001 This Version: January 2002 of New York JEL Classi cation: E52, E3,
More information1 Example of Time Series Analysis by SSA 1
1 Example of Time Series Analysis by SSA 1 Let us illustrate the 'Caterpillar'SSA technique [1] by the example of time series analysis. Consider the time series FORT (monthly volumes of fortied wine sales
More informationA Review of the Literature of Real Business Cycle theory. By Student E XXXXXXX
A Review of the Literature of Real Business Cycle theory By Student E XXXXXXX Abstract: The following paper reviews five articles concerning Real Business Cycle theory. First, the review compares the various
More informationCorporate Defaults and Large Macroeconomic Shocks
Corporate Defaults and Large Macroeconomic Shocks Mathias Drehmann Bank of England Andrew Patton London School of Economics and Bank of England Steffen Sorensen Bank of England The presentation expresses
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationThe Cyclical Behavior of Debt and Equity Finance Web Appendix
The Cyclical Behavior of Debt and Equity Finance Web ppendix Francisco B. Covas and Wouter J. Den Haan December 15, 2009 bstract This appendix gives details regarding the construction of the data set and
More informationReal Business Cycle Models
Real Business Cycle Models Lecture 2 Nicola Viegi April 2015 Basic RBC Model Claim: Stochastic General Equlibrium Model Is Enough to Explain The Business cycle Behaviour of the Economy Money is of little
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationPortfolio selection based on upper and lower exponential possibility distributions
European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department
More informationRandom Walk Expectations and the Forward Discount Puzzle
Random Walk Expectations and the Forward Discount Puzzle Philippe Bacchetta and Eric van Wincoop* Two wellknown, but seemingly contradictory, features of exchange rates are that they are close to a random
More informationChapter 5: The Cointegrated VAR model
Chapter 5: The Cointegrated VAR model Katarina Juselius July 1, 2012 Katarina Juselius () Chapter 5: The Cointegrated VAR model July 1, 2012 1 / 41 An intuitive interpretation of the Pi matrix Consider
More informationThe RBC methodology also comes down to two principles:
Chapter 5 Real business cycles 5.1 Real business cycles The most well known paper in the Real Business Cycles (RBC) literature is Kydland and Prescott (1982). That paper introduces both a specific theory
More informationAlberto Musso at European Central Bank, Kaiserstrasse 29, D60311 Frankfurt am Main, Germany; email: alberto musso@ecb.europa.eu
Acknowledgements The financial support from the Spanish Ministry of Science and Innovation through grant ECO200909847 and the Barcelona Graduate School Research Network is gratefully acknowledged. The
More informationSubspaces of R n LECTURE 7. 1. Subspaces
LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r
More informationShould we Really Care about Building Business. Cycle Coincident Indexes!
Should we Really Care about Building Business Cycle Coincident Indexes! Alain Hecq University of Maastricht The Netherlands August 2, 2004 Abstract Quite often, the goal of the game when developing new
More informationOnline Appendix to Impatient Trading, Liquidity. Provision, and Stock Selection by Mutual Funds
Online Appendix to Impatient Trading, Liquidity Provision, and Stock Selection by Mutual Funds Zhi Da, Pengjie Gao, and Ravi Jagannathan This Draft: April 10, 2010 Correspondence: Zhi Da, Finance Department,
More information1 Another method of estimation: least squares
1 Another method of estimation: least squares erm: estim.tex, Dec8, 009: 6 p.m. (draft  typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i
More informationCardiff Economics Working Papers
Cardiff Economics Working Papers Working Paper No. E015/8 Comparing Indirect Inference and Likelihood testing: asymptotic and small sample results David Meenagh, Patrick Minford, Michael Wickens and Yongdeng
More informationCash in advance model
Chapter 4 Cash in advance model 4.1 Motivation In this lecture we will look at ways of introducing money into a neoclassical model and how these methods can be developed in an effort to try and explain
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationMeasuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices  Not for Publication
Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices  Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationCorporate Income Taxation
Corporate Income Taxation We have stressed that tax incidence must be traced to people, since corporations cannot bear the burden of a tax. Why then tax corporations at all? There are several possible
More informationReal Business Cycle Theory. Marco Di Pietro Advanced () Monetary Economics and Policy 1 / 35
Real Business Cycle Theory Marco Di Pietro Advanced () Monetary Economics and Policy 1 / 35 Introduction to DSGE models Dynamic Stochastic General Equilibrium (DSGE) models have become the main tool for
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationInterlinkages between Payment and Securities. Settlement Systems
Interlinkages between Payment and Securities Settlement Systems David C. Mills, Jr. y Federal Reserve Board Samia Y. Husain Washington University in Saint Louis September 4, 2009 Abstract Payments systems
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More information14.451 Lecture Notes 10
14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2
More informationThe Myth of Financial Innovation and the Great Moderation
The Myth of Financial Innovation and the Great Moderation Wouter J. Den Haan and Vincent Sterk July 28, 21 Abstract Financial innovation is widely believed to be at least partly responsible for the recent
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationVoluntary Voting: Costs and Bene ts
Voluntary Voting: Costs and Bene ts Vijay Krishna y and John Morgan z November 7, 2008 Abstract We study strategic voting in a Condorcet type model in which voters have identical preferences but di erential
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationGeometric Brownian Motion, Option Pricing, and Simulation: Some SpreadsheetBased Exercises in Financial Modeling
Spreadsheets in Education (ejsie) Volume 5 Issue 3 Article 4 November 01 Geometric Brownian Motion, Option Pricing, and Simulation: Some SpreadsheetBased Exercises in Financial Modeling Kevin D. Brewer
More informationConditional guidance as a response to supply uncertainty
1 Conditional guidance as a response to supply uncertainty Appendix to the speech given by Ben Broadbent, External Member of the Monetary Policy Committee, Bank of England At the London Business School,
More informationOur development of economic theory has two main parts, consumers and producers. We will start with the consumers.
Lecture 1: Budget Constraints c 2008 Je rey A. Miron Outline 1. Introduction 2. Two Goods are Often Enough 3. Properties of the Budget Set 4. How the Budget Line Changes 5. The Numeraire 6. Taxes, Subsidies,
More information3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
More informationLongTerm Debt Pricing and Monetary Policy Transmission under Imperfect Knowledge
LongTerm Debt Pricing and Monetary Policy Transmission under Imperfect Knowledge Stefano Eusepi, Marc Giannoni and Bruce Preston The views expressed are those of the authors and are not necessarily re
More informationA Comparison of Option Pricing Models
A Comparison of Option Pricing Models Ekrem Kilic 11.01.2005 Abstract Modeling a nonlinear pay o generating instrument is a challenging work. The models that are commonly used for pricing derivative might
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25Sep02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationWhat s New in Econometrics? Lecture 8 Cluster and Stratified Sampling
What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationDo federal budget deficits cause crowding out?
Do federal budget deficits cause crowding out? Tricia Coxwell Snyder William Paterson University Abstract Currently the U.S. President and congress are debating the size and role of government spending
More informationA note on the impact of options on stock return volatility 1
Journal of Banking & Finance 22 (1998) 1181±1191 A note on the impact of options on stock return volatility 1 Nicolas P.B. Bollen 2 University of Utah, David Eccles School of Business, Salt Lake City,
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationChanging income shocks or changed insurance  what determines consumption inequality?
Changing income shocks or changed insurance  what determines consumption inequality? Johannes Ludwig Ruhr Graduate School in Economics & RuhrUniversität Bochum Abstract Contrary to the implications of
More information11.2 Monetary Policy and the Term Structure of Interest Rates
518 Chapter 11 INFLATION AND MONETARY POLICY Thus, the monetary policy that is consistent with a permanent drop in inflation is a sudden upward jump in the money supply, followed by low growth. And, in
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationMonetary Policy and Credit Cards: Evidence from a SmallOpen Economy
Monetary Policy and Cards: Evidence from a SmallOpen Economy by Hakan Yilmazkuday Department of Economics DETU Working Paper 11 September 21 131 Cecil B. Moore Avenue, Philadelphia, PA 19122 http://www.temple.edu/cla/economics/
More informationThe Decline of the U.S. Labor Share. by Michael Elsby (University of Edinburgh), Bart Hobijn (FRB SF), and Aysegul Sahin (FRB NY)
The Decline of the U.S. Labor Share by Michael Elsby (University of Edinburgh), Bart Hobijn (FRB SF), and Aysegul Sahin (FRB NY) Comments by: Brent Neiman University of Chicago Prepared for: Brookings
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
More informationThe Credit Spread Cycle with Matching Friction
The Credit Spread Cycle with Matching Friction Kevin E. BeaubrunDiant and Fabien Tripier y June 8, 00 Abstract We herein advance a contribution to the theoretical literature on nancial frictions and show
More informationUnderstanding the Effects of a Shock to Government Purchases*
Review of Economic Dynamics 2, 166 206 Ž 1999. Article ID redy.1998.0036, available online at http: www.idealibrary.com on Understanding the Effects of a Shock to Government Purchases* Wendy Edelberg Department
More informationCommon sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.
Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand
More informationDebt, Delinquencies, and Consumer Spending Jonathan McCarthy
February 1997 Volume 3 Number 3 Debt, Delinquencies, and Consumer Spending Jonathan McCarthy The sharp rise in household debt and delinquency rates over the last year has led to speculation that consumers
More informationKeywords: High Frequency Data, Identification, Vector Autoregression, Exchange Rates, Monetary Policy. JEL Classifications: C32, E52, F30.
EUROPEAN CENTRAL BANK WORKING PAPER SERIES WORKING PAPER NO 167 IDENTIFYING THE EFFECTS OF MONETARY POLICY SHOCKS ON EXCHANGE RATES USING HIGH FREQUENCY DATA BY JON FAUST, JOHN H ROGERS, ERIC SWANSON AND
More informationA Critique of Structural VARs Using Business Cycle Theory
Federal Reserve Bank of Minneapolis Research Department A Critique of Structural VARs Using Business Cycle Theory V. V. Chari, Patrick J. Kehoe, Ellen R. McGrattan Working Paper 631 Revised May 2005 ABSTRACT
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems  Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More information