The Analytics of SVARs: A Unified Framework to Measure Fiscal Multipliers

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. The Analytics of SVARs: A Unified Framework to Measure Fiscal Multipliers Dario Caldara and Christophe Kamps 2012-20 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Analytics of SVARs: A Uni(cid:2)ed Framework to Measure Fiscal Multipliers Dario Caldara and Christophe Kamps (cid:3) February 21, 2012 Abstract Does(cid:2)scalpolicystimulateoutput? SVARshavebeenusedtoaddressthisquestionbutno stylizedfactshaveemerged. Wederiveanalyticalrelationshipsbetweentheoutputelasticities of (cid:2)scal variables and (cid:2)scal multipliers. We show that standard identi(cid:2)cation schemes imply different priors on elasticities, generating a large dispersion in multiplier estimates. We then useextra-modelinformationtonarrowthesetofempiricallyplausibleelasticities,allowingfor sharperinferenceonmultipliers. OurresultsfortheU.S.fortheperiod1947-2006suggestthat the probability of the tax multiplier being larger than the spending multiplier is below 0.5 at allhorizons. JELClassi(cid:2)cation: E62;C52. Keywords: FiscalPolicy;Identi(cid:2)cation;VectorAutoregressions. Caldara: Board of Governors of the Federal Reserve System, Division of Research and Statistics, 20th Street (cid:3) and Constitution Avenue, N.W., Washington, DC 20551 (e-mail: dario.caldara@frb.gov). Kamps: European Central Bank, Monetary Policy Strategy Division, Kaiserstr. 29, 60311 Frankfurt am Main, Germany (e-mail: christophe.kamps@ecb.europa.eu). We would like to thank Jesœs FernÆndez-Villaverde, John Hassler, Torsten Persson, John Roberts and Frank Schorfheide for valuable advice. We thank seminar participants at Sveriges Riksbank, ECB, Bank of England, IIES, Stockholm School of Economics, University of Pennsylvania, Indiana University, the FourthOsloWorkshoponEconomicPolicyandtheWorldCongressoftheEconometricSocietyforhelpfulcomments. Theviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpretedasre(cid:3)ecting theviewsoftheBoardofGovernorsoftheFederalReserveSystemortheEuropeanCentralBank. 1

Governmentsoftenuse(cid:2)scalpolicytostabilizeeconomic(cid:3)uctuations. Forexample,duringthe recent recession, the United States Congress approved the American Recovery and Reinvestment Act,whichintroducedincreasesinpublicspendingandcutsintaxesbyapproximately6%ofGDP (CBO, 2010). The rationale for such (cid:2)scal stimulus rests on the assumption that (cid:2)scal interventions do affect economic activity. Yet, the size of (cid:2)scal multipliers, de(cid:2)ned as the dollar response of output to an exogenous dollar spending increase or tax cut, is the subject of a long-standing debate in academia. As Perotti (2008) observes in his survey of the literature: "... perfectly reasonable economists can and do disagree on the basic theoretical effects of (cid:2)scal policy and on the interpretationofexistingempiricalevidence". The presence of competing economic theories has motivated a large body of empirical investigations that measure the size of these (cid:2)scal multipliers. An important share of the literature relies on structural vector autoregressions (SVARs). Prominent examples include Blanchard and Perotti (2002), and Mountford and Uhlig (2009). The appeal of SVARs is that they control for endogenous movements in (cid:2)scal policies by imposing only a minimal set of assumptions, known as identi(cid:2)cation schemes. An alternative methodology identi(cid:2)es exogenous changes in taxation (Mertens and Ravn, 2011b; Romer and Romer, 2010) and public spending (Ramey and Shapiro, 1998; Eichenbaum and Fisher, 2005; Ramey, 2011) from narrative records, and studies their effectsusingVARs. Yet,despitetheirsimplestructureandtheuseofsimilardata,studiesemploying SVARs and narrative records report (cid:2)scal multipliers that are spread over a broad range of values. The lack of consensus prevents the profession from providing clear guidance on important policy choices,suchasthesizeandcompositionof(cid:2)scalinterventions. Motivated by this knowledge gap, our paper asks two questions. Why do SVARs provide different measures of (cid:2)scal multipliers? Can we construct robust measures of (cid:2)scal multipliers usingSVARs? Weanswerthe(cid:2)rstquestionbyderivingauni(cid:2)edanalyticalframeworktocomparecompeting identi(cid:2)cation schemes. We then apply this analysis to a (cid:2)scal VAR for the United States for the period1947-2006. Weshowthatexistingidenti(cid:2)cationschemesimplydifferentrestrictionsonthe 2

outputelasticityoftaxrevenueandgovernmentspending,whichmeasuretheendogenousresponse oftaxandspendingpoliciestoeconomicactivity. We illustrate the framework for comparing different identi(cid:2)cation schemes with a tax policy example. Assume that only two shocks explain contemporaneous co-movements between output and tax revenue: a tax shock and a non-policy shock. The object of interest is the response of outputto thetaxshock. Thenon-policy shockcontrolsforco-movements inthetwovariables due toautomaticmovementsoftaxrevenueoverthebusinesscycle. Inthissetting,theidenti(cid:2)cationof tax and non-policy shocks depends only on the restriction on one structural coef(cid:2)cient: the output elasticityoftaxrevenue. The Blanchard and Perotti (2002) and the Mountford and Uhlig (2009) identi(cid:2)cation schemes implyoutputelasticitiesoftaxrevenueequalto1:7and3:0,respectively. Standardsignrestrictions on impulse response functions imply output elasticities of tax revenue between zero and in(cid:2)nity. Narrative identi(cid:2)cation of tax shocks imply an output elasticity of tax revenue above 3. Different restrictions on the output elasticity of tax revenue generate a large dispersion in the estimates of tax multipliers. For instance, we (cid:2)nd that the impact tax multiplier is 0:17 dollars for an output elasticity of tax revenue equal to 1:7, whereas it is more than (cid:2)ve times as large - at 0:93 dollars - for an output elasticity of tax revenue equal to 3:0. The impact tax multiplier is negative for all output elasticities of tax revenue smaller than 1:5. More in general, thanks to the analytical relations, we can readily map beliefs of policy-makers and economists about plausible values of theoutputelasticityoftaxrevenueintotaxmultipliers. These (cid:2)ndings lead to the second question. We propose to measure (cid:2)scal multipliers more robustly by imposing restrictions on the output elasticities of (cid:2)scal variables in the form of probabilitydistributions. Incontrasttotheexistingliterature,weimposedistributionsthatencompassthe existing empirical evidence on elasticities. The distribution of the output elasticity of tax revenue that we obtain ranges between 1:7 and 3. The distribution of the output elasticity of government spendingrangesbetween 0:1and0:1. Theserestrictionsarerobustbecausetheyaregeneratedby (cid:0) differentapproachesandempiricalstrategiesand,hence,arelesslikelytobeaffectedbyparticular 3

assumptionsorobservations. We apply this robust identi(cid:2)cation scheme to measure tax and spending multipliers associated with unexpected (cid:2)scal shocks.1 We document three (cid:2)ndings. First, the median tax multiplier is 0:65 on impact, and it becomes larger than 1 (cid:2)ve quarters after the policy intervention. Second, the median spending multiplier is larger than 1 at all horizons. Third, the probability that the tax multiplierislargerthanthespendingmultiplierisbelow0:5atallhorizons. We also document a high probability that private consumption increases following a spending shock. Competing macroeconomic theories have different theoretical predictions regarding the effects of government spending shocks on private consumption. The standard neoclassical and New Keynesian models predict a decline in consumption (Baxter and King, 1993; Linnemann andSchabert,2003). Arecentbranchoftheliterature(Gal(cid:237),Lopez-SalidoandValles,2007;Ravn, Schmitt-GrohØandUribe,2006)proposesmodelsthatgenerateanincreaseinprivateconsumption. Theevidenceisinlinewiththelatterclassofmodels. The focus on the identi(cid:2)cation problem, as opposed to the estimation and speci(cid:2)cation of the reduced-form VAR model, is based on evidence provided by Chahrour, Schmitt-GrohØ and Uribe (forthcoming)andCaldaraandKamps(2008). Chahrour,Schmitt-GrohØandUribe(forthcoming) employ a DSGE-model approach to reject the hypothesis that the different tax multipliers estimated by the SVAR and narrative approaches are due to differences in the assumed reduced-form transmissionmechanisms. CaldaraandKamps(2008)(cid:2)ndthat,controllingforthespeci(cid:2)cationof the reduced-form VAR model, different identi(cid:2)cation schemes provide different estimates of tax andspendingmultipliers. The remainder of the paper is organized as follows. Section I derives the analytical relation between output elasticities of (cid:2)scal variables and impulse response functions. It also characterizes theoretical properties of such relations. Section II reinterprets (cid:2)ve alternative identi(cid:2)cation 1Theestimationstrategyaddressesthewell-knownmisspeci(cid:2)cationproblemofSVARsinthepresenceofanticipated(cid:2)scalshocks(Leeper,WalkerandYang,2008). Weincludeasetofvariablesthatreactstosignalsaboutfuture policies, suchasconsumption, investment, andvariousmeasuresofprices. Laggedvaluesofthesevariablespredict future policy actions and, consequently, help to identifying truly unexpected (cid:2)scal shocks (Giannone and Reichlin, 2006;ForniandGambetti,2010). 4

schemes used in the literature as restrictions on the output elasticities of (cid:2)scal variables. Section III analyses the implications of alternative priors on elasticities for (cid:2)scal multipliers. Section IV reviews the existing empirical evidence on elasticities and reports evidence on (cid:2)scal multipliers based on prior distributions that encompass the full range of elasticity estimates documented in the literature. Section V sheds light on two debates in the literature on the effects of government spending shocks: the response of private consumption and the role of (cid:2)scal foresight. Section VI concludes. 1 The Analytics of SVARs Considerthereduced-formVARmodel: X (cid:22) B.L/X u ; (1) t t 1 t D C (cid:0) C where X is a vector of n endogenous variables, (cid:22) is a constant, B.L/ is a lag polynomial of t order L,andu isavectorofone-step-aheadpredictionerrorswithzeromeanandpositivede(cid:2)nite t covariancematrix6 [(cid:27) ]. u ij D The reduced-form disturbances will in general be correlated with each other and consequently do not have any economic interpretation. It is thus necessary to model the contemporaneous relation between reduced-from disturbances u to identify structural shocks e with an economic t t interpretation: u Fe ; (2) t t D where F isafactormatrixholdingthestructuralcoef(cid:2)cients. Weassumethatthestructuralshocks e have zero mean, have unit variance, and are uncorrelated with each other, i.e. the covariance t matrix of structural shocks 6 is the identity matrix. We restrict attention to the class of juste identi(cid:2)ed SVAR models for which FF 6 , which nests the SVAR identi(cid:2)cation approaches 0 u D usedintheliteraturetoidentifytheeffectsof(cid:2)scalpolicyshocks. Columnsofmatrix F areknown 5

asimpulsevectors(Uhlig,2005),withthei; j elementof F givingthecontemporaneouseffecton variablei ofshock j. Numerical results presented in the paper are based on the estimation of an 8-equation VAR model in the logarithm of output, tax revenue, government spending (sum of government consumption and investment), private consumption, private non-residential investment (all in real, per-capita terms), CPI in(cid:3)ation, the 3-month T-bill rate, and a measure of stock prices. We use quarterly data for the United States from 1947 to 2006.2 To illustrate our main results on identi(cid:2)cation uncertainty, we abstract from sampling uncertainty and present results based on OLS point estimates; the evidence on (cid:2)scal multipliers reported later in the paper accounting both for identi(cid:2)cationandsamplinguncertaintyinsteadreliesonBayesianestimates.3 The estimation strategy addresses the well-known misspeci(cid:2)cation problem of SVARs in the presence of anticipated (cid:2)scal shocks (Leeper, Walker and Yang, 2008). Variables such as consumption, investment, and prices react to signals about future policies. Lagged values of these variables predict future policy actions and, consequently, help to identify truly unexpected (cid:2)scal shocks(GiannoneandReichlin,2006;ForniandGambetti,2010).4 Byincludingthesevariablesin our reduced-form VAR model, we ensure that measures of anticipated (cid:2)scal shocks derived from narrativerecords(seeRamey(2011)formeasuresofanticipatedgovernmentspendingshocks,and MertensandRavn(2011b)formeasuresofanticipatedtaxshocks)donotGranger-causethe(cid:2)scal shocksidenti(cid:2)edbyourSVARmodels(seeAppendixA). 2Thesampleendsin2006becauseitisthelastyearforwhichnarrativemeasuresofunexpectedtaxshockthatwe introducelaterinbthepaperareavailable. 3FortheBayesianestimatesweimposeaMinnesotaprioronthereduced-formVARcoef(cid:2)cientsfollowingDelNegroandSchorfheide(2011). Priorsarebasedonhyper-parametersandpre-sampledata. Ourpre-sampleis1947-1951. Estimation results (including OLS estimates) are based on data from 1952-2006. Details on the dataset and on the Bayesianframeworkarereportedintheappendix. 4Leeper, Walker and Yang (2008) show that simple macroeconomic models where agents receive signals about future(cid:2)scalpolicydonothaveaVARrepresentation. Thesemodelsarenon-invertible. GiannoneandReichlin(2006) and Forni and Gambetti (2010) suggest that forward-looking variables mitigate the non-invertibility problem. If the econometrician observes a large number of forward-looking variables, the model should become close to invertible, andthebiasininferenceshouldbesmall.Foradetaileddiscussiononnon-invertibility,seeFernÆndez-Villaverdeetal. (2007). 6

1.1 Deriving the Analytical Relationship Between Output Elasticities of Fiscal Variables and Fiscal Multipliers Tounderstandhowthechoiceofidentifyingrestrictionsaffectsinference,weconsiderasimpli(cid:2)ed example. Weassumethatthemodelconsistsofanon-policyvariableandofapolicyvariable. The non-policyvariableisoutput(Y ). Thepolicyvariable,denotedby P;iseithertaxrevenue(T ),or t t t governmentspending(G ). t Therelationbetweenreduced-formdisturbancesu andstructuralshockse canbewrittenas: t t u a u d e ; (3) Y;t Y;P P;t Y Y;t D C u a u d e ; (4) P;t P;Y Y;t P P;t D C where u and u are the one-step prediction errors for output and the policy variable, respec- Y;t P;t tively,andd andd arethestandarddeviationsofthestructuraloutputandpolicyshocks,respecy P tively. Equation (3) states that unexpected movements in output are due to either unexpected movements in the policy variable (a u ) or sources of business cycle (cid:3)uctuations unrelated to the Y;P P;t policy under investigation (e ). Equation (4) states that unexpected changes in the policy vari- Y;t ableareeitherendogenoustothebusinesscycle(a u )orexogenoustothebusinesscycleand P;Y Y;t uncorrelated with non-policy sources of (cid:3)uctuations (e ). Endogeneity of policy can arise either P;t becausepolicy-makersreacttocontemporaneousdevelopmentsineconomicactivity,orbecauseof automatic feedback from activity to tax revenue and government spending. We follow Blanchard and Perotti (2002), B&P henceforth, and assume that the (cid:2)rst channel is eliminated by the use of quarterly data. This is plausible due to information lags, legislative lags, and implementation lags faced by policy makers. Consequently, the coef(cid:2)cient a captures the automatic response P;Y of(cid:2)scalvariablestochangesineconomicactivity,measuredastheoutputelasticityoftaxrevenue ((cid:17) )andtheoutputelasticityofgovernmentspending((cid:17) ),respectively. T;Y G;Y In the bivariate case, we need to impose one identi(cid:2)cation restriction to identify the SVAR 7

model: here this boils down to a restriction on a .5 To highlight the restricted coef(cid:2)cient, we P;Y denote throughout the paper a as (cid:17) . In the public (cid:2)nance literature, there is a long tradition P;Y P;Y of measurement of the output elasticity of (cid:2)scal variables in the context of the cyclical adjustment of budget balances. The output elasticity of tax revenue (cid:17) is the most familiar measure T;Y of sensitivity of taxes to income changes. This elasticity serves as an indicator of the tax system’s overall progressivity. A proportional income tax has an elasticity of 1.0. Progressive tax systems,forwhichtax-to-incomeratiosallotherthingsequalincreasewithincome,haveanelasticity larger than 1.0. As far as the output elasticity of government spending (cid:17) is concerned, most G;Y studies - including B&P - assume its value to be zero, based on the observation that government consumptionandinvestmenthaveatmostweakcyclicalcomponents. We view numerical restrictions as priors of the economist regarding a plausible value, or a set of plausible values, for the elasticities. As we describe in Section 2, in the literature economists haveformedandimplementedpriorson(cid:17) usingavarietyofmethods. P;Y ThesystemdescribedbyEquations(3)and(4)canbewrittenintermsofimpulsevectorsas: u Y;t 1 1 a Y;P d Y 0 e Y;t : (5) 2 3 2 32 32 3 D 1 a (cid:17) u Y;P P;Y (cid:17) 1 0 d e P;t (cid:0) P;Y P P;t 6 7 6 76 76 7 4 5 4 54 54 5 Theobjectofinterestisthecontemporaneousresponseofoutputtoapolicyshock:6 @u a Y;t Y;P : (6) @.d e / D 1 a (cid:17) P P;t Y;P P;Y (cid:0) The denominator of this expression measures the strength of macroeconomic feedback. In the specialcasesinwhicheitheroneofa or(cid:17) iszerothereisnofeedback. Y;P P;Y Whatweareinterestedinistoknowhowtheoutputresponsetoapolicyshockdependsonthe 5The necessary condition for exact identi(cid:2)cation states that in an n-variable model, there is a need for n.n (cid:0) 1/=2 restrictions. Rubio-Ram(cid:237)rez, Waggoner and Zha (2010) derive necessary and suf(cid:2)cient conditions for global identi(cid:2)cationofexactlyidenti(cid:2)edmodelswhich,inadditiontothecountingcondition,requirethatrestrictionsfollowa certainequationbyequationpattern.TheSVARsstudiedinthispapersatisfytheseconditionsforglobalidenti(cid:2)cation. 6Thesizeofthepolicyshockiscalibratedsuchthatu wouldincreasebyoneunitintheabsenceofmacroeco- P;t nomicfeedback,i.e. e 1=d . P;t p D 8

prior of the econometrician about (cid:17) . In the bivariate model, there exists a simple-closed form P;Y solution.7 Thecontemporaneousresponseofoutputtoapolicyshockcanberewrittenas:8 @u (cid:27) (cid:17) (cid:27) Y;t YP P;Y YY (cid:0) : (7) @.d P e P;t / D (cid:17)2 P;Y (cid:27) YY C (cid:27) PP (cid:0) 2(cid:17) P;Y (cid:27) YP Equation (7) reveals that for a given reduced-form model (i.e. given 6 ), the contemporaneous u response of output to a policy shock is a function of the identi(cid:2)cation restriction on the output elasticity of the policy variable ((cid:17) ). To obtain (cid:2)scal multipliers, we divide the contempora- P;Y neous output responses by the policy variable to output ratio.9 The key properties of the impact multiplier are summarized by Proposition 1 in the appendix. Furthermore, in the appendix we derive expression (7) and its properties in a multivariate model. To this end, we need to assume that equation (4) holds. That is, we assume that the shock e is enough to control for co-movements Y;t in Y and P unrelated to the policy of interest. Mountford and Uhlig (2009), M&U henceforth, t t identify, in addition to a non-policy shock, a monetary policy shock, and they (cid:2)nd that the identi(cid:2)cation of this shock has no impact on the (cid:2)scal multipliers. In a similar vein, Perotti (2005) (cid:2)nds that the contemporaneous responses of (cid:2)scal variables to in(cid:3)ation and interest rates have a negligibleimpacton(cid:2)scalmultipliers. Weinterpretthisevidenceassupportiveofourassumption. Finally, in the appendix we derive analytical expressions for the response to a policy shock of all modelvariablesatanyhorizon. 7Thissolutionalsoholdsforthetrivariatesystemwithonenon-policyvariable(output)andtwonon-policyvariables(taxesandgovernmentspending)studiedbyB&P. 8Theassumptionthat6 ispositivede(cid:2)niteensuresthatthedenominatorof(7)isstrictlylargerthanzero. This u guaranteesthatimpulseresponsefunctionsarede(cid:2)nedforalloutputelasticitiesovertherealline. 9Weconvertpercentchangesintodollarchanges,thelatterbeingtheunitinwhichmultipliersareusuallyreported, by dividing the output response to a (cid:2)scal shock by the tax-to-output or government spending-to-output ratio. We followB&Pinevaluating(cid:2)scalmultipliersatthesamplemeanofthetaxratioandspendingratio. Thisrescalingdoes notchangetheanalyticalpropertiesofexpression(7). 9

2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Output Elasticity of Tax Revenue reilpitluM xaT tcapmI BP MU CHOL NARR 3 2 1 0 -1 -2 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Output Elasticity of Spending reilpitluM gnidnepS tcapmI BP-CHOL MU Figure 1: Impact tax and spending multipliers as a function of the output elasticity of taxes and spending. 1.2 An Illustration of the Analytical Relationship Between Output Elasticities of Fiscal Variables and Fiscal Multipliers Tofacilitatethecomparisonbetweentaxmultipliersandspendingmultipliers,wecompareshocks thatareintendedtostimulateoutput,i.e. weanalyzetheeffectsofstructuralspendingincreasesbut structural tax cuts. Figure 1 plots the impact (cid:2)scal multiplier as a function of the output elasticity of the respective (cid:2)scal variable, evaluating the covariance matrix 6 at the OLS estimates of the u taxandspendingmodel,respectively. Figure1highlightstwoimportantpropertiesofexpression(7). First,thesetofoutputresponses to a policy shock is bounded, and bounds have opposite signs. An important implication is that if theeconometriciandoesnothaveanyinformationtolimitthesetofplausiblevaluesfortheoutput elasticity, the sign of the output response cannot be determined. Second, the output response to a policyshockiszeroifandonlyif(cid:17) (cid:17) (cid:27) =(cid:27) . Hence(cid:17) isthethresholdelasticity P;Y P;Y YP YY P;Y D (cid:17) todeterminethesignoftheoutputresponsetoapolicyshock. ThetoppanelofFigure1showsthat,iftheeconometriciantakesanagnosticviewofplausible 10

valuesfortheoutputelasticityoftaxes,theimpacttaxmultiplierlieswithinarangebetween 1:07 (cid:0) and 1:07 dollars. However, typically at least some extra-model information may be available to narrow down the set of plausible assumptions about the output elasticity of taxes. For example, it appears implausible to assume that the business cycle has a negative effect on tax revenue. Yet excluding negatives values of (cid:17) would still be insuf(cid:2)cient to pin down the sign - let alone the T;Y size-oftheimpacttaxmultiplier. Indeed,rulingout(cid:17) < 0,theimpacttaxmultiplierlieswithin T;Y a range between 0:92 and 1:07 dollars. Furthermore, even excluding (cid:17) < 1, i.e. assuming T;Y (cid:0) that the tax system is at least proportional, is insuf(cid:2)cient to pin down the sign of the impact tax multiplier. In this case, the impact tax multiplier lies within a range between 0:39 and 1:07 (cid:0) C dollars. To ensure that the impact tax (cut) multiplier is non-negative the econometrician has to assume (cid:17) (cid:27) =(cid:27) (cid:17) . In our application, the output elasticity of taxes has to be at T;Y YT YY T;Y (cid:21) (cid:17) least1:5. Turningtospendingshocks,thebottompanelofFigure1showsthat-again,iftheeconometrician takes an agnostic view of plausible values for the output elasticity of government spending theimpactspendingmultiplierlieswithinarangebetween2:26and 2:26dollars. However,neg- (cid:0) ative spending multipliers only occur if governmentspendingis procyclical ((cid:17) > (cid:27) =(cid:27) G;Y YG YY (cid:17) (cid:17) ), given that in empirical applications the correlation between output and government spend- G;Y ingresiduals(cid:27) ispositiveingeneral. Inourapplication,forallvaluesoftheoutputelasticityof YG governmentspendingsmallerthan0:38theimpactspendingmultiplierispositive. Summing up, we have shown that the sign and size of spending and tax multipliers depend on the choice of the output elasticity of tax revenue and government spending. We have also characterizedanalyticallytheidenti(cid:2)cationproblem. Inthenextsection,weshowhowthealternative identi(cid:2)cation schemes used in the existing literature can be mapped into priors of economists regardingtheoutputelasticitiesof(cid:2)scalvariables. 11

2 Identi(cid:2)cation Restrictions as Priors on Elasticities In this section, we use analytical results to reinterpret identi(cid:2)cation schemes as priors on output elasticities of (cid:2)scal variables. We show that these priors are different enough to produce widely divergent (cid:2)scal multipliers. We examine (cid:2)ve identi(cid:2)cation schemes used in the literature: the recursiveapproach,thetraditionalSVARanalysisimplementedbyB&P,the(cid:147)pure(cid:148)signrestriction approach,thepenaltyfunctionapproachtosignrestrictions,andthenarrativeapproach. 2.1 The Recursive Approach We (cid:2)rst analyze the recursive approach, proposed by Sims (1980). The recursive approach imposes a dogmatic prior either on the impact multiplier (tax policy) or on the output elasticity of government spending (spending policy). In the recursive approach the ordering of the variables in the reduced-form VAR model determines the contemporaneous effects of shocks: the variable ordered (cid:2)rst in the VAR system is only affected contemporaneously by the (cid:2)rst shock but not by the secondshock,whereasthevariableorderedsecondiscontemporaneouslyaffectedbybothshocks. The recursive VAR approach is applied via the lower-triangular Cholesky decomposition of the covariancematrix6 . u Tax Shocks. In our implementation of the recursive approach, we order output (cid:2)rst and tax revenuesecondintheVARsystem.10 Ontheonehand,assumingazerocontemporaneousresponse of output to a tax shock is restrictive. On the other hand, the alternative ordering, equivalent to assuming that tax revenue does not react at all contemporaneously to the business cycle, would be even more implausible. For the chosen order, the second property of the impact multiplier mentioned in Section 1 gives the result: the impact tax multiplier is zero if and only if the output elasticity of taxes is equal to (cid:17)CHOL (cid:27) =(cid:27) (cid:17) .11 In our VAR, (cid:17) is 1.5, a value T;Y D YT YY (cid:17) T;Y T;Y 10Inann equationmodel,aslongasequation(4)holds,theorderingofoutputandtheremainingn 2variables (cid:0) (cid:0) inthesystemisirrelevantfortheidenti(cid:2)cationofthepolicyshock. 11Fortheimplementationoftherecursiveapproachouranalyticalresultsarenotneeded. Itiswell-knownthatthe Cholesky factorization has an analytical solution which relies on a simple recursive algorithm, with the elements of theCholeskyfactorbeingfunctionsoftheelementsof6 . Ourcontributionistoshowthatananalyticalsolutionto u theidenti(cid:2)cationproblemisfeasiblenotonlyfortherestrictiverecursiveVARassumptionsbutmoregenerally,with 12

which,aswediscussinSection4,lieswithintherangeofempiricallyplausibleelasticities. Thisis the point denoted ‘CHOL’ in the top panel of Figure 1. This point is a useful reference point: the impacttax(cut)multiplierwillbepositiveifandonlyif(cid:17) > (cid:17)CHOL. T;Y T;Y SpendingShocks. Weordergovernmentspending(cid:2)rstandoutputsecond. Thatis,weassume thatgovernmentspendingisacyclical,i.e. (cid:17)CHOL 0. AsdiscussedinSection4,thisassumption G;Y D isinlinewiththeconsensusviewthatintheU.S.thecontemporaneousoutputelasticityofgovernment spending is zero. The point denoted ‘BP-CHOL’ in the bottom panel of Figure 1 shows that the impact spending multiplier amounts to 1:25 dollars for this value of the elasticity. The label ‘BP-CHOL’ reveals that in the case of the spending model this recursive formulation is equivalent totheB&Papproach,towhichweturnnext. 2.2 The Blanchard-Perotti Approach TheB&Papproachreliesoninstitutionalinformationaboutthetaxandtransfersystemsandabout the timing of tax collections in order to form a dogmatic prior about plausible out elasticities of (cid:2)scalvariables. WeprovideadetailedanalysisoftheB&Pmethodologytocalculateelasticitiesin Section4. Tax Shocks. The point denoted ‘BP’ in the top panel of Figure 1 gives the value of the impact tax multiplier for the point estimate of the output elasticity of taxes constructed according to the B&P methodology ((cid:17)BP 1:7). For this value of the output elasticity of taxes the impact tax T;Y D multiplier amounts to 0:17 dollars. Notice that in our sample, the B&P elasticity is only slightly larger than the elasticity implied by the recursive approach, with the implication that the B&P tax multiplierisonlyslightlylargerthanzero. Spending Shocks. As discussed in the previous subsection, B&P assume that government spending is acyclical, i.e. (cid:17)BP 0, which is equivalent to the lower-triangular Cholesky decom- G;Y D position with government spending ordered (cid:2)rst. Accordingly, the B&P and recursive approaches provideidenticalestimatesofspendingmultipliers(1:25dollarsonimpact). therecursiveVARbeingaspecialcasenestedinthemoregeneralformulation. 13

2.3 The Pure Sign Restriction Approach An alternative approach to identi(cid:2)cation is to impose sign restrictions on impulse responses. We base the discussion of this approach on the work by M&U. M&U impose sign restrictions on impulse responses in combination with a criterion function, discussed in the next subsection. The exerciseinthissubsectionunveilswhattheinferenceon(cid:2)scalmultipliersinM&Uwithoutpenalty function would have been. Other studies identifying (cid:2)scal shocks using the pure sign-restriction approach include Canova and Pappa (2007) and Pappa (2009). For the sake of simplicity, we only impose sign restrictions on impact responses.12 We continue to focus the theoretical discussion on a bivariate model, while presenting numerical results for a multivariate model. We provide intuition for the generalization of the theoretical results to multivariate models (see the appendix fordetails). We follow Uhlig’s suggestion to decompose the factor matrix F into the lower- triangular Cholesky factor of the reduced-form covariance matrix, denoted P, and an orthogonal matrix, denoted Q;with QQ I. Thatis,forthepuresignrestrictionapproachwehave FSR PQ. 0 D D Thesystemdescribingtherelationshipbetweenreduced-formdisturbancesandstructuralshocks can be written in compact form as u PQe . As shown in the appendix, this system (cid:150) using the t t D analyticalsolutionfortheCholeskyfactorization(cid:150)canbeexpressedasfollows: u (cid:27) cos(cid:18) (cid:27) sin(cid:18) e Y;t Y Y Y;t (cid:0) ; (8) 2 3 2 32 3 D u (cid:27) cos.(cid:18) ’ / (cid:27) sin.(cid:18) ’ / e P;t P YP P YP P;t (cid:0) (cid:0) (cid:0) 6 7 6 76 7 4 5 4 54 5 where (cid:18) [ (cid:25);(cid:25)] is a rotation angle, and ’ is the angle representation of the correlation YP 2 (cid:0) coef(cid:2)cientbetweenpolicy-variableandoutputdisturbances.13 Contrary to the B&P and the recursive identi(cid:2)cation strategies, the pure sign-restriction approach does not impose a dogmatic prior on the output elasticities or the impact multipliers. Instead, it places restrictions on the sign of impulse responses to the shock(s) of interest. These 12Thereisagrowingconsensusintheliteraturethatimposingsignrestrictionsonlyonimpactresponsesispreferable toimposingsignrestrictionsalsoatlongerhorizons(FryandPagan,2011;Kilian,forthcoming). 13’ arccos(cid:26) ,where(cid:26) (cid:27) =.(cid:27) (cid:27) /. YP YP YP YP Y P (cid:17) (cid:17) 14

restrictions translate into restrictions on the set of admissible rotation angles (cid:18):14 In general, there are (in(cid:2)nitely) many structural models satisfying the sign restrictions, each of which has the same likelihood. To map the factorization of the covariance matrix 6 described in (8) into the identi(cid:2)cation u framework described in Section 1, we map restrictions on the rotation angle (cid:18) into restrictions on theoutputelasticityofthepolicyvariable: FSR (cid:27) cos.(cid:18) ’ / (cid:27) (cid:17)SR 21 P (cid:0) YP (cid:17) P sin’ tan(cid:18): (9) P;Y (cid:17) FSR D (cid:27) cos(cid:18) D P;Y C (cid:27) YP Y Y 11 Tax Shocks. We apply the basic assumptions of M&U who identify a non-policy shock labelled ‘business cycle shock’ - and a tax shock. They assume that the business cycle shock drives up both output and tax revenue and that the tax shock is orthogonal to the business cycle shock. M&Uleavetheresponseofoutputtoataxshock(theobjectofinterest)unrestricted. Inour framework,theseassumptionsimplythefollowingrestrictionsontheelementsofthefactormatrix FSR:15 ? FSR C 2 3 D C C 6 7 4 5 Using the analytical expression for FSR, Proposition 2 in the appendix characterizes the set of all output elasticities of tax revenue that satisfy this sign pattern. We show in particular that all elasticities(cid:17)SR betweenzeroandplusin(cid:2)nitysatisfytheabovesignrestrictions. Hence,inthetop T;Y panelofFigure1allpointsonthelinesegmentwithnon-negativevaluesoftheoutputelasticityof taxrevenueareelementsofthesetofpuresignrestrictionsolutions. Is this large set of pure sign restriction solutions an artefact of our dataset and reduced-form VAR model? This is very unlikely. It is generally accepted that output can be viewed as a proxy 14Itisstandardintheliteraturetoimplementthepuresignrestrictionapproachdrawingtherotationangles(cid:18) from auniformdistribution. Iftheimpulseresponsesassociatedtotheproposeddrawsatisfysignrestrictions,thedrawis kept,otherwiseitisdiscarded. 15Theorthogonalityassumptionisautomaticallysatis(cid:2)ed. MultiplyingtheCholeskyfactorbyanorthogonalmatrix results in a factor matrix with orthogonal columns, thus satisfying the assumption that the business cycle shock and thetaxshockareorthogonal. 15

for the tax base. When the tax base expands, so does tax revenue (for constant tax rates). This is whyempiricallythecorrelationcoef(cid:2)cientbetweenoutputandtaxresidualsispositive(veryoften, strongly positive). In this case, sign restrictions alone are insuf(cid:2)cient to pin down the sign of the taxmultiplier. The sign restrictions given above pose a challenge in the bivariate context. For all values of the elasticity between 0 and (cid:17) , we obtain two shocks with identical sign pattern, which puts T;Y in question identi(cid:2)cation unless additional restrictions are imposed. The above result, however, is useful in that so far it has been hard to understand the implications of such sign restrictions for the sign of the response of output to a tax shock. For example, it would have been reasonable to assume that the sign restrictions on the business cycle shock and the orthogonality assumption are suf(cid:2)cient conditions to rule out negative impact tax cut multipliers (Mountford and Uhlig, 2009: 965). Probability Density Function Probability Density Function 0.4 5 4.5 0.35 4 0.3 3.5 0.25 3 0.2 2.5 2 0.15 1.5 0.1 1 0.05 0.5 0 0 2 4 6 0 0.5 1 Output Elasticity of Tax Revenue Impact Tax Multiplier Figure 2: Kernel densities of the output elasticity of tax revenue and impact tax multiplier satisfyingsignrestrictionsevaluatedatOLSestimates. To rule out the subset of pure sign restriction solutions with negative impact tax cut multipliers, at least one additional assumption is needed. In the next subsection, we discuss the M&U 16

approach, which adds a criterion function to the pure sign-restriction approach. In a second approach, discussed below, we impose an additional sign restriction on the response of output to a taxshock: FSR C (cid:0) 2 3 D C C 6 7 4 5 Proposition 3 in the appendix characterizes the set of all output elasticities of tax revenue that satisfy this sign pattern. With the additional sign restriction on the response of output to a tax shock all output elasticities of taxes between (cid:17) , i.e. the elasticity implied by the Cholesky T;Y factorization,and(cid:17) (cid:27) =(cid:27) satisfythesignrestrictions. T;Y TT YT D Figure 2 plots the empirical distributions of the output elasticity of tax revenue and of the tax multiplier over the set of sign restriction solutions evaluated at the OLS estimate, assuming (cid:150) as it is common practice in the literature (cid:150) that the rotation angle is uniformly distributed over the range satisfying the sign restrictions. Imposing this additional sign restriction "reduces" the set of admissibleelasticitiestoallelasticitiesbetween1:5and6:15,andthesetofadmissiblemultipliers to [0;1:07] dollars. Of course, a drawback of the additional assumption is that in principle we wouldliketoleaveopenthesignoftheresponseofoutputtoataxshock. Finally, we can ask how the above results are affected if we move beyond the bivariate setting and impose restrictions on additional variables. For example, M&U identify the business cycle shock assuming that such shock increases not only output and taxes but also private consumption and non-residential investment. Would these additional assumptions restrict the set of admissible elasticities? Would it make the additional assumption on the output response to a tax shock redundant? Ingeneral,theanswerstothesequestionsdependonthecorrelationstructurebetweenthesignrestricted variables (see the appendix for details). In our application, at the OLS estimates, all output elasticities of taxes between 0 and 73 remain admissible. In other words, the additional assumptions have only a minor effect on the set of pure sign restriction solutions identi(cid:2)ed in the bivariate case. The intuition is that consumption and investment are positively correlated with 17

output and tax revenue. Hence, business cycle shocks that drive up output and tax revenue are very likely to also drive up consumption and investment.16 Turning to the second question, our application reveals that the lower bound of the set of admissible elasticities remains unaffected by the additional sign restrictions on the responses to a business cycle shock. These additional restrictionsare,thus,bythemselvesinsuf(cid:2)cienttoruleoutnegativetaxcutmultipliers. Spending shocks. Similar to the identi(cid:2)cation of the tax shock, M&U identify the spending shock as a shock that increases spending and that is orthogonal to the business cycle shock. The M&U tax and spending models differ in one crucial dimension: there is no sign restriction on the responseofgovernmentspendingtothenon-policyshock.17 The implication of the lack of restriction on the spending response to a business cycle shock is that government spending can be pro-cyclical, a-cyclical, or counter-cyclical. In fact, all output elasticities of government spending ranging between minus and plus in(cid:2)nity satisfy these loose restrictions.18 As can be seen in Figure 1, in our empirical application, the impact spending multiplier can range anywhere between 2:26 and 2:26 dollars, because all points on the line are (cid:0) elementsofthesetofpuresign-restrictionsolutions. This minimal set of assumptions therefore does not rule out solutions for which the responses tothetwoshocksfollowthesamesignpattern. Toruleoutthosesolutionsimplyingthesamesign pattern for the two shocks, it is necessary to restrict the set of admissible output elasticities to the rangebetweenminusin(cid:2)nityandzero. Forthisrange,theimpactspendingmultiplierispositive. 16Intheappendixweformalizethisargument. Ithastobekeptinmindthatthesetofpuresignrestrictionsolutions identi(cid:2)edinthebivariatecaseconstitutesasubspaceofallsolutionsinthemultivariatecontext-albeitaveryinterestingsubset. Furtherextendingtheanalysisbyconsideringalsorotations/re(cid:3)ectionsbeyondtheoutput-taxsubspace canonlyfurtherenlargethesetofadmissibleelasticities. 17M&U do not restrict the sign of the response of government spending to the business cycle shock because it is hardtojustifyempiricallyortheoreticallysuchrestriction. Yet,ifwehadtoaddazerorestrictionontheresponseof governmentspendingtotheexistingsignrestrictions,wewouldgobacktotheB&P-Choleskyidenti(cid:2)cation. 18In analogy to the tax model we can ask how imposing restrictions on the responses of other variables to the businesscycleshockaffectstheresults. M&Uidentifythebusinesscycleshockbyrestrictingtheresponsesofoutput, taxes, consumption and investment to be positive. In our application, at the OLS estimates, all output elasticities of spendingbetweenminusin(cid:2)nityand5.7remainadmissible,withthesignrestrictiononinvestmentbeingthebinding restriction. Again, the additional assumptions have only a minor effect on the set of pure sign restriction solutions identi(cid:2)edinthebivariatecase.Theseareverylooserestrictions,asempiricallyplausiblevaluesoftheoutputelasticity of government spending range in a neighborhood of zero (i.e. close to the assumptions of the B&P and recursive approaches). 18

Summing up, the sets of pure sign restriction solutions are very large in (cid:2)scal VAR models. Standard sign restrictions applied in the literature are insuf(cid:2)cient to pin down the sign, let alone thesize,ofimpacttaxandspendingmultipliers. Topindownthesignoftheimpactmultiplier,itis necessaryeithertodirectlyrestricttheobjectofinterest(themultiplier)ortoaugmentthepuresign restriction approach with a selection criterion, such as the one embodied in the penalty function approachtowhichweturnnext. 2.4 Penalty Function Approach to Sign Restrictions Pure sign restrictions alone are insuf(cid:2)cient to pin down the sign of the multiplier. To address this limitation, M&U augment the pure sign restriction approach with a penalty function, as proposed byUhlig(2005). Tax Shocks. In a bivariate model, the M&U penalty function translates into the following objectivefunction,maximizedwithrespectto(cid:18): FSR FSR (cid:127)MU 11 21 cos(cid:18) cos.(cid:18) ’ /: (10) T YT (cid:17) (cid:27) C (cid:27) D C (cid:0) Y T Proposition 4 in the appendix provides the analytical solution for this maximization problem. Importantly,weprovethattheimpacttaxcutmultiplier,evaluatedatthepenaltyfunctionsolution, ispositiveforalladmissiblevaluesofthecorrelationcoef(cid:2)cientbetweenoutputandtaxresiduals. An important implication is that the application of the M&U penalty function is equivalent to imposing the restriction that the output response to a tax increase is negative, as discussed in the previoussubsection. The penalty function solution in the bivariate setup maximizes the fraction of covariance betweentheoutputandtaxdisturbancesexplainedbythebusinesscycleshock. Suchpenaltyfunction summarizes the belief of M&U, well grounded in the evidence provided by the DSGE literature, that (cid:2)scal shocks do not contribute substantially to business cycle (cid:3)uctuations. Consequently, the role of the business cycle shock is to explain as much variability as possible in the restricted vari- 19

ables,letting(cid:2)scalshocksexplaintheresidualvariance. Incomparison,theCholeskyfactorization withoutputordered(cid:2)rstmaximizesthefractionofoutputvarianceexplainedbythebusinesscycle shock, explaining 100% of both the output variance and the covariance on impact. Recall that for this Cholesky factorization the impact tax multiplier is zero. For the impact multiplier to be positive,thebusinesscycleshockhastoexplainmorethan100%ofthecovariance. Whenthiscondition is ful(cid:2)lled, the conditional covariance generated by the tax shock has to be negative, which is only possible if output declines in response to a tax shock meant to increase tax revenue. The penaltyfunctioninoursetupselectsabusinesscycleshockthatexplains151%ofthecontemporaneous covariance between output and tax disturbances, while the tax shock explains -51%. Hence the penalty function, which maximizes the covariance between output and tax revenue explained bythebusinesscycleshock,favorslargepositivetaxcutmultipliers. In the top panel of Figure 1, the penalty function sign restriction solution is denoted ‘MU’. In our example, this point - compared to the B&P and recursive approaches - corresponds to a large value of the output elasticity of tax revenue ((cid:17)MU 3:04) and to a value of the impact tax T;Y D cut multiplier of 0:93 dollars. Note that the penalty function solution satis(cid:2)es any additional sign b restrictions on the impact responses of private consumption and investment to a business cycle shock(seeappendixfordetails). Spending Shocks. To explain the identi(cid:2)cation of government spending shocks in the bivariate setting, we assume (cid:150) consistent with the subsection on the pure sign restriction approach (cid:150) that the business cycle shock drives up output only (leaving the response of government spending unrestricted) and that the government spending shock is orthogonal to the business cycle shock. Trivially, the solution to the penalty function associated to these restrictions is the Cholesky factorization with output ordered (cid:2)rst and government spending ordered second. For this Cholesky factorization, government spending is procyclical; in our example, (cid:17)MU 0:38, and the impact G;Y D spendingmultiplieriszero,comparedto1:25dollarsfortheB&Papproach. b What would happen if, following M&U, we identify a business cycle shock imposing restrictionsonoutputandtaxrevenue,whilekeepingtheresponseofgovernmentspendingunrestricted? 20

As we show in the appendix the penalty function solution under these assumptions implies a zero impact spending multiplier (while the penalty function solution picks the same - large - impact taxcutmultiplierasinthebivariatetaxmodel). Inaddition,thepenaltyfunctionsolutionselectsa positivevalueoftheoutputelasticityofspending(inourapplicationtheoutputelasticityofgovernment spending goes down slightly compared to the one obtained for the Cholesky decomposition, to0:36). Summing up, the penalty-function approach to sign restrictions can be interpreted as an additional identifying assumption beyond pure sign restrictions. Moreover, in (cid:2)scal VAR models, the penalty function as speci(cid:2)ed by M&U picks a solution favoring large tax multipliers and low spending multipliers. This explains the main result of M&U, namely that tax cuts are more effectivethanspendingincreases. 2.5 Narrative Approach An alternative methodology for estimating the effects of (cid:2)scal policy shocks using VARs is the narrative approach. Prominent examples are Romer and Romer (2010), who identify tax shocks studying narrative records of tax policy decisions, and Ramey (2011), who identi(cid:2)es government spending shocks using changes in military spending associated with wars. Multipliers estimated using SVAR models are different from multipliers estimated using the narrative approach. Differences are in part due to the fact that most studies using the narrative approach identify anticipated (cid:2)scal shocks. Yet, Mertens and Ravn (2011b) construct a series of unanticipated tax shocks based on Romer and Romer (2010) narrative records, and (cid:2)nd larger multipliers than SVARs. To understandwhatdrivessuchdifferences,weconductthefollowingexercise: 1. We regress the reduced-form VAR residuals on the Mertens and Ravn (2011b) narrative seriesofunanticipatedtaxshocks,whichwedenotebyeMR: T;t u (cid:11) eMR (cid:24) ; Y;t Y T;t Y;t D C 21

u (cid:11) eMR (cid:24) ; P;t T T;t P;t D C where(cid:11) isthecontemporaneousresponseofoutputtoanarrativetaxshockofsizeonestan- Y dard deviation. This step follows the empirical speci(cid:2)cation for estimating tax multipliers usingnarrativemeasuresoftaxshocksproposedbyFaveroandGiavazzi(forthcoming).19 2. Wecomputetheresponseofoutputtoa1%increaseintaxrevenue,(cid:11) =(cid:11) . Thiscoef(cid:2)cient Y T isequivalenttoa inequation(3). Y;T 3. We invert the analytical function a f.(cid:17) 6 / to obtain the output elasticity of tax Y;T T;Y u D I revenue consistent with the narrative measure of the effects of tax policy on output, which wedenoteby(cid:17)MR.20 T;Y In our VAR model, (cid:17)MR 3:10, a value remarkably close to the elasticity associated to the T;Y D M&Upenaltyfunctionapproachtosignrestrictions.21 Theassociatedimpacttaxmultiplieris0:95 b dollars. In Figures 1 and 2 the results for our application of the narrative approach are denoted ‘NARR’. Thenarrativetaxmultiplierspresentedinthispaperareconsiderablysmallerthantheestimates reported by Romer and Romer (2010) and Mertens and Ravn (2011b). This is due to differences in the scaling of shocks. Narrative studies consider tax shocks that increase tax revenue by 1%. SVARs instead consider 1% tax shocks that increase tax revenue by 1=.1 a (cid:17) / < 1; i.e. Y;P P;Y (cid:0) 19FaveroandGiavazzi(forthcoming)estimatejointlythecoef(cid:2)cientvector(cid:11)associatedtothenarrativeshocksand the reduced-form VAR coef(cid:2)cients in (1). Under the orthogonality assumption E.eMRX / 0 and the assumption T;t t D that eMR is orthogonal to non-policy and policy shocks other than tax shocks, the two procedures deliver identical T;t estimates. FaveroandGiavazzi(forthcoming)pointoutthatRomerandRomer(2010)taxepisodesarenotorthogonal tothedevelopmentofgovernmentdebt. Yet,theyshowthatintheUnitedStatescontrollingforgovernmentdebthas littleeffectonresults. 20Impact tax multipliers estimated assuming (cid:17) (cid:17)MR in an SVAR or simply propagating narrative shocks T;Y D T;Y throughtheequationsinstep1. areidentical(uptoascalingfactordiscussedinthesecond-to-lastparagraphofthis subsection). In a bivariate model in output and tax revenue, the two procedures produce identical multipliers at any horizon. Inmultivariatemodelsdynamicmultipliersmightbedifferent. Yet,inour8-equationmodel,we(cid:2)ndthatthe twoproceduresdelivernearlyidenticalmultipliers. Resultsareavailableuponrequest. 21Inanindependentstudy,MertensandRavn(2011a)computeanoutputelasticityoftaxrevenueconsistentwith theirnarrativerecordsofunanticipatedtaxshocksof3.13,whichisinlinewiththeback-of-the-envelopecalculations presentedhere. 22

SVARs account for macroeconomic feedback: if exogenous tax increases depress output, tax revenue will increase by less than one-for-one. To compare the SVAR and narrative approaches, all shocksarerescaledfollowingtheSVARconvention.22 2.5 2 1.5 1 0.5 0 1 2 3 4 Output Elasticity of Tax Revenue ytisneD lenreK BP MU CHOL NARR 15 10 5 0 -0.5 0 0.5 1 Impact Tax Multiplier ytisneD lenreK 5 4 3 2 1 0 -0.2 0 0.2 0.4 0.6 0.8 Output Elasticity of Spending ytisneD lenreK BP-CHOL MU 1.5 1 0.5 0 -1 -0.6 -0.2 0.2 0.6 1 1.4 1.8 Impact Spending Multiplier ytisneD lenreK Figure3: Kerneldensities. 3 Implications of Different Priors on Elasticities In the previous section, we have concentrated on identi(cid:2)cation uncertainty to clarify the main properties of the alternative identi(cid:2)cation approaches. We now broaden the analysis, accounting alsoforsamplinguncertaintyandlookingatdynamicmultipliersbeyondtheimpactperiod. We start with the implications of sampling uncertainty on estimates of elasticities and impact multipliers. Figure 3 plots kernel densities of output elasticities and (cid:2)scal multipliers. The B&P and recursive approaches provide lower estimates of the output elasticity of tax revenue, and consequently of the impact tax multiplier, than the M&U and narrative approaches. The B&P and recursiveapproachesalsoprovidelowerestimatesoftheoutputelasticityofgovernmentspending 22Perotti (2011) makes a similar point when allowing for differences in the effects of exogenous and endogenous movementsintaxrevenue. 23

than the M&U approach. This translates into a larger impact spending multiplier for the B&P and recursive approaches. All the identi(cid:2)cation approaches are dogmatic as regards structural uncertainty, i.e. for a given reduced-form estimate, they imply a single elasticity as well as multiplier. Distributionsofelasticitiesandimpactmultipliersareuniquelyduetosamplinguncertainty. GDP Tax Revenue 2 0.5 1.5 0 1 -0.5 0.5 0 BP -1 MU -0.5 -1.5 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Government Spending Private Consumption 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Figure 4: Responses of output, tax revenue, government spending, and private consumption to a 1 dollartaxcut. Figures4and5plotdynamictaxandspendingmultipliersfortheB&PandM&Uapproaches.23 Focusing on the output multiplier, differences across approaches are substantial up to three years after the policy intervention, although differences across approaches diminish in the long-run, despite being very persistent. The top-right and bottom-left panel of Figure 4 plot the response of taxes and spending to a tax shock. For up to the three years after the policy intervention, the B&P approach predicts a larger effect of a tax cut on the de(cid:2)cit than does the M&U approach. The reason is that tax cuts identi(cid:2)ed with the M&U approach are partly self-(cid:2)nancing, as they boost GDP and consequently the tax base.24 The opposite holds for the response of the de(cid:2)cit to 23We do not plot multipliers associated to the recursive and narrative approaches as they are very similar to the resultsfortheB&PandM&Uapproaches,respectively. 24We show in the appendix that for the maximum of the objective function (10) the degree of self-(cid:2)nancing is exactly50%,i.e. taxrevenuedropsbyonly0.50dollarsonimpactinresponsetoa1$taxcut. 24

spending shocks. As shown in Figure 5, the spending shock associated to the B&P approach is partlyself-(cid:2)nancing,duetotheboostintaxrevenueassociatedtotheincreaseinGDP. Thebottom-rightpanelofFigure5plotstheresponseofconsumptiontoaspendingshock. For the B&P approach, the response is positive at all horizons. For the M&U approach, the response is not signi(cid:2)cant for the (cid:2)rst (cid:2)ve quarters, and turns positive thereafter. We provide a detailed analysisoftheresponseofconsumptiontoaspendingshockinSection5. GDP Tax Revenue 2.5 0.6 2 0.4 1.5 0.2 1 0 0.5 0 BP -0.2 MU -0.5 -0.4 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Government Spending Private Consumption 1.5 0.4 0.3 1 0.2 0.1 0.5 0 0 -0.1 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Figure 5: Responses of output, tax revenue, government spending, and private consumption to a 1 dollarspendingincrease. Figure 6 plots the probability of the tax multiplier being larger than the spending multiplier for all four SVAR-based approaches.25 The policy implication is very different: according to the B&P and the recursive approaches, tax multipliers are very likely to be smaller than spending multipliers for the entire forecast horizon. This probability reaches at most 0.2 two years after the policy intervention, and falls thereafter. The M&U approach instead (cid:2)nds that tax multipliers are very likely to be larger than spending multipliers. The probability is 1 for the (cid:2)rst two years after the policy intervention. It declines to 0.45 after 5 years, and increases again thereafter. The pure 25Forthisexercisewejointlyidentifytaxandspendingshockstoensureorthogonalitybetweenthem. Inparticular, weassumethatspendingaffectscontemporaneouslytaxrevenueonlythroughoutput. 25

sign restriction approach (cid:2)nds probabilities close to 0.5, re(cid:3)ecting the large structural uncertainty associatedwiththisapproach. 1.2 1 0.8 0.6 0.4 0.2 0 1 5 9 13 17 21 25 29 33 37 Time ytilibaborP BP MU CHOL SR Figure6: Probabilitythattaxmultipliersarelargerthanspendingmultipliers 4 Robust Fiscal Multipliers In the previous sections, we have shown that differences in priors on elasticities implicit in alternative identi(cid:2)cation schemes translate into large differences in (cid:2)scal multipliers. Some of the identi(cid:2)cation schemes appear very dogmatic, selecting a single value of the relevant output elasticity. Others appear quite loose, imposing almost no restriction on the relevant elasticity. In this section we strike a balance between these two extremes, surveying the existing literature on automatic stabilizers to derive distributions on elasticities that encompass the existing empirical evidence. Thenweestimate(cid:2)scalmultipliersbasedonthesepriordistributions. Output elasticity of tax revenue. The size of automatic stabilizers is the subject of many empirical studies in the macro public (cid:2)nance literature. Several international organizations and national agencies estimate the output elasticity of tax revenue for different tax categories, using 26

these elasticities to construct cyclically-adjusted measures of the budget balance. Results for the B&P approach presented in this paper are based on elasticity estimates provided by Follette and Lutz(2010).26 Theseauthorsestimatetheoutputelasticityoftaxrevenueusingmicrodataforfour different tax categories: personal income tax, social security contributions, corporate income tax, andindirecttaxes. Weaggregatetheseelasticitiestoobtainapointestimatefortheoutputelasticity (cid:17) accordingtothefollowingaggregator: T;Y T i (cid:17) (cid:17) ; (11) T;Y D Ti;Y T i X where i denotes the tax category, T denotes the level of tax revenue, T denotes total tax revi enue, and (cid:17) denotes the output elasticity of tax category i. Following B&P, we evaluate T Ti;Y i and T at their sample mean. The point estimate for the period 1947-2006 is 1:71. Caldara (2011) showsthatsamplinguncertaintyaroundthepointestimateissmallandcanbesafelyneglected. The NBER also estimates the output elasticity of personal income taxes and social security contributions using the TAXSIM model (Feenberg and Coutts, 1993). This model implements a micro-simulation of the U.S. federal income tax system. The model is based on a large sample of actual tax returns prepared by the Statistics of Income Division of the Internal Revenue Service. The average elasticity of personal income taxes and social security contributions estimated using TAXSIM is 1:65, which would increase the estimate for the overall elasticity to 1:8: The OECD27 also estimates the output elasticity of tax revenue for the United States. Following the OECD methodology,theaggregateelasticityis1:2. All these estimates of the output elasticity of taxes are lower than the value of 2:08 reported by B&P for their sample period 1947-1997. This difference is mainly due to differences in the de(cid:2)nition of tax aggregates considered. In line with the literature cited above we use total tax revenue as tax variable. B&P instead use a concept of net taxes, subtracting transfers and net interest payments from tax revenue. This procedure mechanically increases the output elasticity of (net) 26TheCongressionalBudgetOf(cid:2)ceadoptsasimilarestimationmethodology. 27Seee.g. GirouardandAndrØ(2005) 27

taxes. The reason is that subtracting transfers and net interest payments (cid:150) whose elasticity B&P set to 0:1 and 0, respectively (cid:150) increases the weight associated to the other sub-elasticities28, in (cid:0) turnimplyingthattheoutputelasticityofnettaxesismuchlargerthantheoutputelasticityoftotal tax revenue. Considering only genuine tax revenue, i.e. the four tax categories mentioned above, the B&P elasticity of taxes amounts to 1:5 for their sample period and their assumptions about sub-elasticities. This latter (cid:2)gure is in line with the evidence reported in the cyclical-adjustment literaturecitedabove. Elasticityestimatestakenfromthiscyclical-adjustmentliteratureareoftenusedinDSGEmodeling when authors want to move beyond the simplistic assumption of a proportional tax system (output elasticity of taxes equal to 1.0). A prominent example for the United States is Leeper, PlanteandTraum(2010)whorelyonelasticityestimatesfromB&PandtheOECDtocalibrateor setpriorsontheoutputelasticityofdifferenttaxcategories.29 Inanindependentstudy,MertensandRavn(2011a)argueinfavorofvaluesoftheoutputelasticityoftaxrevenuearound3. Theirestimatesarebasedonnarrativemeasuresoftaxshocks. They make three arguments to support their (cid:2)nding. First, elasticity estimates from public (cid:2)nance studiesarebasedonregressionsthat,althoughbasedonmicrodata,mightbesubjecttoasimultaneity bias. The narrative measure of tax shocks is exogenous to the state of the economy, and hence is not subject to such bias. Second, conditional on observing output, an SVAR identi(cid:2)ed assuming an output elasticity of tax revenue of 3 has greater explanatory power for the dynamics of tax revenues than an SVAR identi(cid:2)ed assuming a smaller value of the elasticity. Third, an elasticity of 3 generates an endogenous drop in tax revenue in 2008-2009 consistent with the drop observed in thedata. All in all, the macro public (cid:2)nance literature consistently documents output elasticities of tax revenuerangingfrom1:2to1:8. Studiesbasedonnarrativemeasuresoftaxshocks(cid:2)ndelasticities 28Overtheperiod1947-1997consideredbyB&P,onaverage,theshareoftransfersinnettaxesamountsto(minus) 47% and the share of net interest payments to (minus) 14%, while the sum of the shares of the four tax categories mentionedaboveamountsto161%. 29Caldara (2011) shows that the uncertainty from prior distributions on the deep parameters of DSGE models translatesintosmalluncertaintyfortheoutputelasticityoftaxrevenue. 28

around3. Toencompassthisempiricalevidence,wedrawtheoutputelasticityoftaxrevenuefrom twonormaldistributionscenteredattheB&PandM&Unarrativeelasticityestimatesof1:7and3. Both distributions have a standard deviation of 1 to ensure a wide coverage around their mean.30 The top panel of Figure 7 shows that for this prior distribution the median tax multiplier is 0:65 dollarsonimpact. Itstartstoexceedonedollar(cid:2)vequartersafterthepolicyintervention. Output elasticity of government spending. Most authors in the VAR literature assume that the output elasticity of government consumption and investment is zero. In a similar vein, the cyclical-adjustment literature (cid:150) e.g. the OECD (cid:150) shares this assumption and does not attempt to estimatethiselasticity. There are some studies in the political-economy literature that estimate the output elasticity of government spending to assess whether (cid:2)scal policymakers behave pro-cyclically. Examples include Lane (2003) and Rodden and Wibbels (2010). Aggregate elasticities are not statistically signi(cid:2)cant in general. Yet, these papers document that some components of government consumption, such as public wages or state and local spending, are mildly pro-cyclical. Furthermore, international evidence on spending elasticities suggests that in some countries government spending is pro-cyclical. Finally, Leeper, Plante and Traum (2010) model government consumption and investmentasmildlycounter-cyclical. TheexistingevidencetendstosupporttheB&PassumptionthatintheU.S.theoutputelasticity ofgovernmentspendingiszero. However,itcanalsonotberuledoutthatgovernmentspendingis mildly cyclical. Therefore, we implement a prior on the output elasticity of government spending centered at zero. We set the standard deviation to 0:1 to allow for some uncertainty. The middle panel of Figure 7 shows that for this prior distribution the median spending multiplier is 1 dollar onimpactandstaysabove1dollarovertheentirehorizon. The bottom panel of Figure 7 shows that the probability of the tax multiplier being larger than the spending multiplier remains below 0:5 at all horizons. Hence, for these prior distributions on 30We draw from both distributions assuming a weight of 0:5. The 5th and 95th percentiles are 1:5 and 3:2. This choiceofdistributionsandparameterizationsisoneofmanypossibleplausiblechoices. Forinstance,wecouldhave assumed that the elasticity are uniformly distributed. Our point is that economists should use priors, even dogmatic priors,aslongastheyareconsistentwiththeirbeliefsabouttheelasticity. 29

the output elasticity of (cid:2)scal variables, there is no evidence to support the view that tax policy providesalargerstimulustooutputthanspendingpolicy. GDP multiplier - Tax Shock 2 1 Mixture of Normals 0 4 8 12 16 20 24 28 32 36 40 GDP Multiplier - Spending Shock 3 2 1 0 4 8 12 16 20 24 28 32 36 40 Prob(ΠY,T>ΠY,G) 1 0.5 0 4 8 12 16 20 24 28 32 36 40 Figure7: GDPmultipliersaftertaxandspendingshocksforalternativepriorsonoutputelasticities of(cid:2)scalvariables. 5 Shedding Light on Two Debates on the Effects of Spending Shocks Theanalyticalresultspresentedintheprevioussectionscanhelpshedlightontwoongoingdebates intheliteratureontheeffectsofspendingshocks: theresponseofprivateconsumptionandtherole of(cid:2)scalforesight. 5.1 On the Effects of Spending Shocks on Private Consumption Standard RBC and New Keynesian models predict that, due to a negative wealth effect, consumption falls after a spending shock (Baxter and King, 1993; Linnemann and Schabert, 2003). Yet, 30

SVARsmodelconsistently(cid:2)ndthatconsumptionincreases.31 Assumingthatgovernmentspending isacyclical((cid:17) 0/,theimpactconsumptionmultiplieris: G;Y D (cid:27) 1 G;C CG 5 .(cid:17) 0 6 / . (12) 0 G;Y D I u D (cid:27) GG G=C The response of consumption is positive as long as (cid:27) > 0. The sample covariance between CG consumption and government spending is robustly positive across VAR speci(cid:2)cations, samples and dataset used in the (cid:2)scal VAR literature. Figure 9 in the Appendix plots the response of consumptionasfunctionoftheoutputelasticityofspendingatdifferenthorizons. Themedianimpact responseofconsumptionispositiveaslongastheoutputelasticityofspendingissmallerthan0:35. Atlongerhorizons,theconsumptionresponseremainspositiveforevenlargervaluesoftheoutput elasticity of government spending. However, as argued in the previous section, positive quarterly output elasticities of government spending are not plausible for the U.S. Hence, our (cid:2)ndings supportDSGEmodelscapableofgeneratinganincreaseinprivateconsumptionfollowingaspending shock,asforinstanceGal(cid:237),Lopez-SalidoandValles(2007)whorelyoncredit-constrained agents, andRavn,Schmitt-GrohØandUribe(2006)whorelyonhabitformationinprivateconsumption. 5.2 The Role of Anticipation What is the effect of (cid:2)scal foresight on the estimated response of output and consumption to a spending shock? Similarly to equation (12), we can write the response of output to a spending shockas: (cid:27) 1 G;Y YG 5 .(cid:17) 0 6 / . (13) 0 G;Y D I u D (cid:27) GG G=Y Letusassumethat,dueto(cid:2)scalforesight,thereduced-formresidualu (whichequalse since G;t G;t (cid:17) 0), is not truly unpredictable, but contains some anticipated components. Let us further G;Y D assume that we add variables to the VAR that help predict future changes in government spending andtomitigatethebiasassociatedto(cid:2)scalforesightassuggestedbyGiannoneandReichlin(2006) 31Seee.g. BlanchardandPerotti(2002);CaldaraandKamps(2008);Gal(cid:237),Lopez-SalidoandValles(2007). 31

and Forni and Gambetti (2010). Anticipated government spending shocks are de(cid:2)ned as shocks thathaveanimmediateeffectonmacrovariablessuchasoutputandconsumptionuponannouncement, while leaving current government spending unchanged until the moment of implementation (Ramey, 2011). Hence, anticipated spending shocks can by de(cid:2)nition not be the source of contemporaneous co-movements between output and spending ((cid:27) ), and between consumption and YG spending((cid:27) ). Instead,theyhelppredictfuturevaluesofgovernmentspending,i.e. inclusionof CG anticipated spending shocks should reduce the variance of one-step-ahead government spending forecast errors ((cid:27) ). Consequently from Equation (13) we see that adding variables to the VAR GG model will tend to increase the impact response of output to unanticipated spending shocks, i.e. increasetheimpactspendingmultiplier. Thispredictionissatis(cid:2)edforourVARmodel. Theimpactspendingmultiplierestimatedusing a3-equationVARinoutput,taxrevenue,andgovernmentspendingis1:09dollars,insteadof1:25 dollarsinthefull8-equationVARmodelincludingadditionalvariableslikelytocaptureforesight. Similarly, the impact response of consumption in a 4-equation VAR is 0:06 dollars, compared to 0:09dollarsinthe8-equationVARmodel.32 6 Conclusions We providecomprehensive evidenceon (cid:2)scal multipliersfor theU.S. basedon data forthe period 1947-2006. Our novel analytical framework allows us to reveal the core properties of the alternativeidenti(cid:2)cationschemesusedinthe(cid:2)scalVARliterature. Weshowthatdifferencesinestimates of (cid:2)scal multipliers documented in the literature by Blanchard and Perotti (2002), Mountford and Uhlig (2009) and Romer and Romer (2010) are due mostly to different restrictions on the output 32It should be noted that while (cid:2)scal foresight will not affect the contemporaneous comovement between output andgovernmentspending,theadditionofvariablestotheVARmodelcanaffectcovarianceestimatesduetoreasons unrelatedtoforesight, e.g. totheextentthataddingvariablescuresotherformsofmisspeci(cid:2)cationsuchasomittedvariable bias. In our VAR model, the estimate of (cid:27) , goes down somewhat as variables are added to the model, YG althoughbylessthan(cid:27) . Keepingthelatterconstantatthe3-equationestimatewouldgenerateanimpactspending YG multiplier of 1:37 dollars in the 8-equation model. Instead, the estimate of (cid:27) is unaffected by the addition of CG variablesinourapplication. 32

elasticities of tax revenue and government spending. We use extra-model information to narrow the set of empirically plausible values of the elasticities, which in turn allows us to sharpen the inference on (cid:2)scal multipliers. Our results suggest that spending multipliers tend to be larger than taxmultipliers. The analytical framework developed in this paper can be applied to study identi(cid:2)cation problems in a large class of time-series models, including VARs with time-varying reduced-form coef(cid:2)cients, regime-switching VARs and factor models. The use of such models can help unveil whether the transmission of (cid:2)scal policy shocks has changed over time33 or depends on the state of the economy (Auerbach and Gorodnichenko, forthcoming). Finally, the analytical framework developed here can be easily adapted to the study of other topics in empirical macroeconomics, suchastheidenti(cid:2)cationofmonetarypolicyshocks. References Auerbach, Alan J., and Yuriy Gorodnichenko. forthcoming. (cid:147)Measuring the Output Responses toFiscalPolicy.(cid:148)AmericanEconomicJournal: EconomicPolicy. Baxter,Marianne,andRobertG.King.1993.(cid:147)FiscalPolicyinGeneralEquilibrium.(cid:148)American EconomicReview,83(3):315(cid:150)34. Blanchard, Olivier, and Roberto Perotti. 2002. (cid:147)An Empirical Characterization of the Dynamic Effects of Changes in Government Spending and Taxes on Output.(cid:148) Quarterly Journal of Economics,117(4):1329(cid:150)68. Caldara, Dario. 2011. (cid:147)Essays on Empirical Macroeconomics.(cid:148) Monograph Series, 71, Institute forInternationalEconomicStudies,Stockholm. Caldara, Dario, and Christophe Kamps. 2008. (cid:147)What Are the Effects of Fiscal Shocks? A VAR-basedComparativeAnalysis.(cid:148)EuropeanCentralBankWorkingPaper877. Canova, Fabio, and Evi Pappa. 2007. (cid:147)Price Differentials in Monetary Unions: The Role of FiscalShocks.(cid:148)EconomicJournal,117(520):713(cid:150)37. CBO.2010.(cid:147)EstimatedImpactoftheAmericanRecoveryandReinvestmentActonEmployment and Economic Output From April 2010 Through June 2010.(cid:148) Congressional Budget Of(cid:2)ce Report. 33SeePrimiceri(2005)foranapplicationtomonetarypolicy. 33

Chahrour, Ryan, Stephanie Schmitt-GrohØ, and Mart(cid:237)n Uribe. forthcoming. (cid:147)A Model-Based Evaluation of the Debate on the Size of the Tax Multiplier.(cid:148) American Economic Journal: EconomicPolic. Del Negro, Marco, and Frank Schorfheide. 2011. (cid:147)Bayesian Macroeconometrics.(cid:148) Handbook of Bayesian Econometrics, ed. John Geweke, Gary Koop and Herman van Dijk, Oxford Oxford UniversityPress:Chapter7. Eichenbaum, Martin, and Jonas D.M. Fisher. 2005. (cid:147)Fiscal Policy in the Aftermath of 9/11.(cid:148) JournalofMoney,Credit,andBanking,37(1):1(cid:150)22. Faust, Jon. 1998. (cid:147)The Robustness of Identi(cid:2)ed VAR Conclusions about Money.(cid:148) Carnegie- RochesterConferenceSeriesonPublicPolicy,49(Dec):207(cid:150)44. Favero, Carlo A., and Francesco Giavazzi. forthcoming. (cid:147)Measuring Tax Multipliers: The NarrativeMethodinFiscalVARs.(cid:148)AmericanEconomicJournal: EconomicPolicy. Feenberg,Daniel,andElisabethCoutts.1993.(cid:147)AnIntroductiontotheTAXSIMModel.(cid:148)Journal ofPolicyAnalysisandManagement,12(1):189(cid:150)94. FernÆndez-Villaverde,Jesœs,JuanF.Rubio-Ram(cid:237)rez,ThomasJ.Sargent,andMarkW.Watson.2007.(cid:147)ABCs(andDs)ofUnderstandingVARs.(cid:148)AmericanEconomicReview,97(3):1021(cid:150) 1026. Follette, Glenn, and Byron Lutz. 2010. (cid:147)Fiscal Policy in the United States: Automatic Stabilizers, Discretionary Fiscal Policy Actions, and the Economy.(cid:148) Board of Governors of the Federal ReserveSystemFinanceandEconomicsDiscussionSeries2010-43. Forni,Mario,andLucaGambetti.2010.(cid:147)FiscalForesightandtheEffectsofGovernmentSpending.(cid:148)CentreforEconomicPolicyResearchDiscussionPaper7840. Fry,Renee,andAdrianPagan.2011.(cid:147)SignRestrictionsinStructuralVectorAutoregressions: A CriticalReview.(cid:148)JournalofEconomicLiterature,49(4):938(cid:150)60. Gal(cid:237),Jordi,J.DavidLopez-Salido,andJavierValles.2007.(cid:147)UnderstandingtheEffectsofGovernmentSpendingonConsumption.(cid:148)JournaloftheEuropeanEconomicAssociation,5(1):227(cid:150) 70. Giannone, Domenico, and Lucrezia Reichlin. 2006. (cid:147)Does Information Help Recovering Structural Shocks from Past Observations?(cid:148) Journal of the European Economic Association, 4(2- 3):455(cid:150)65. Girouard, Nathalie, and Christophe AndrØ. 2005. (cid:147)Measuring Cyclically-adjusted Budget BalancesforOECDCountries.(cid:148)OECDEconomicsDepartmentWorkingPaper434. Golub,GeneH.,andCharlesF.vanLoan.1996.MatrixComputations,3rded.Baltimore: Johns HopkinsUniversityPress. Judd,KennethL.1998.NumericalMethodsinEconomics.Cambridge,MA:MITPress. 34

Kilian, Lutz. forthcoming. (cid:147)Structural Vector Autoregressions.(cid:148) Handbook of Research Methods andApplicationsonEmpiricalMacroeconomics,edNigarHashimzadeandMichaelThornton. Lane, Philip R. 2003. (cid:147)The Cyclical Behaviour of Fiscal Policy: Evidence from the OECD.(cid:148) JournalofPublicEconomics,87(12):2661(cid:150)75. Leeper,EricM.,MichaelPlante,andNoraTraum.2010.(cid:147)DynamicsofFiscalFinancinginthe UnitedStates.(cid:148)JournalofEconometrics,156(2):304(cid:150)21. Leeper,EricM.,ToddB.Walker,andShu-ChunSusanYang.2008.(cid:147)FiscalForesight: AnalyticsandEconometrics.(cid:148)NationalBureauofEconomicAnalysisWorkingPaper14028. Linnemann,Ludger,andAndreasSchabert.2003.(cid:147)FiscalPolicyintheNewNeoclassicalSynthesis.(cid:148)JournalofMoney,CreditandBanking,35(6):911(cid:150)29. Mertens, Karel, and Morten O. Ravn. 2011a. (cid:147)A Reconciliation of SVAR and Narrative EstimatesofTaxMultipliers.(cid:148)CornellUniversity. Mertens, Karel, and Morten O. Ravn. 2011b. (cid:147)Understanding the Aggregate Effects of AnticipatedandUnanticipatedTaxPolicyShocks.(cid:148)ReviewofEconomicDynamics,14(1):27(cid:150)54. Mountford, Andrew, and Harald Uhlig. 2009. (cid:147)What Are the Effects of Fiscal Policy Shocks?(cid:148) JournalofAppliedEconometrics,24(6):960(cid:150)992. Pappa, Evi. 2009. (cid:147)The Effects of Fiscal Shocks on Employment and the Real Wage.(cid:148) InternationalEconomicReview,50(1):217(cid:150)244. Perotti, Roberto. 2005. (cid:147)Estimating the Effects of Fiscal Policy in OECD Countries.(cid:148) Centre for EconomicPolicyResearchDiscussionPaper4842. Perotti, Roberto. 2008. (cid:147)In Search of the Transmission Mechanism of Fiscal Policy.(cid:148) NBER Macroeconomics Annual 2007, Volume 22, ed. Daron Acemoglu, Kenneth Rogoff and Michael Woodford.Cambridge,Mass: MITPress:169(cid:150)226. Perotti, Roberto. 2011. (cid:147)The Effects of Tax Shocks on Output: Not So Large, But Not Small Either.(cid:148)NationalBureauofEconomicResearchWorkingPaper16786. Primiceri,Giorgio.2005.(cid:147)TimeVaryingStructuralVectorAutoregressionsandMonetaryPolicy.(cid:148) ReviewofEconomicStudies,72(3):821(cid:150)52. Ramey,ValerieA.2011.(cid:147)IdentifyingGovernmentSpendingShocks: It’sallintheTiming.(cid:148)QuarterlyJournalofEconomics,126(1):1(cid:150)50. Ramey, Valerie A., and Matthew D. Shapiro. 1998. (cid:147)Costly capital reallocation and the effects of government spending.(cid:148) Carnegie-Rochester Conference Series on Public Policy, 48(1): 145(cid:150) 194. Ravn, Morten O., Stephanie Schmitt-GrohØ, and Mart(cid:237)n Uribe. 2006. (cid:147)Deep Habits.(cid:148) Review ofEconomicStudies,73(1):195(cid:150)218. 35

Rodden, Jonathan, and Erik Wibbels. 2010. (cid:147)Fiscal Decentralization And The Business Cycle: AnEmpiricalStudyOfSevenFederations.(cid:148)EconomicsandPolitics,22(1):37(cid:150)67. Romer, Christina D., and David H. Romer. 2010. (cid:147)The Macroeconomic Effects of Tax Changes: Estimates Based on a New Measure of Fiscal Shocks.(cid:148) American Economic Review, 100(3):763(cid:150)801. Rubio-Ram(cid:237)rez, Juan F., Daniel F. Waggoner, and Tao Zha. 2010. (cid:147)Structural Vector Autoregressions: TheoryofIdenti(cid:2)cationandAlgorithmsforInference.(cid:148)ReviewofEconomicStudies, 77(2):665(cid:150)96. Simon, Carl P., and Lawrence Blume. 1994. Mathematics for Economists. New York: W.W. Norton&Company. Sims,ChristopherA.1980.(cid:147)MacroeconomicsandReality.(cid:148)Econometrica,48(1):1(cid:150)48. Uhlig, Harald. 2005. (cid:147)What Are the Effects of Monetary Policy on Output? Results from an AgnosticIdenti(cid:2)cationProcedure.(cid:148)JournalofMonetaryEconomics,52(2):381(cid:150)419. 36

Appendix A Data and Estimation WeestimatetheVARmodelusingBayesiantechniques. Inparticular,weimposepriordistributions onthereduced-formcoef(cid:2)cients B.L/and6 followingthemethodologydiscussedinDelNegro u andSchorfheide(2011). Weimplementthisprior,whichisavariantofthewell-knownMinnesota prior, through dummy observations. The hyper-parameters are chosen to impose a fairly loose prior, so that the comparison to the existing literature does not depend on the choice of the prior. FollowingthenotationinDelNegroandSchorfheide(2011),thehyper-parametersare:(cid:21) 0:01; 1 D (cid:21) 6;(cid:21) 0;(cid:21) (cid:21) 0:001: 2 3 4 5 D D D D All components of national income are taken from the NIPA Tables published by the U.S. BureauofEconomicAnalysis. Theyareinrealpercapitatermsandaretransformedfromthenominal values by dividing them by the GDP de(cid:3)ator (NIPA Table 1.1.4, Line 1) and the population measure(NIPATable 2.1,Line38). The remainingseriesaredownloadedfromFederal ReserveBank of St. Louis FRED database. The table and row numbers given below refer to the organization of thedatabytheBEA.Dataareataquarterlyfrequencyfrom1947Ito2006IV.Weusethelogarithm ofallnationalincomevariables. GDP:GrossDomesticProduct(NIPATable1.1.5,Line1). (cid:15) Government Spending: Government consumption (NIPA Table 3.1, Line 16) expenditures (cid:15) andgrossinvestment(NIPATable3.1. Line35). TaxRevenue: Governmentcurrentreceipts(NIPATable3.1,Line1). (cid:15) PrivateConsumption: Personalconsumptionexpenditures(NIPATable1.1.5,Line2). (cid:15) Non-Residential Investment: Private (cid:2)xed investment - non-residential (NIPA Table 1.1.5, (cid:15) Line9). CPI (Fred Series ID: CPIAUCSL): Consumer Price Index For All Urban Consumers (All (cid:15) Items). 37

HypothesisTests p-valueinparenthesis F-test 1952-2006 (1) No(0.93) 0.30 (2) No(0.79) 0.52 (3)1981:3-2008:3 No(0.28) 1.28 (4)1981:3-2008:3 No(0.70) 0.63 (5) No(0.23) 1.37 1947-2006 (1) No(0.12) 1.71 (2) Yes(0.08) 1.91 (5) No(0.26) 1.28 Table1: Granger-CausalityTests StockMarketIndex: S&P500Compositew/GFDextension. (cid:15) InterestRate(FredSeriesID:TB3MS):3-MonthTreasuryBill: SecondaryMarketRate. (cid:15) A.1 Granger-Causality Tests FollowingRamey(2011)werunthefollowingGrangerCausalitytests: 1. DowardatesGrangercauseVARshocks? 2. DoDefensenewsGrangercauseVARspendingshocks? 3. Do1-quarteraheadprofessionalforecastsGrangercauseVARspendingshocks? 4. Do4-quarteraheadprofessionalforecastsGrangercauseVARspendingshocks? 5. DoMertensandRavnanticipatedtaxshocksGrangercauseVARtaxshocks? Werunthefollowingregression: 6 6 shock (cid:11) (cid:12) shock (cid:13) news (cid:23) t i t i i t i t D C (cid:0) C (cid:0) C i 1 i 1 XD XD where tax and spending shocks are identi(cid:2)ed by the B&P approach at the OLS estimates. The followingtablereportstheF-testforthenullhypothesis(cid:13) 0;i 1;:::;6. i D D 38

B Details of Analytical Results in Section I The relation between reduced-form residuals u and structural shocks e presented in (2) can also t t bewrittenas: Au D1=2e ; t t D where A is a .n n/ matrix of structural coef(cid:2)cients, and D is a diagonal matrix containing the (cid:2) variances of the structural shocks, and F A 1D1=2.We denote the standard deviation of the (cid:0) D structuralshockse asd ,withd pd ;fori 1;:::;n. i;t i i ii (cid:17) D Pre-multiplyingequation(1)bymatrix A givesthestructuralformoftheVARmodel: AX AB.L/X e : t t 1 t D (cid:0) C Finally,therelationbetweenstructuralcoef(cid:2)cients.A;D/andreduced-formcoef(cid:2)cients6 is u givenby: E u t u 0t E A (cid:0) 1e t e t0 A (cid:0) 1 0 (B.1) D E (cid:2)u t u 0t (cid:3) A (cid:0) h1 E e t e t0 A (cid:0) 1i 0 D (cid:2) 6(cid:3) A 1D(cid:2)A 1(cid:3); u (cid:0) (cid:0) 0 D whichdescribesasystemofn.n 1/=2independentnon-linearequations. (cid:0) B.1 Bivariate Models In the bivariate model, we solve a system of three equations (as many as the distinct elements of 6 )inthreeunknowns.a ;d ;d /: u Y;P YY PP 6 A 1DA 1; u (cid:0) (cid:0) 0 D 39

where 1 a Y;P A (cid:0) ; D 2 3 (cid:17) 1 P;Y (cid:0) 4 5 Thesolutionofthissystemis: (cid:17) (cid:27) (cid:27) P;Y YY YP a (cid:0) Y;P D (cid:17) (cid:27) (cid:27) P;Y YP PP (cid:0) (cid:27) (cid:17)2 (cid:27) 2(cid:17) (cid:27) (cid:27) (cid:27) (cid:27)2 PP C P;Y YY (cid:0) P;Y YP PP YY (cid:0) YP d YY D (cid:16) (cid:27) (cid:17) (cid:27) (cid:17) (cid:0) 2 (cid:1) PP P;Y YP (cid:0) (cid:0) (cid:1) d (cid:27) (cid:17)2 (cid:27) 2(cid:17) (cid:27) : PP PP P;Y YY P;Y YP D C (cid:0) Substituting the analytical solution for a in matrix A 1,we obtain the following analytical Y;P (cid:0) expressionfortheimpactimpulseresponses: 1 (cid:27)YP (cid:0) (cid:17) P;Y (cid:27)YY A (cid:0) 1 (cid:17) P;Y ;6 u D (cid:17)2 P;Y (cid:27) (cid:27) YY P C P (cid:0) (cid:27) (cid:17) P P P ; (cid:0) Y (cid:27) 2(cid:17) Y P P ;Y (cid:27)YP 2 (cid:17) P;Y (cid:27)PP (cid:0) (cid:17) 1 P;Y (cid:27)YP 3 : (cid:0) (cid:1) 4 5 Theassumptionthat6 ispositivede(cid:2)niteensuresthatthedenominatorofallimpactresponses u is strictly larger than zero. This guarantees that impulse response functions are de(cid:2)ned for all outputelasticities(cid:17) .34 P;Y 34Letabea2 1vector. Thefunctiona6 a iscalledaquadraticformina. Thematrix6 ispositivede(cid:2)niteif u u a6 a > 0foral (cid:2) la 0. Fora (cid:17) ;1 we 0 canwritethisconditionas(cid:17)2 (cid:27) (cid:27) 2(cid:17) (cid:27) > 0. See u 0 6D D P;Y P;Y YY C PP (cid:0) P;Y YP GolubandvanLoan(1996). (cid:2) (cid:3) 40

B.2 Multivariate Models Inthebivariatemodel,wecanrewritetheelementioftheimpulsevectorassociatedwiththepolicy shocke as: P;t (cid:27) (cid:17) (cid:27) A 1 iP (cid:0) P;Y iY ; (B.2) i(cid:0);P D (cid:27) (cid:17)2 (cid:27) 2(cid:17) (cid:27) PP C P;Y YY (cid:0) P;Y YP fori Y; P. D Letusintroduceathirdvariableintothesystem: u a u a u e Y;t Y;P P;t Y;3 3;t Y;t D C C u (cid:17) u a u e P;t P;Y Y;t P;3 3;t P;t D C C u a u a u e : 3;t 3;Y Y;t 3;P P;t 3;t D C C Inathree-equationVARmodel,weneedthreerestrictionstoidentifythesystem(B.1). Without loss of generality, we assume that restrictions are imposed on (cid:17) ;a ; and a . In the interest P;Y P;3 Y;3 ofspace,wedonotreportthesolutiontothethree-equationsystem.35 Theelementi oftheimpulse vectorassociatedwiththepolicyshocke P;t canbewrittenas: (cid:27) (cid:17) (cid:27) a (cid:27) A 1 iP (cid:0) P;Y iY (cid:0) P;3 i3 : (B.3) i(cid:0);P D (cid:27) (cid:17)2 (cid:27) a2 (cid:27) (cid:17) (cid:27) 2a (cid:27) 2(cid:17) a (cid:27) PP C P;Y YY C P;3 33 (cid:0) P;Y YP (cid:0) P;3 P3 C P;Y P;3 Y3 Notice that the impulse vector is independent of the restriction on a that we impose to identify 13 e : Y;t Ifa 0,thesolutionfortheimpulsevector(B.3)collapsestoexpression(B.2),thesolution P;3 D for the impulse vector found in the bivariate model. This result generalizes to VAR models of dimensionn underthefollowingassumptions: If, without loss of generality, the policy variable is ordered (cid:2)rst in the system, matrix A is (cid:15) 35Theresultsareavailableuponrequest. 41

blockrecursive: 1 (cid:17) 0 P;Y (cid:0) A 2 3; D (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) 6 7 Block 2 6 7 6 7 4 5 where Block 2 is an .n 1 n/ submatrix of structural coef(cid:2)cients for output and the (cid:0) (cid:2) additionaln 2variablesintheVAR,and0isan1 n 2vectorofzeros. (cid:0) (cid:2) (cid:0) Thecontemporaneousresponseofallvariablestothepolicyshockisleftunrestricted. (cid:15) Under these assumptions, expression (B.2) describes the impact response of variable i to a policy shock, for i Y; P;:::;n. The intuition is simple: what matters for the identi(cid:2)cation of D the policy shock is the output elasticity of the policy variable (cid:17) and the reduced-form residual P;Y u : How the remaining n 2 structural shocks generate (cid:3)uctuations in u is irrelevant for the Y;t Y;t (cid:0) identi(cid:2)cation of e : This is the same argument that explains why in a Cholesky decomposition, P;t impulse responses associated to the shock ordered (cid:2)rst do not depend on the identi(cid:2)cation of the remainingn 1shocks,ieontheorderingoftheremainingn 1variables. (cid:0) (cid:0) To derive analytical expressions for impulse responses at longer horizon we start from the MovingAverage(MA)representationoftheSVARmodel: 1 X 2 e ; (B.4) t j t j D (cid:0) j 0 XD where 2 8 A 1 .j 0;1;2;:::/. The elements of matrices 8 ’s are functions of the j j (cid:0) j D D autoregressive coef(cid:2)cients contained in the lag polynomial B.L/. 36 The matrix 2 contains j impulse responses j quarters after the shock, which are linear combination of impact responses B.2. We do not inspect the analytical expression for impulse responses at horizon j 1. The (cid:21) above assumptions ensures that, for given 2 and 6 , impulse responses to a policy shock e j u P;t only depend on the output elasticity of the policy variable at any horizon. Hence we can plot impulse responses as non-linear functions of the elasticities for any value of the elasticities, as we did for the impact responses.37 For instance, Figure 8 plots the tax and spending multiplier 4, 8, 36Impactresponsescanbewrittenas X 2 A 1e . Since8 I,weobtaintheexpressionspresentedabove. t 0 (cid:0) t 0 D D 37If we did not have analytical expressions, we should have solved a system of 55 non-linear equations for each valueoftheelasticitywewantedtostudy,andthencomputeimpulseresponsesatdifferenthorizons. Theanalytical procedureissubstantiallyfasterandpossiblymoreaccurate. 42

1.5 0.5 -0.5 -1.5 -2 -1 0 1 2 3 4 5 Output Elasticity of Tax Revenue tluM T tcapmI 3 Median 1 68% C.S. -1 -3 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM G tcapmI 1.5 0.5 -0.5 -1.5 -2 -1 0 1 2 3 4 5 Output Elasticity of Tax Revenue tluM T trQ-4 Median 68% C.S. 1.5 0.5 -0.5 -1.5 -2 -1 0 1 2 3 4 5 Output Elasticity of Tax Revenue tluM T trQ-8 1.5 0.5 -0.5 -1.5 -2 -1 0 1 2 3 4 5 Output Elasticity of Tax Revenue tluM T trQ-21 3 1 -1 -3 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM G rtQ-4 3 1 -1 -3 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM G rtQ-8 3 1 -1 -3 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM G rtQ-21 Figure 8: Tax and spending multipliers at different horizons as function of the output elasticity of taxrevenueandspending. 43

and12quartersaftertheshock. Proposition1 Theoutputresponsetoapolicyshock(7)hasthefollowingproperties: 1. Ithasaglobalminimumandaglobalmaximumsuchthat: A 1 (cid:17)min;6 < 0 (cid:0)Y;P P;Y u A 1 (cid:0)(cid:17)max;6 (cid:1) > 0 (cid:0)Y;P P;Y u (cid:0) (cid:1) where(cid:17)min argminA 1 (cid:17) ; ,(cid:17)max argmaxA 1 (cid:17) ; ,and P;Y D (cid:17) (cid:0)Y;P P;Y (cid:1) P;Y D (cid:17) (cid:0)Y;P P;Y (cid:1) P;Y P;Y (cid:0) (cid:1) (cid:0) (cid:1) (cid:17)max < (cid:17)min: P;Y P;Y 2. Itequalszeroifandonlyif(cid:17) (cid:27) =(cid:27) (cid:17) . P;Y YP YY P;Y D (cid:17) 3. It is strictly decreasing for (cid:17) (cid:17)max; (cid:17)min , and strictly increasing for (cid:17) < (cid:17)max P;Y 2 P;Y P;Y P;Y P;Y or(cid:17) > (cid:17)min. h i P;Y P;Y Proof of Proposition 1. First, we prove existence of a global minimum and maximum of A 1 (cid:17) ;6 . Notethat A 1 (cid:17) ;6 belongstothefamilyofrationalfunctions,whichare (cid:0)Y;P P;Y u (cid:0)Y;P P;Y u contin(cid:0)uous and(cid:1)differentiable. So(cid:0)in order t(cid:1)o (cid:2)nd the global extrema of A (cid:0)Y; 1 P (cid:17) P;Y ;6 u we have toinvestigateits(cid:2)rstandsecondderivatives. Withsomeabuseofnotation,deno(cid:0)te A (cid:0)Y; 1 P (cid:17)(cid:1) P;Y ;6 u by f (cid:17) . Equating the (cid:2)rst derivative to zero we obtain two points that satisfy the(cid:0)necessary(cid:1) P;Y condi(cid:0)tionsf(cid:1)oranextremumof f (cid:17) : P;Y (cid:0) (cid:1) (cid:26) (cid:27) 1 (cid:26)2 (cid:26) (cid:27) 1 (cid:26)2 (cid:17)min YP C P (cid:0) YP ; (cid:17)max YP (cid:0) P (cid:0) YP : P;Y D (cid:27)q P;Y D (cid:27)q Y Y Itisimmediatetoseethat(cid:17)min > (cid:17)max. Thesuf(cid:2)cientconditionforextremumischeckedderiving P;Y P;Y 44

thesecondderivativesof f (cid:17) andevaluatingitat(cid:17)min and(cid:17)max: P;Y P;Y P;Y (cid:0) (cid:1) (cid:27)3 1 (cid:26)2 f (cid:17) Y (cid:0) PY > 0 00 P;Y j(cid:17) P;Y D (cid:17)m P; i Y n D q2(cid:27)3 P (cid:0) (cid:1) (cid:27)3 1 (cid:26)2 f 00 (cid:17) P;Y j (cid:17) P;Y D (cid:17)m P; a Y x D (cid:0) Y q2(cid:27) (cid:0) 3 PY < 0; P (cid:0) (cid:1) providedthat (cid:26) < 1. PY Finally,the(cid:12) (cid:12) glob(cid:12) (cid:12) alminimumandmaximumof A (cid:0)Y; 1 P (cid:17) P;Y ;6 u are: (cid:0) (cid:1) (cid:27) A 1 (cid:17)min;6 Y < 0 (cid:0)Y;P P;Y u D (cid:0) 2(cid:27) 1 (cid:26)2 (cid:0) (cid:1) P (cid:0) YP A 1 (cid:17)max;6 (cid:27)qY > 0: (cid:0)Y;P P;Y u D 2(cid:27) 1 (cid:26)2 (cid:0) (cid:1) P (cid:0) YP q ThesecondstatementinProposition1canbeeasilyprovedusingthede(cid:2)nitionof A 1 (cid:17) ;6 : (cid:0)Y;P P;Y u (cid:0) (cid:1) (cid:27) A 1 (cid:17) ;6 0 (cid:27) (cid:17) (cid:27) 0 (cid:17) YP : (cid:0)Y;P P;Y u D () YP (cid:0) P;Y YY D () P;Y D (cid:27) YY (cid:0) (cid:1) The third statement in Proposition 1 states that A 1 (cid:17) ;6 is strictly decreasing for (cid:17) (cid:0)Y;P P;Y u P;Y 2 (cid:17)max; (cid:17)min andstrictlyincreasingfor(cid:17) < (cid:17)max (cid:0) (cid:17) >(cid:1)(cid:17)min. Thisstatementcanbeeasily P;Y P;Y P;Y P;Y _ P;Y P;Y phrovedbyanialyzingthesignof f (cid:17) . 0 P;Y (cid:0) (cid:1) C Analytical Results for the Sign Restriction Approach This part of the Appendix provides formal derivations of the results for the pure sign restriction approach and the penalty function approach to sign restrictions cited in the main text of the paper. WestartwiththeanalyticalsolutiontotheCholeskydecompositionofthecovariancematrix,which willproveusefulinthederivationoftheanalyticalresultsforthesignrestrictionapproach. 45

0.5 0.25 0 -0.25 -0.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM C tcapmI 0.5 0.25 0 -0.25 -0.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM C rtQ-4 0.5 0.25 0 -0.25 -0.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM G rtQ-8 0.5 0.25 0 -0.25 -0.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 Output Elasticity of Spending tluM C rtQ-21 Figure 9: Consumption response to a spending shock as function of the output elasticity of governmentspending. 46

C.1 The Symbolic Cholesky Decomposition We assume that the prediction error covariance matrix 6 is a real symmetric positive de(cid:2)nite u matrix. The positive de(cid:2)niteness of6 implies(cid:27) > 0 for alli and (cid:27) < (cid:27) (cid:27) or, equivalently, u i ij i j (cid:26) < 1 for i j. This assumption guarantees that the Cholesky f(cid:12)acto(cid:12)rization of the covariance ij 6D (cid:12) (cid:12) (cid:12)matr(cid:12)ix exists, i.e. a unique lower triangular matrix P Rn n with positive elements on the (cid:2) (cid:12) (cid:12) 2 principal diagonal exists such that 6 PP (Golub and van Loan (1996), Theorem 4.2.5, p. u 0 D 143). We write the individual elements of the covariance matrix in terms of standard deviations and correlation coef(cid:2)cients, which is useful for the presentation of the analytical results in the paper: (cid:27) ::: (cid:27) (cid:27) (cid:26) ::: (cid:27) (cid:27) (cid:26) 11 1 j 1j 1 n 1n 2 : : : ::: : : : : : : 3 6 6 (cid:27) (cid:27) (cid:26) ::: (cid:27) ::: (cid:27) (cid:27) (cid:26) 7: (C.1) u D 6 6 1 j 1j ii j n nj 7 7 .n n/ 6 6 : : : : : : ::: : : : 7 7 (cid:2) 6 7 6 7 6 (cid:27) 1 (cid:27) n (cid:26) 1n ::: (cid:27) j (cid:27) n (cid:26) nj ::: (cid:27) nn 7 6 7 4 5 The key to our analytical approach is the existence of an analytical expression for the lowertriangular Cholesky decomposition of the covariance matrix 6 . The Cholesky decomposition u has a recursive structure, which greatly simpli(cid:2)es the derivation of the individual elements of the Cholesky factor matrix. Denoting the Cholesky factor of 6 by P [p ] the algorithm for the u ij D computationofitsindividualelementscanbeexpressedasfollows: 0 for : i < j; 8 p ij D > > > > > (cid:27) ii (cid:0) i k(cid:0) D 1 1 p i 2 k for : i D j; (C.2) <q > p 1 jj (cid:27) ij P (cid:0) k j D (cid:0) 1 1 p ik p jk for : i > j: > > (cid:16) (cid:17) > > P : The computation starts from the upper left corner of P and proceeds to calculate the matrix either row by row (Cholesky-Banachiewicz algorithm) or column by column (Cholesky-Crout algorithm). The lower-triangular Cholesky decomposition of the covariance matrix 6 yields the u following expression for the Cholesky factor P, where to save space we report only the elements 47

ofthe(cid:2)rstandsecondcolumnsexpressedas p.1/ and p.2/,respectively: (cid:27) 0 1 2 (cid:27) (cid:26) (cid:27) 1 (cid:26)2 3 2 12 2 (cid:0) 12 : : 6 : q : 7 : : 6 7 6 7 p.1/ p.2/ D 6 6 (cid:27) j (cid:26) 1j (cid:27) j (cid:26) 2j (cid:0) (cid:26) 1j (cid:26) 12 7 7 : (C.3) 6 1 (cid:26)2 7 h i 6 (cid:0) 12 7 : : 6 : q: 7 6 : : 7 6 7 6 (cid:26) (cid:26) (cid:26) 7 6 (cid:27) n (cid:26) 1n (cid:27) n 2n (cid:0) 1n 12 7 6 1 (cid:26)2 7 6 (cid:0) 12 7 4 q 5 We use the Cholesky decomposition because this allows for the derivation of simple analytical expressions. Note, however, that the inference does not depend on the use of this particular decomposition. Any other exact factorization of 6 will yield the same inference (see Uhlig (2005), u AppendixB). C.2 The Pure Sign Restriction Approach: Bivariate Model C.2.1 DerivingEquation(8) We start with the derivation of Equation (8) in the main text, which gives the factor matrix for the pure sign restriction approach. Recall that the system can be written u PQe in compact t t D form. ForthebivariatesystemtheCholeskyfactor,withoutputordered(cid:2)rstandthepolicyvariable orderedsecond,canbeexpressedasfollows: (cid:27) 0 (cid:27) 0 Y Y P ; D 2 (cid:27) (cid:26) (cid:27) 1 (cid:26)2 3 D 2 (cid:27) cos’ (cid:27) sin’ 3 P YP P (cid:0) YP P YP P YP 4 q 5 4 5 where, to facilitate the derivation of analytical results, without loss of generality, we express the error correlation coef(cid:2)cient (cid:26) as angle. With ’ arccos.(cid:26) / being the angle represention YP YP YP (cid:17) oftheerrorcorrelationcoef(cid:2)cientwecanwrite(cid:26) cos.’ /and 1 (cid:26)2 sin.’ /.38 YP D YP (cid:0) YP D YP q 38Notethat(cid:26) . 1;1/implies’ .0;(cid:25)/. Theangle’ isstrictlydecreasinginthecorrelationcoef(cid:2)cient YP YP YP (cid:26) ,with’ (cid:25) 2 a (cid:0) s(cid:26) 1,’ 2 (cid:25) for(cid:26) 0and’ 0as(cid:26) 1. YP YP ! YP !(cid:0) YP D 2 YP D YP ! YP ! 48

Theorthogonalmatrix Q canbeexpressedasfollowsinthebivariatecase: cos(cid:18) sin(cid:18) Q (cid:0) ; D 2 3 sin(cid:18) cos(cid:18) 4 5 where (cid:18) [ (cid:25);(cid:25)] is a rotation angle. Using these de(cid:2)nitions the factor matrix for the pure sign 2 (cid:0) restrictionapproach FSR canbeexpressedasfollows: (cid:27) cos(cid:18) (cid:27) sin(cid:18) FSR Y (cid:0) Y : D 2 3 (cid:27) .cos’ cos(cid:18) sin’ sin(cid:18)/ (cid:27) .cos’ sin(cid:18) sin’ cos(cid:18)/ P YP YP P YP YP C (cid:0) (cid:0) 4 5 Usingbasictrigonometricidentitiesthisexpressioncanbefurthersimpli(cid:2)edtoyieldEquation (8)inthemaintext39: (cid:27) cos(cid:18) (cid:27) sin(cid:18) FSR Y (cid:0) Y : D 2 3 (cid:27) cos.(cid:18) ’ / (cid:27) sin.(cid:18) ’ / P YP P YP (cid:0) (cid:0) (cid:0) 4 5 C.2.2 Thesetofpuresignrestrictionsolutionsforthestandard(loose)setofrestrictions Wenextcharacterizethesetofpuresignrestrictionsolutionsunderthebaselineassumptionsgiven inthemaintext(forthesakeofbrevityweconcentrateonthemoreinterestingcaseoftheidenti(cid:2)cationoftaxshocks): ? FSR C : (C.4) D 2 3 C C 4 5 Proposition2 Let S bethesetofallsolutionssatisfyingthesignrestrictionsgivenby(C.4). Then, theset S,forgiven’ .0;(cid:25)/,is YP 2 (cid:25) (cid:25) S (cid:18) [ (cid:25);(cid:25)] : ’ (cid:18) : YP (cid:17) 2 (cid:0) (cid:0)2 C (cid:20) (cid:20) 2 n o This set is non-empty for all ’ .0;(cid:25)/, i.e. for less than perfect correlation between the YP 2 one-stepaheadpredictionerrors,(cid:26) . 1;1/. YP 2 (cid:0) 39Theexpressionfor FSR usestheangledifferenceidentitycos.(cid:18)/cos.’ / sin.(cid:18)/sin.’ / cos.(cid:18) ’ /, 21 YP C YP D (cid:0) YP whiletheexpressionfor FSR usestheangledifferenceidentitysin.(cid:18)/cos.’ / cos.(cid:18)/sin.’ / sin.(cid:18) ’ /. 22 YP (cid:0) YP D (cid:0) YP 49

Proof. Abstracting from the sign restrictions, note that the interval [ (cid:25);(cid:25)] describes the set of (cid:0) all possible rotation angles, with angles measured in radians. The angle (cid:18) (cid:25) corresponds to a D (cid:0) clockwiserotationby 180 ,whiletheangle(cid:18) (cid:25) correspondstoacounterclockwiserotationby (cid:14) (cid:0) D 180 . Inotherwords,theintervaldescribestheunitcircle. Withintheselimits,thesignrestrictions (cid:14) furtherrestrictthesetofadmissiblerotationangles: (cid:25) (cid:25) FSR 0 (cid:18) ; 11 (cid:21) , (cid:0)2 (cid:20) (cid:20) 2 (cid:25) (cid:25) FSR 0 ’ (cid:18) ’ ; 21 (cid:21) , (cid:0)2 C YP (cid:20) (cid:20) 2 C YP FSR 0 (cid:25) ’ (cid:18) ’ : 22 YP YP (cid:21) , (cid:0) C (cid:20) (cid:20) ThisonitsownwouldsuggestthatthesetofpuresignrestrictionsolutionsisS (cid:18) [ (cid:25);(cid:25)] : (cid:25) ’ (cid:18) min.’ ; (cid:25)/ . (cid:17) 2 (cid:0) (cid:0)2 C YP (cid:20) (cid:20) YP 2 However,wealsoneedtolookatre(cid:3)ectionsbecausethesignoftheshocksisjus(cid:8)tanormalization. (cid:9) The assumptions given in (C.4) will also be satis(cid:2)ed if FSR < 0 and FSR < 0 for the (cid:2)rst shock 11 21 and FSR < 0 for the second shock, simply requiring a sign-(cid:3)ipping of the respective column of 22 FSR. First, note that the inclusion of re(cid:3)ections means that the sign restriction on FSR can be dis- 22 regarded as it will be satis(cid:2)ed, after sign-(cid:3)ipping where needed for all (cid:18) [ (cid:25);(cid:25)]. Disregard- 2 (cid:0) ing the sign restriction on FSR implies that the set of pure sign restriction solutions grows to 22 S (cid:18) [ (cid:25);(cid:25)] : (cid:25) ’ (cid:18) (cid:25) . (cid:17) 2 (cid:0) (cid:0)2 C YP (cid:20) (cid:20) 2 S(cid:8)econd, consider the (cid:2)rst shock for w(cid:9)hich the inclusion of re(cid:3)ections leads to the following conditions: FSR 0 (cid:25) (cid:18) (cid:25) and (cid:25) (cid:18) (cid:25); 11 (cid:20) , (cid:0) (cid:20) (cid:20) (cid:0)2 2 (cid:20) (cid:20) FSR 0 (cid:25) (cid:18) (cid:25) ’ and (cid:25) ’ (cid:18) (cid:25): 21 (cid:20) , (cid:0) (cid:20) (cid:20) (cid:0)2 C YP 2 C YP (cid:20) (cid:20) Note, however, that these conditions do not add solutions to the sign restriction set, once the sign-(cid:3)ipping is taken into consideration. The reason is that the sign-(cid:3)ipping can be implemented through a phase shift by 180 ( (cid:25)/. Shifting the subsets satisfying FSR 0 by (cid:25) (adding (cid:25) to (cid:14) (cid:6) 11 (cid:20) C (cid:25) (cid:18) (cid:25) gives the subset 0 (cid:18) (cid:25)/ or by (cid:25) (substracting (cid:25) from (cid:25) (cid:18) (cid:25) gives the (cid:0) (cid:20) (cid:20) (cid:0)2 (cid:20) (cid:20) 2 (cid:0) 2 (cid:20) (cid:20) subset (cid:25) (cid:18) 0/,respectively,andtakingtheunionofthetworesultingsubsetsgivesthesame (cid:0)2 (cid:20) (cid:20) e set as the one satisfying FSR 0. The same holds for the sign-(cid:3)ipping of the solutions satisfying e 11 (cid:21) 50

FSR 0: shifting the subsets satisfying FSR 0 by (cid:25) (adding (cid:25) to (cid:25) (cid:18) (cid:25) ’ 21 (cid:20) 21 (cid:20) C (cid:0) (cid:20) (cid:20) (cid:0)2 C YP gives the subset 0 (cid:18) (cid:25) ’ / or by (cid:25) (substracting (cid:25) from (cid:25) ’ (cid:18) (cid:25) gives the (cid:20) (cid:20) 2 C YP (cid:0) 2 C YP (cid:20) (cid:20) subset (cid:25) ’ (cid:18) 0/, respectively, and taking the union of the two resulting subsets gives (cid:0)2 C YP (cid:20)e (cid:20) thesamesetastheonesatisfying FSR 0. e 21 (cid:21) Takentogetherthesetofallpuresignrestrictionsolutionsisgivenby S. Thissetisnon-empty foralladmissiblevaluesoftheerrorcorrelationcoef(cid:2)cientas (cid:25) ’ < (cid:25) forall’ .0;(cid:25)/. (cid:0)2 C YP 2 YP 2 ThiscompletestheproofofProposition2. It is worthwhile to quickly review the implications of Proposition 2 for our application. First, for the sign restrictions given by (C.4) the set will be the larger the smaller the value of ’ , YP i.e. the larger the value of the correlation coef(cid:2)cient between reduced-form output and policyvariable disturbances.40 In our application - but also in the VARs estimated by B&P and M&U - the correlation coef(cid:2)cient is positive and large: for our VAR - evaluated at the OLS estimate - (cid:26)OLS 0:49 (and (cid:26)OLS 0:29). This implies that the set S is very large in terms of the range YT D YG D ofadmissible(cid:18). However,regardlessofthesizeof S,itisalwaystruethatalloutputelasticitiesof b b taxes (cid:17)SR between zero and plus in(cid:2)nity will satisfy the sign restrictions given by (C.4). To see T;Y thisrecallthede(cid:2)nitionof(cid:17)SR givenbyEquation(9)inthemaintext: P;Y FSR (cid:27) cos.(cid:18) ’ / (cid:17)SR 21 P (cid:0) YP : P;Y (cid:17) FSR D (cid:27) cos(cid:18) Y 11 Forthelowerboundof S((cid:18) (cid:25) ’ )theelasticityiszerobecausethenumerator FSR 0 D (cid:0)2C YP 21 D while the denominator FSR > 0 for all ’ .0;(cid:25)/. For the upper bound of S ((cid:18) (cid:25)) the 11 YP 2 D 2 elasticity goes to plus in(cid:2)nity because the denominator FSR 0 while the numerator FSR > 0 11 ! 21 for all ’ .0;(cid:25)/. In addition, (cid:17)SR is strictly increasing in (cid:18) over S. To see this note that YP 2 P;Y the expression for the elasticity can be transformed as follows (yielding the expression shown in 40Recallthat’ istheanglerepresentationofthecorrelationcoef(cid:2)cient(cid:26) .Thecorrelationcoef(cid:2)cientisde(cid:2)ned YP YP overtheinterval. 1 1/,whereweruleoutthecasesofperfectcorrelation. Theintervalforthecorrelationcoef(cid:2)cient (cid:0) I translatesinto’ .0;(cid:25)/. Notethat’ isdecreasingin(cid:26) ,with’ (cid:25) for(cid:26) 1,’ (cid:25)=2for YP YP YP YP YP YP 2 ! ! (cid:0) D (cid:26) 0and’ 0for(cid:26) 1. YP YP YP D ! !C 51

Equation(9)inthemaintext): (cid:27) cos’ cos(cid:18) sin’ sin(cid:18) (cid:17)SR P YP C YP P;Y D (cid:27) cos(cid:18) Y (cid:27) (cid:27) P P (cid:26) sin’ tan(cid:18) YP YP D (cid:27) C (cid:27) Y Y (cid:27) P (cid:17) sin’ tan(cid:18): P;Y YP D C (cid:27) Y Toshowthat(cid:17)SR isstrictlyincreasingin(cid:18) over S itissuf(cid:2)cienttoshowthatthe(cid:2)rstderivative P;Y of tan(cid:18) with respect to (cid:18) is positive for all (cid:18), because (cid:27)P sin’ > 0 in any case. Now, the (cid:2)rst (cid:27)Y YP derivativeoftan(cid:18) withrespectto(cid:18) is1 tan2(cid:18),whichispositiveforall(cid:18). C Note also that the expression for the impact tax cut multiplier for the sign restriction approach canbederivedinanalogytoEquation(6)inSection1ofthemaintext: aSR 1 T;Y Y;T 5 .(cid:18) 6 / 0 I u D (cid:0)1 aSR (cid:17)SR T=Y (cid:0) Y;T T;Y FSR=FSR 1 12 22 D (cid:0)1 FSR=FSR(cid:17)SR T=Y (cid:0) 12 22 T;Y (cid:27) =(cid:27) sin(cid:18)=sin.(cid:18) ’ / 1 Y T YT (cid:0) D (cid:0) 1 tan(cid:18)=tan.(cid:18) ’ / T=Y YT (cid:0) (cid:0) (cid:27) 1 sin2(cid:18) 1 Y : D (cid:27) T 2sin’ YT T=Y The main properties of the impact tax cut multiplier for the pure sign restriction approach and forthesignrestrictionsgivenby(C.4)canbesummarizedasfollows: 1. The impact multiplier is negative for (cid:18) [ (cid:25)=2 ’;0/. The reason is that sin2(cid:18) < 0 2 (cid:0) C for this range of (cid:18). This subset is empty if and only if output and tax disturbances are negativelycorrelated(’ (cid:25)=2). Thissubsetimplies0 (cid:17)SR < (cid:17) ;inourapplication YT (cid:21) (cid:20) T;Y T;Y 0 (cid:17)SR < 1:5. (cid:20) T;Y 2. The impact multiplier is zero for (cid:18) 0 (the Cholesky factorization with output ordered D (cid:2)rst). The Cholesky factor is a particular pure sign restriction solution as long as the correlation between output and tax disturbances is non-negative (’ (cid:25)=2). This Cholesky YT (cid:20) factorizationimplies(cid:17)SR (cid:17) ( 1:5inourapplication). T;Y D T;Y D 52

3. The impact multiplier is positive for (cid:18) .0;(cid:25)=2]. The reason is that sin2(cid:18) > 0 for this 2 range of (cid:18). This subset is non-empty for all admissible values of the correlation coef(cid:2)cient betweenoutputandtaxdisturbances(’ .0;(cid:25)/,i.e. (cid:26) . 1;1/). Thissubsetimplies YT YT 2 2 (cid:0) (cid:17) < (cid:17)SR ;inourapplication1:5 < (cid:17)SR < . T;Y T;Y (cid:20) C1 T;Y C1 4. The impact multiplier reaches its maximum over S for (cid:18) (cid:25)=4.41 The impact tax cut D multiplier,evaluatedat(cid:18) (cid:25),is D 4 (cid:25) (cid:27) 1 1 1 1 p 1 T;Y Y 11 5 .(cid:18) 6 / ; 0 D 4I u D (cid:27) T 2sin’ YT T=Y D 2 p 22 T=Y andtheoutputelasticityoftaxes,evaluatedat(cid:18) (cid:25),is D 4 (cid:25) (cid:27) p p (cid:17)SR .(cid:18) 6 / T .cos’ sin’ / 21 C 22 : T;Y D 4I u D (cid:27) YT C YT D p Y 11 Note the simplicity of these expressions: all that it is needed for the calculation of the maximum impact multiplier and the associated elasticity is knowledge of the elements of the Cholesky factorization. In our application - with 6 evaluated at the OLS estimate - the u maximumimpacttaxcutmultiplierisequalto1:07dollarsandtheassociatedoutputelasticityoftaxesisequalto4:15. C.2.3 The set of pure sign restriction solutions for the alternative (more restrictive) set of restrictions Wenextturntothesetofpuresignrestrictionsolutionsunderthealternativeassumptionsgivenin themaintext(againweconcentrateonthemoreinterestingcaseoftheidenti(cid:2)cationoftaxshocks) underwhichwerestricttheobjectofinterest(theoutputresponsetoataxshock)tobenegativein 41Notethatmaximizing5 T;Y .(cid:18) 6 /withrespectto(cid:18) over S isequivalenttomaximizingsin2(cid:18) withrespectto(cid:18) 0 I u ! overS. The(cid:2)rst-orderconditionisdsin2(cid:18)=d(cid:18) 2cos2(cid:18) 0. Thishastwosolutions: (cid:18) (cid:25)=4and(cid:18) (cid:25)=4. 1 2 D D DC D(cid:0) ! Thesecond-orderconditionforamaximumisdsin22(cid:18)=d(cid:18)2 4sin2(cid:18) <0,whichissatis(cid:2)edonlyfor(cid:18) (whereas 1 D(cid:0) (cid:18) satis(cid:2)es the conditions for a minimum). The maximum is an interior maximum if and only if (cid:26) < 3=4(cid:25), i.e. 2 YT (cid:26) > 1=p2 0:71. Thisconditioniseasilysatis(cid:2)edfor(cid:2)scalVARmodelsgiventhatthecorrelationbetween YT taxand (cid:0) outputre (cid:25) sid (cid:0) ualsistypicallystronglypositive(inourapplication(cid:26)OLS 0:49). For(cid:26) 3=4(cid:25) instead YT D C YT (cid:21) themaximumimpacttaxmultiplierobtainsforthelowerboundofS. b 53

ordertoruleoutnegativeimpacttaxcutmultipliers: FSR C (cid:0) : (C.5) D 2 3 C C 4 5 Proposition3 Let S bethesetofallsolutionssatisfyingthesignrestrictionsgivenby(C.5). Then, 2 theset S ,forgiven’ .0;(cid:25)/,is 2 YP 2 (cid:25) (cid:25) S (cid:18) [ (cid:25);(cid:25)] : max. ’ 0/ (cid:18) min. ’ / : 2 YP YP (cid:17) 2 (cid:0) (cid:0)2 C I (cid:20) (cid:20) 2I n o This set is non-empty for all ’ .0;(cid:25)/, i.e. for less than perfect correlation between the YP 2 one-stepaheadpredictionerrors,(cid:26) . 1;1/. YP 2 (cid:0) Proof. Considering rotations (cid:2)rst, note that the sign restrictions restrict the set of admissible anglesto: (cid:25) (cid:25) FSR 0 (cid:18) ; 11 (cid:21) , (cid:0)2 (cid:20) (cid:20) 2 (cid:25) (cid:25) FSR 0 ’ (cid:18) ’ ; 21 (cid:21) , (cid:0)2 C YP (cid:20) (cid:20) 2 C YP FSR 0 0 (cid:18) (cid:25); 12 (cid:20) , (cid:20) (cid:20) FSR 0 (cid:25) ’ (cid:18) ’ : 22 YP YP (cid:21) , (cid:0) C (cid:20) (cid:20) TheintersectionofthesubsetssatisfyingtheindividualsignrestrictionsgivesthesetS (cid:18) [ (cid:25);(cid:25)] : max. (cid:25) ’ 0/ (cid:18) min.(cid:25) ’ / . 2 (cid:17) 2 (cid:0) (cid:0)2 C YP I (cid:20) (cid:20) 2I YP The additional consideration of re(cid:3)ections does not affect this subset. To see this con(cid:8)sider the (cid:9) followingtwosubcases: First, the subcase in which FSR 0 and FSR 0 but FSR 0 and FSR 0. The individual 11 (cid:21) 21 (cid:21) 12 (cid:21) 22 (cid:20) restrictionsaresatis(cid:2)edforthefollowingangles: FSR 0 (cid:25) (cid:18) (cid:25); 11 (cid:21) , (cid:0)2 (cid:20) (cid:20) 2 FSR 0 (cid:25) ’ (cid:18) (cid:25) ’ ; 21 (cid:21) , (cid:0)2 C YP (cid:20) (cid:20) 2 C YP FSR 0 (cid:25) (cid:18) 0; 12 (cid:21) , (cid:0) (cid:20) (cid:20) FSR 0 (cid:25) (cid:18) (cid:25) ’ and (cid:25) ’ (cid:18) (cid:25): 22 (cid:20) , (cid:0) (cid:20) (cid:20) (cid:0)2 C YP 2 C YP (cid:20) (cid:20) 54

Takingtheintersectionofthesubsetssatisfyingtheindividualrestrictionsgivestheset (cid:25) ’ (cid:0)2C YP (cid:20) (cid:18) min. (cid:25) ’ 0/, which consists of one element ((cid:18) (cid:25) ’ ) for all ’ (cid:25) and is (cid:20) (cid:0)2 C YP I D (cid:0)2 C YP YP (cid:20) 2 emptyotherwise. Thisisalreadyincludedin S . 2 Second, consider the subcase in which FSR 0 and FSR 0 but FSR 0 and FSR 0. 12 (cid:20) 22 (cid:21) 11 (cid:20) 21 (cid:20) Theserestrictionsindividuallyaresatis(cid:2)edforthefollowingangles: FSR 0 (cid:25) (cid:18) (cid:25) and (cid:25) (cid:18) (cid:25); 11 (cid:20) , (cid:0) (cid:20) (cid:20) (cid:0)2 2 (cid:20) (cid:20) FSR 0 (cid:25) (cid:18) (cid:25) ’ and (cid:25) ’ (cid:18) (cid:25); 21 (cid:20) , (cid:0) (cid:20) (cid:20) (cid:0)2 C YP 2 C YP (cid:20) (cid:20) FSR 0 0 (cid:18) (cid:25); 12 (cid:20) , (cid:20) (cid:20) FSR 0 (cid:25) ’ (cid:18) ’ : 22 (cid:21) , (cid:0) C YP (cid:20) (cid:20) YP Theintersectionofthesubsetssatisfyingtheseindividualrestrictionsisanemptyset. Taken togetherthe set of allpure sign restrictionsolutions is given by S . For’ (cid:25)=2, i.e. 2 YP (cid:20) for(cid:26) 0,thesetisgivenby(cid:18) [0;’ ];thissetisnon-emptyforalladmissiblevaluesofthe YP YP (cid:21) 2 error correlation coef(cid:2)cient as ’ > 0 for all ’ .0;(cid:25)/. For ’ > (cid:25)=2, i.e. for (cid:26) < 0, YP YP YP YP 2 the set is given by (cid:18) [ (cid:25)=2 ’ ;(cid:25)=2]; this set is also non-empty for all admissible values YP 2 (cid:0) C of the error correlation coef(cid:2)cient as ’ < (cid:25) for all ’ .0;(cid:25)/. This completes the proof of YP YP 2 Proposition3. It is worthwhile to quickly review the implications of Proposition 3 for our application. First, for the sign restrictions given by (C.5) the set is largest for ’ (cid:25)=2, i.e. for (cid:26) 0, and YP YP D D shrinksasthe(absolute)valueofthecorrelationcoef(cid:2)cientbetweenoutputandpolicydisturbances increases. In our application (cid:26)OLS 0:49, i.e. the set of pure sign restriction solutions satisfying YT D the restrictions (C.5) is given by (cid:18) [0;’ ]. The lower bound of this interval corresponds to YT b 2 the Cholesky factorization with output ordered (cid:2)rst, for which as shown above the impact tax cut multiplier is zero and the output elasticity of taxes is equal to 1:5. At the upper bound of this intervaltheoutputelasticityoftaxesis (cid:27) 1 (cid:27) 1 (cid:17)SR .(cid:18) ’ 6 / T T ; T;Y D YT I u D (cid:27) cos’ D (cid:27) (cid:26) Y YT Y YT 55

whichin ourapplication-evaluated attheOLSestimate -yieldsavalue oftheoutput elasticity of taxesof6:15. Theimpacttaxmultiplierattheupperboundofthisintervalis (cid:27) 1 (cid:27) 1 T;Y Y Y 5 .(cid:18) ’ 6 / cos’ (cid:26) ; 0 D YT I u D (cid:27) T YT T=Y D (cid:27) T YT T=Y which in our application - evaluated at the OLS estimate - yields a value of the impact tax cut multiplier of 0.89 dollars. Note that the impact tax multiplier at the upper bound of the pure sign restrictionsetsatisfying(C.5)istheinverseoftheoutputelasticityoftaxesatthispoint(abstracting from the scaling by the inverse of the tax-to-output ratio necessary to convert percent changes to dollarchanges). C.3 The Penalty Function Approach to Sign Restrictions: Bivariate Model As an alternative to the pure sign restriction approach, the literature has used the penalty function approach to select one particular solution out of the set of pure sign restriction solutions (see e.g. Faust (1998), Uhlig (2005) and Mountford and Uhlig (2009)). More generally, the penalty function approach is used to numerically solve constrained nonlinear optimization problems for which closed-form analytical solutions may be hard to obtain or may not exist at all (see Judd (1998), pp. 123-25). The idea is to replace the constraints (cid:150) here: the sign restrictions (cid:150) with a continuouspenaltyfunctionthatpermitsbut(heavily)penalizeschoicesthatviolatetheconstraints. Asaresult,theconstrainedproblemisreplacedwithanunconstrainedone. Weshowherethatitispossibletoanalyticallysolvethenonlinearoptimizationproblemunderlying the sign-restriction penalty function approach for the bivariate case. In particular, we show that the standard penalty function used in the literature implies that the optimum is an element of thesubsetrulingoutnegativeimpacttaxcutmultipliersthatobtainswhenaddingasignrestriction on the object of interest (the output response to a tax shock) as in (C.5). This shows what should beintuitivelyclear: thestandardpenaltyfunctionisanidentifyingassumption. The standard penalty function has the sum of some or all impact impulse responses to a given shock as its arguments, where in general the impulse response of variable i to shock j is scaled by the standard deviation of this variable’s one-step-ahead prediction error, (cid:27) .42 In our notation, i 42Recallthatwerestrictourattentiontocontemporaneoussignrestrictions. 56

theindividualargumentsofthepenaltyfunctioncanthusbewrittenas FSR=(cid:27) . Thisexpressionis ij i equivalent to the square root of the fraction of variable i’s one-step-ahead forecast error variance explainedbyshock j. The literature, in general, proceeds as follows: (cid:2)rst, numerically minimize a penalty function havingsomeorallimpulseresponsestothe(cid:2)rstshockasitsarguments. Second,ifasecondshock has to be identi(cid:2)ed, minimize a penalty function having some impulse responses to the second shockasarguments,imposingthefurtherconstraintthatthesecondshockisorthogonaltothe(cid:2)rst shock. Theidenti(cid:2)cationoffurthershocksproceedsanalogously. In the bivariate case, the problem simpli(cid:2)es as there is a maximum of two shocks. There can be only one penalty function as the second shock imperatively has to explain all (cid:3)uctuation not explained by the (cid:2)rst shock. In line with M&U we assume that the penalty function has the impulseresponsestothe(cid:2)rstshockofthesign-restrictedvariablesasitsarguments. Wehereagain focus on the more interesting case of the identi(cid:2)cation of the tax shock. In this case the penalty functionhastheimpulseresponsesofoutputandtaxestothebusinesscycleshockasitsarguments. Thisyieldsthefollowingobjectivefunction(10)giveninthemaintext: FSR FSR (cid:127)MU 11 21 cos(cid:18) cos.(cid:18) ’ /; T YT (cid:17) (cid:27) C (cid:27) D C (cid:0) Y T whichistobemaximizedwithrespectto(cid:18). Expressing the problem in terms of trigonometric functions greatly facilitates the analytical solution to this optimization problem. First, the orthonormality restriction on the rotation matrix Q isautomaticallysatis(cid:2)edbecausecos2.!/ sin2.!/ 1holdsforany! R. Second,sincewe C D 2 can analytically characterize the set of all pure-sign restriction solutions, S; (see Proposition 2), weknowthatthedomainof(cid:127)MU isaclosedandboundedintervalonR. Third,asweshowinthe T prooftothefollowingproposition,since(cid:127)MU isacontinuousandconcavefunctionon S,wehave T byWeierstrass’stheoremthat(cid:127)MU achievesitsglobalmaximumonitsdomain. Furthermore,this T globalmaximumisunique. Insum,weneitherhavetoexplicitlyaccountfortheequalityconstraint implied by the orthonormality assumption, nor for the inequality constraints associated with the signrestrictions. Bothsetsofconstraintswillbeautomaticallysatis(cid:2)ed. Thefollowingpropositiongivestheanalyticalsolutiontotheoptimizationproblemunderlying 57

thepenaltyfunctionapproach: Proposition4 Let(cid:127)MU F 1 S 1 R F 2 S 1 R betheconcaveobjectivefunctiontobemaximized,de(cid:2)ned T (cid:17) (cid:27)Y C (cid:27)T ontheconvexsubset(cid:18) 2 [ (cid:0) (cid:25) 2 C ’ YT ; (cid:25) 2 ]ofRgivenbythesetofallpuresignrestrictionsolutions S (see Proposition 2). Then for given ’ .0;(cid:25)/, (cid:18)MU.’ / is the unique global maximizer of YT YT 2 (cid:127)MU on S,with T ’ (cid:18)MU.’ / YT : YT D 2 The global maximizer is the mid-point of the set of all pure sign restriction solutions S and the mid-point of the subset S satisfying the additional restriction that the output response to a tax 2 shockbenegative. Finally,thegobalmaximizeristhepuresignrestrictionsolutionthatmaximizes the fraction of one-step ahead forecast error covariance explained by the (cid:2)rst shock if the error correlationispositiveandminimizesitiftheerrorcorrelationisnegative. Proof. First, we prove that the maximum is unique and global. Since we can analytically characterizethesetofallpure-signrestrictionsolutions, S;(seeProposition2),weknowthatthedomain of(cid:127)MU isaclosedandboundedintervalonR. Moreover,(cid:127)MU isacontinuousandconcavefunc- T T tion on S. The cosine function is continuous for all real numbers. To show that (cid:127)MU is concave T on S it is suf(cid:2)cient to show that cos.(cid:18)/ and cos.(cid:18) ’ / are both concave on S (see Simon and YT (cid:0) Blume (1994), Theorem 21.8, p. 519). Using the second-derivative test it is straightforward to show that cos.(cid:18)/ is concave on the interval [ (cid:25); (cid:25)] and cos.(cid:18) ’ / is concave on the interval (cid:0)2 2 (cid:0) YT [ (cid:25) ’; (cid:25) ’ ]. The set of all pure sign-restriction solutions S is a subset of both intervals (cid:0)2 C 2 C YT for all ’ .0;(cid:25)/. Then, by Weierstrass’s theorem (cid:127)MU achieves its global maximum on its YT 2 T domain. To establish uniqueness we need to show that (cid:127)MU is strictly concave d2(cid:127)1 < 0 on its T d(cid:18)2 entire domain. Note, (cid:2)rst, that cos.(cid:18)/ and cos.(cid:18) ’ / have continuous deriva(cid:16)tives with r(cid:17)espect YT (cid:0) to (cid:18) of every order, implying that (cid:127)MU is also in(cid:2)nitely continuously differentiable. Concavity T implies that d2 d co (cid:18) s 2 .(cid:18)/ D (cid:0) cos.(cid:18)/ (cid:20) 0 on S and d2cos d .(cid:18) (cid:18) (cid:0) 2 ’ YT / D (cid:0) cos.(cid:18) (cid:0) ’ YT / (cid:20) 0 on S for all ’ YT 2 .0;(cid:25)/. Furthermore, d2 d co (cid:18) s 2 .(cid:18) Q / D d2cos d .(cid:18) (cid:18) Q(cid:0) 2 ’ YT / D 0 for identical angle of rotation (cid:18) Q 2 S if andonlyif’ (cid:25) (perfectnegativecorrelation),whichisruledoutbyassumption. Thus,(cid:127)MU YT D T isstrictlyconcaveon S forall’ .0;(cid:25)/,ensuringuniquenessofitsglobalmaximum. YT 2 Second,weprovethat(cid:18)MU.’ /isthemaximizerof(cid:127)MU on S. The(cid:2)rst-orderconditionfor YT T 58

amaximumis d(cid:127)MU T sin(cid:18) sin.(cid:18) ’ / ! 0; YT d(cid:18) D (cid:0) (cid:0) (cid:0) D which after some simple derivations yields (cid:18)MU.’ / ’ YT as its unique solution. The second- YT D 2 orderconditionforamaximumis d2(cid:127)MU ! T cos(cid:18) cos.(cid:18) ’ / < 0; d(cid:18)2 D (cid:0) (cid:0) (cid:0) YT whichevaluatedat(cid:18) (cid:18)MU.’ /yields YT D d2(cid:127)MU ’ ’ T cos YT cos. YT / d(cid:18)2 (cid:12) D (cid:0) 2 (cid:0) (cid:0) 2 (cid:12)(cid:18) (cid:18)MU.’ YT / (cid:12) D ’ (cid:12) 2cos YT (cid:12) D (cid:0) 2 < 0forall’ .0;(cid:25)/: YT 2 Third,weprovethat(cid:18)MU.’ /isthemid-pointofboththeset S andthesubset S . First,recall YT 2 that the set S of all solutions satisfying the sign restrictions (C.4) is given by (cid:18) [ (cid:25) ’ ; (cid:25)]. 2 (cid:0)2 C YT 2 The mid-point of this set is equal to half the sum of the lower and upper bound of S, i.e. 1. (cid:25) 2 (cid:0)2 C ’ (cid:25)/ ’ YT (cid:18)MU.’ /. Second, recall that the subset S of all solutions satisfying the YT C 2 D 2 D YT 2 more restrictive sign restrictions (C.5) is given by (cid:18) [max. (cid:25) ’ 0/;min.(cid:25) ’ /]. For 2 (cid:0)2 C YT I 2I YT ’ (cid:25), i.e. for non-negative error correlation, this subset is given by (cid:18) [0;’ ]. It is easy YT (cid:20) 2 2 YT to see that the mid-point of this interval is ’ YT (cid:18)MU.’ /. For ’ (cid:25), i.e. for non-positive 2 D YT YT (cid:21) 2 error correlation, this subset is given by (cid:18) [ (cid:25) ’ ; (cid:25)], i.e. it is identical to S, for which we 2 (cid:0)2 C YT 2 alreadyshowedthatitsmid-pointisequalto(cid:18)MU.’ /. YT Finally, we prove that (cid:18)MU.’ / maximizes the fraction of one-step ahead forecast error co- YT varianceexplainedbythe(cid:2)rstshockiftheerrorcorrelationispositiveandminimizesitiftheerror correlationisnegative. Thefractionofcovarianceexplainedbythe(cid:2)rstshockisgivenby FSRFSR cos(cid:18) cos.(cid:18) ’ / (cid:127) 11 21 (cid:0) YT ; (cid:17) (cid:27) D (cid:26) YT YT e 59

whichistobemaximizedwithrespectto(cid:18). The(cid:2)rst-orderconditionis d(cid:127) 1 1 ! .sin(cid:18) cos.(cid:18) ’ / cos(cid:18) sin.(cid:18) ’ // sin.2(cid:18) ’ / 0; YT YT YT d(cid:18) D (cid:0)(cid:26) (cid:0) C (cid:0) D (cid:0)(cid:26) (cid:0) D YT YT e which has (cid:18)MU.’ / ’ YT as its unique solution. The second derivative of (cid:127), evaluated at YT D 2 (cid:18) (cid:18)MU.’ /,is YT D e < 0for(cid:26) > 0; YT d2(cid:127) 2 2 cos.2(cid:18) ’ / 8 0for(cid:26) 0; d(cid:18)2 D (cid:0)(cid:26) YT (cid:0) YT D (cid:0)(cid:26) YT > > > D YT D e < < 0for(cid:26) < 0; YT > > > : con(cid:2)rming that (cid:18)MU.’ / maximizes the error covariance if the error correlation is positive and YT minimizesitiftheerrorcorrelationisnegative. ThiscompletestheproofofProposition4. ItisworthwhiletoquicklyreviewtheimplicationsofProposition4forourapplication. 1. Themostinteresting(cid:2)ndingiscertainlythateventhoughtheobjectivefunctionismaximized over the set of pure sign restriction solutions (C.4) leaving open the sign of the response of output to a tax shock (the object of interest), the solution to this maximization problem alwayssatis(cid:2)estheadditionalsignrestrictionimpliedby(C.5). 2. Theoutputelasticityoftaxesevaluatedat(cid:18)MU is (cid:27) cos. ’ =2/ (cid:27) (cid:17)MU (cid:17)SR .(cid:18) (cid:18)MU/ T (cid:0) TY T ; T;Y D T;Y D (cid:17) (cid:27) cos.’ =2/ D (cid:27) Y TY Y which is larger than the output elasticity of taxes evaluated at (cid:18) 0 (the Cholesky fac- D torization), given by (cid:17) , for all admissible values of (cid:26) . The difference between these T;Y YT elasticitesisthelargerthelowerthevalueofthecorrelationcoef(cid:2)cient(cid:26) . Inourapplica- YT tion, evaluated at the OLS estimate, the elasticity (cid:17)MU 3:04 is roughly twice as large as T;Y D theelasticityobtainingfortheCholeskyfactorization. b 3. Theimpacttaxcutmultiplier,evaluatedat(cid:18)MU,ispositiveforalladmissiblevaluesof(cid:26) . YT 60

Infact,itdoesnotdependon(cid:26) : YT 1(cid:27) 1 5 T;Y .(cid:18)MU 6 / Y . 0 I u D 2(cid:27) T T=Y Theimpacttaxcutmultiplieristhelargerthelargerthestandarddeviationoftheoutputerror (cid:27) and the lower the standard deviation of the tax error (cid:27) . In our application, evaluated Y T at the OLS estimate, the impact tax cut multiplier for the penalty function solution is 0.93 dollars. 4. The degree of self-(cid:2)nancing of a tax cut is 50% on impact for the penalty function solution. To see this note that from Equation (5) in Section 1 the response of taxes to a one unit tax shockisgivenby @u 1 T;t : @.d e / D 1 a (cid:17) T T;t Y;T T;Y (cid:0) At the penalty function solution we have aMU FMU=FMU (cid:27) =(cid:27) and (cid:17)MU Y;T (cid:17) 12 22 D (cid:0) Y T T;Y (cid:17) FMU=FMU (cid:27) =(cid:27) ,whichgives 21 11 D T Y @u 1 T;t : @.d e / D 2 T T;t (cid:12)(cid:18) (cid:18)MU (cid:12) D (cid:12) (cid:12) C.3.1 Abriefsummaryofresultsforthespendingmodel We close the bivariate section with a summary of results for the spending model, which in the bivariatecontextistrivial. RecallthatM&Udonotsignrestricttheresponseofgovernmentspending toabusinesscycleshock,whichimpliesthefollowingsignpattern ? FSR C : D 2 3 ? C 4 5 For these restrictions the set of pure sign restriction solutions is simply the entire interval of admissiblerotationangles(cid:18) [ (cid:25);(cid:25)]. 2 (cid:0) Theobjectivefunctionforthespendingmodelis FSR (cid:127)MU 11 cos(cid:18): G (cid:17) (cid:27) D Y 61

It is easy to see that the solution is (cid:18)MU 0, i.e. the penalty function solution is nothing else G D than the Cholesky factorization with output ordered (cid:2)rst. For this solution the impact spending multiplier is zero. The output elasticity of government spending implied by the penalty function solutionisgivenby: (cid:27) (cid:17)MU (cid:17) G (cid:26) : G;Y D G;Y D (cid:27) YG Y Inourapplication,asistruealsoforotherVARstudiesinthisliterature,theoutputandgovernmentspendingresidualsarestronglypositivelycorrelated((cid:26)OLS 0:29/. Forthepenaltyfunction YG D solution - and the Cholesky factorization with output ordered (cid:2)rst - 100% of the correlation beb tween one step-ahead output and government spending errors is explained by the business cycle shock,implyingstronglyprocyclicalgovernmentspending((cid:17)MU 0:38).43 G;Y D b C.4 Multivariate extensions Weextendtheanalysistoathree-variablesystemwithoutput,taxesandathirdvariable. Thethird variable, denoted Z, will be private consumption, denoted C, private investment, denoted I, or government spending, G, depending on the object of interest. We restrict attention to the set of pure sign restriction solutions S derived for the bivariate tax model. In particular, we answer the followingquestions: 1. How does an additional sign restriction on the response of a third variable to the business cycle shock affect the set S derived in the bivariate tax model? M&U, for example, restrict the responses of consumption and investment to a business cycle shock to be positive. Does thisstronglyaffecttheset S? 2. Underwhichconditionsdoesthepenaltyfunctionsolution(cid:18)MU derivedforthebivariatetax T modelsatisfytheadditionalrestrictiononathirdvariable? 3. What is the impact spending multiplier implied by the set S when the model is extended to include G asthirdvariable? 43An alternative interpretation of the penalty function in the case of the spending model is that it maximizes the fractionoftheone-stepaheadoutputerrorvarianceexplainedbythe(cid:2)rstshock( FSR 2=(cid:27) ). Thisisexactlywhat 11 YY theCholeskyfactorizationdoes, forwhichthe(cid:2)rstshockexplains100%oftheone-stepaheaderrorvarianceofthe (cid:0) (cid:1) variableordered(cid:2)rstinthesystem. 62

We should clarify that in this section we look at a subspace of the set of all solutions in the multivariate context. However, as we will see this is a very interesting subset, clarifying that the wide range of possible results that we derived for the bivariate context persists even if we don’t lookattheentirespaceofsolutionsinthemultivariatecontext. Inthetrivariatecontext,theorthogonalmatrix Q canbewrittenastheproductofthreeGivens matrices Q , Q and Q , each rotating a different pair of columns of the matrix to be trans- 12 13 23 formed: cos(cid:18) sin(cid:18) 0 cos(cid:18) 0 sin(cid:18) 1 0 0 12 12 13 13 (cid:0) (cid:0) 2 32 32 3 Q D 6 6 sin(cid:18) 12 cos(cid:18) 12 0 7 7 6 6 0 1 0 7 7 6 6 0 cos(cid:18) 23 (cid:0) sin(cid:18) 23 7 7 : 6 76 76 7 6 76 76 7 6 0 0 1 76 sin(cid:18) 0 cos(cid:18) 76 0 sin(cid:18) cos(cid:18) 7 6 76 13 13 76 23 23 7 6 76 76 7 4 54 54 5 In the following we focus attention on those sign restriction solutions that obtain when setting (cid:18) (cid:18) 0,inwhichcase Q Q I and Q Q . 13 23 13 23 3 12 D D D D D C.4.1 Theeffectofadditionalsignrestrictionsonthirdvariables Weconsiderthefollowingsignrestrictionsonthefactormatrix FSR: ? ? C FSR 2 ? 3: D C C 6 7 ? 6 7 6 C C 7 4 5 These restrictions are comparable to the restrictions imposed by M&U: in the tax model we require the responses of output, taxes and the third variable (either C or I) to be positive, whereas for the tax shock (second shock) we require the response of taxes to be positive and orthogonality tothebusinesscycleshock. WeextendthebivariatesystemgivenbyEquation(8)toatrivariatesystem,whereu denotes Z;t 63

theresidualofthethirdvariable: u p 0 0 cos(cid:18) sin(cid:18) 0 e Y;t 11 12 12 Y;t (cid:0) 2 u 3 2 p p 0 32 sin(cid:18) cos(cid:18) 0 32 e 3; (C.6) T;t 21 22 12 12 T;t D 6 7 6 76 76 7 u p p p 0 0 1 e 6 Z;t 7 6 31 32 33 76 76 Z;t 7 6 7 6 76 76 7 4 5 4 54 54 5 where p are elements of the lower-triangular Cholesky factor P in the trivariate system. Now, ij recall from Proposition 2 that in the bivariate model the set of all pure sign restriction solutions required that (cid:18) [ (cid:25) ’ ; (cid:25)]. The question is how the additional sign restriction on the 12 2 (cid:0)2 C YT 2 third variable’s contemporaneousresponse to the business cycleshock affects this set. The impact response of the third variable to the business cycle shock, under the assumption that Q Q , is 12 D givenby FSR p cos(cid:18) p sin(cid:18) . 31 D 31 12 C 32 12 Wearegoingtoassumeinthefollowingthatthecross-correlationsbetweenthethreeresiduals are all positive, i.e. (cid:26) ;(cid:26) ;(cid:26) > 0. This assumption facilitates the exposition of the results YT YZ TZ but is not very restrictive as it is in line with the evidence for the (cid:2)scal VAR models used in the literature. In these models all candidate third variables (private consumption, private investment andgovernmentspending)arerobustlypositivelycorrelatedwithoutputandtaxes. Itcaneasilybe shownthatforthisassumptionallelementsoftheCholeskyfactorexceptfor p arenon-negative 32 (see Equation C.3) and the sign of p is equal to the sign of (cid:26) (cid:26) (cid:26) . Depending on the 32 TZ YT YCZ (cid:0) signof p thesetofpuresignrestrictionsolutionsbecomes 32 [ (cid:25) ’ ;arctan. p31/] if p < 0; (cid:0)2 C YT (cid:0)p32 32 (cid:18) 12 2 8 > > [ (cid:0) (cid:25) 2 C ’ YT ; (cid:25) 2 ] if p 32 D 0; (C.7) > < [max. (cid:25) ’ arctan. p31//; (cid:25)] if p > 0: (cid:0)2 C YT I (cid:0)p32 2 32 > > > : Wecanchecktheimplicationsofthisconditionforourempiricalapplication. Consider (cid:2)rst the case in which private consumption is the third variable. In this case, - with the covariance matrix 6 evaluated at the OLS estimate - the p element of the Cholesky factor u 32 is negative, i.e. the (cid:2)rst case in (C.7) applies. This implies that the additional restriction on the sign of the consumption response shrinks the set of pure sign restriction solutions. How large is this effect? In our empirical application the set still covers all empirically plausible models: at the 64

upper bound of the sign restriction set ((cid:18) arctan. a31/) the output elasticity of taxes is equal 12 D (cid:0)a32 to73:2. Inotherwords,theadditionalsignrestrictionontheconsumptionresponsehasonlyavery minoreffectonthesetofpuresignrestrictionsolutionsderivedforthebivariatemodel. Consider next the case in which private investment is the fourth variable. In this case, - with the covariance matrix 6 evaluated at the OLS estimate -, the p element of the Cholesky factor u 32 is positive, i.e. the third case in (C.7) applies. To know whether the additional restriction on the investmentresponseshrinkstheset S weneedtoverywhetherarctan. p31/ > (cid:25) ’ . Inour (cid:0)p32 (cid:0)2 C YT application this condition is not ful(cid:2)lled, i.e. the sign restriction set S is not affected at all by the additionalsignrestriction. Note also that the above results do not depend on the analysis of a trivariate model. If instead we look at a four-variable model with output, taxes, consumption and investment, the same results hold, with the sign restriction on the consumption response being the binding restriction in our empirical application. This is because the contemporaneous responses of consumption and investment to a business cycle shock do not depend on which one is ordered third or fourth in the system Finally, we can ask whether the penalty function solution derived for the bivariate model, (cid:18)MU ’ YT satis(cid:2)estheadditionalsignrestrictiononthethirdvariable: D 2 ’ ’ ’ FSR.(cid:18) YT / p cos YT p sin YT 0: 31 12 D 2 D 31 2 C 32 2 (cid:21) ’ ’ Note that we have p > 0 by assumption and cos YT > 0 and sin YT > 0 for all ’ 31 2 2 YT 2 .0;(cid:25)/. If p 0 then the sign restriction solution for the bivariate model automatically satis(cid:2)es 32 (cid:21) the sign restriction on the third variable. If p < 0 (which holds for (cid:26) (cid:26) (cid:26) < 0) the 32 TZ YT YZ (cid:0) 65

’ ’ penaltyfunctionsolutionneedstosatisfytheconditionthat p sin YT p cos YT (cid:0) 32 2 (cid:20) 31 2 ’ ’ p sin YT p cos YT; (cid:0) 32 2 (cid:20) 31 2 sin ’Y 2 T p31; () cos ’YT (cid:20) (cid:0)p32 2 sin’ cos’ sin’ YT YZ YT ; () 1 cos’ YT (cid:20) (cid:0)cos’ TZ cos’ YT cos’ YZ C (cid:0) cos’ .1 cos’ / 1 YZ C YT ; () (cid:20) (cid:0)cos’ TZ cos’ YT cos’ YZ (cid:0) cos’ cos’ cos’ cos’ .1 cos’ /; TZ YT YZ YZ YT () (cid:0) (cid:21) (cid:0) C (cid:26) (cid:26) 0: TZ YZ () C (cid:21) Thisconditionissatis(cid:2)edundertheassumptionofpositivecross-correlationbetweenthereducedformerrors. Aswehavearguedabovethisassumptionisempiricallyplausible. ForallVARmodels we are aware of output, taxes, private consumption and investment are robustly positively crosscorrelated. C.4.2 Thetrivariatemodelwithgovernmentspending We last turn to the triviate model with government spending, for which - in line with M&U - we donotsignrestricttheresponseofgovernmentspendingtoabusinesscycleshockwhichgivesthe followingsignrestrictionsonthefactormatrix FSR: ? ? C FSR 2 ? 3: D C C 6 7 ? ? 6 7 6 C 7 4 5 In this case all pure sign restrictions solutions derived for the bivariate model remain sign restrictions in the trivariate model. The sign restriction on the spending shock is automatically satis(cid:2)ed because postmultiplying the Cholesky factor P by Q leaves the third column of P 12 unchangedand p > 0forpositivede(cid:2)nite6 . Thislatterpropertyhasanimportantimplication: 33 u FSR 0 for all (cid:18) S, i.e. the output response to a government spending shock is zero. Of 13 D 12 2 course, this result would not hold if we broadened the analysis to include rotations beyond the output-taxsubspace. ButthisresultisimportantbecauseitalsoholdsfortheM&Upenaltyfunctionif-inaccordance 66

withtheassumptionsmadebyM&U-onlythoseresponsestoabusinesscycleshockthataresign restricted enter as arguments in the objective function to be maximized. Under this assumption the objective function to be maximized with respect to (cid:18) is still the one given by (10) and the 12 maximumstillobtainsfor(cid:18)MU.’ / ’ YT.44 12 YT D 2 Theoutputelasticityofgovernmentspendingimpliedbythissolutionis ’ FSR p cos ’ YT p sin ’ YT p ’ (cid:17)MU.(cid:18) YT / 31 31 2 C 32 2 (cid:17) 32 tan YT : G;Y D 2 D FSR D p cos ’ YT D G;Y C p 2 11 11 2 11 ’ The output elasticity of government spending, evaluated at (cid:18) YT, will be larger than (cid:17) D 2 G;Y for p > 0 and smaller than (cid:17) for p < 0, recalling that (cid:17) > 0 for (cid:26) > 0. In 32 G;Y 32 G;Y GY our application - evaluated at the OLS estimate - p < 0 but the deviation from (cid:17) is small. 32 G;Y The output elasticity for the penalty function solution is 0:36, compared to 0:38 for the Cholesky decomposition. But most importantly, the penalty function solution for standard sign restrictions impliesazeroimpactspendingmultiplier. 44Inthiscaseevenbroadeningtheanalysistotheorthogonalmatrix Q Q Q Q andmaximizing(10)with 12 13 23 respectto(cid:18) ,(cid:18) and(cid:18) doesnotaffecttheresultbecausethemaximum D obtainsfor(cid:18)MU (cid:18)MU 0. 12 13 23 13 D 23 D 67