Are Spectral Estimators Useful for Implementing Long-Run Restrictions in SVARs?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Are Spectral Estimators Useful for Implementing Long-Run Restrictions in SVARs? Elmar Mertens 2010-09 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.

Are Spectral Estimators Useful for Implementing Long-Run Restrictions in SVARs?∗ Elmar Mertens† Federal Reserve Board July 2009 ∗This paper is based on the second chapter of my dissertation written at the University of Lausanne. An earlier draftofthispaperhasbeenwrittenwhilevisitingtheFederalReserveBankofMinneapolis. Iwouldliketothankthe Bank and its stafffor their kind hospitality. In particular, I would liketo thank V.V. Chari, PatrickKehoeand Ellen McGrattanfordiscussionsandsharingtheircomputercodes. InadditionIamgratefulfordiscussionswithLawrence Christiano, Bill Dupor, Mark Watson as well as seminar participants at the University of Basel, the University of Zurich, the YoungSwiss Economist Meetings, and the European Economics Association meeting 2007 in Budapest andthemembersofmydissertationcommittee,Jean-PierreDanthine,PhilippeBacchetta,RobertG.King,andPeter Kugler. All remaining errors are of course mine. This research has been carried out within the National Center of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK). NCCR FINRISK is a researchprogramsupportedbytheSwissNationalScienceFoundation. †Forcorrespondence:ElmarMertens,BoardofGovernorsoftheFederalReserveSystem,WashingtonD.C.20551. emailelmar.mertens@frb.gov. Tel.:+(202)4522916. Theviewsinthispaperdonotnecessarilyrepresentthe viewsoftheFederalReserveBoard,oranyotherpersonintheFederalReserveSystemortheFederalOpenMarket Committee. Anyerrorsoromissionsshouldberegardedasthosesolelyoftheauthor. 1

Are Spectral Estimators Useful for Implementing Long-Run Restrictions in SVARs? Abstract No, not really, since spectral estimators suffer from small sample and misspecification biasesjustasVARsdo. Spectralestimatorsarenopanaceaforimplementinglong-runrestrictions. Inaddition,whencombiningVARcoefficientswithnon-parametricestimatesofthespectral density, care needs to be taken to consistently account for information embedded in the non-parametricestimatesaboutserialcorrelationinVARresiduals. Thispaperusesaspectral factorization to ensure a correct representation of the data’s variance. But this cannot overcome the fundamental problems of estimating the long-run dynamics of macroeconomic data insamplesoftypicallength. JELClassification: C32,C51,E17,E32 Keywords: StructuralVAR,Long-RunIdentification,Non-parametricEstimation,SpectralFactorization 2

1 Introduction VARs have been criticized for failures in estimating the responses to long-run shocks. A crucial element for long run identification is the spectral density at zero-frequency, also known as “longrun variance”. But OLS estimates of VAR coefficients are concerned with minimizing forecast error variance, not estimating the long run variance. This has motivated Christiano, Eichenbaum, and Vigfusson (2006a, 2006b), henceforth “CEV”, to propose a new way of estimating structural VARsusingacombinationofOLSandanon-parametricestimator. Theyarguethattheirestimator virtuallyeliminatesthebiasassociatedwiththestandardOLSestimator. Thispapershowsthatnon-parametricestimatesofthespectraldensity,henceforthcalled“spectral estimators”, are no panacea for the implementation of long-run restrictions in finite sample. Macroeconomictimeseriesdisplayafairamountofpersistence,posingtwoseriouschallengesfor long-runidentification. First,anaccuraterepresentationofthetruemodeltypicallyrequiresaVAR withahighlagorder,muchhigherthanwhatisaffordableinasampleoftypicallengthandresulting in a sizable truncation bias (Chari, Kehoe, and McGrattan 2008, henceforth “CKM”). Second, there is the small sample bias in estimated coefficients known from Hurwicz (1950), which becomes ever more severe the smaller the sample, and the more persistent the data. As will be shown, both issues affect not only VARs in the time domain, but also spectral estimators in the frequencydomain. TheconventionalVARtechniqueaswellasdifferentcombinationswithspectralestimatorsare evaluated in the context of a simple two-shock RBC model, which has also been used by CEV and CKM. When using the various procedures to estimate the response of hours to technology, or to decompose the variance of fluctuations in output or hours, none of the procedures clearly dominatestheothers. Furthermore, CEV do not consider some conceptual pitfalls in combining VAR coefficients with spectral estimates. Non-parametric estimates of the spectral density allow for non-iid residuals in the finite-order VAR, which is good since the underlying model is likely of infinite order. In what may be called “mixing and matching”, the CEV approach plugs these estimates into the 3

standard VAR formula alongside with coefficients from the finite-order VAR. This approach uses theextrainformationaboutomittedlagsintheVARtocomputethelong-runresponsesofvariables to shocks—but not when mapping these back into impact responses. To retain a consistent representationof thedata, that wouldhoweverbenecessary. Otherwise, thetotal varianceof the datais misrepresented. In the simulations reported here, this misrepresentation is quantitatively relevant. As a related issue, when the relationship between forecast errors and structural shocks is inverted withtheCEVcoefficients,oneobtainsatimeserieswhichisidenticaltotheshockestimatesfrom OLSuptoascalefactor. Allinall,thisisofconcernforanyresearcherwantingtoadopttheCEV strategy. The CEV framework is amended here by recognizing that the non-parametric estimate containsinformationaboutomittedlagsintheVAR.ThismisspecificationhasbeenstressedbyCKM, Erceg, Guerrieri, and Gust (2005), Ravenna (2007) and Cooley and Dywer (1998). The adjusted procedure retains the OLS estimates and fills up the omitted lags with a spectral factorization of the spectral density’s non-parametric estimate. By construction, this adjusted SVAR—in fact an SVARMA—matches the sample variance of the data just as OLS does. Overall, this corrected proceduresuffersfromthesamebasicproblemsastheotherlong-runidentificationmethods: truncationandsmallsamplebias. The remainder of this paper is structured as follows: Section 2 describes the model economy against which the various estimation routines will be evaluated. Section 3 describes the various SVAR methods, including a new spectral factorization procedure. Section 4 presents the Monte CarloresultsandSection 5concludesthepaper. 2 A Model Economy This section describes a simple model economy, which will be used to illustrate and quantify the issues associated with various long-run identification schemes. None of the conceptual concerns relatedtospectralestimatesraisedinSection3willbespecifictothismodel. Themodelisidentical 4

tothetwo-shockeconomyusedbyCKMandCEV. Themodelisacommonone-sectorRBCeconomydrivenbytwoshocks: First,aunitrootshock to technology, z . This is the permanent shock to be estimated by the VAR. Second, a transitory t non-technology shock, τ , which drives a wedge between private household’s labor-consumption lt decision. The representative household maximizes his lifetime utility over (per-capita) consumption, c , t andlaborservices, l t (cid:88)∞ E (β(1+γ))tu(c ,l ) 0 t t t=0 andfacesthebudgetconstraintc +(1+γ)k −(1−δ)k = (1−τ )w l +r k +T wherek is t t+1 t lt t t t t t t theper-capitastockofcapital,w thewagerate,r therentalrateofcapital,T arelumpsumtaxes, t t t γ isthegrowthrateofpopulation,δ thedepreciationrateofcapital(γ > 0,0 ≤ δ ≤ 1andβ < 1). The non-technology shock τ is an exogenous labor tax. As discussed by CKM, it need not lt be literally interpreted as a tax levy, but stands in for the effects of a variety of non-technology shocks introduced into second-generation RBC models. Mechanically, it distorts the first-order conditionforconsumptionandlabor. ItworkssimilartoastochasticpreferenceshocktotheFrisch elasticity of labor supply. Chari, Kehoe, and McGrattan (2007) show how this labor “wedge” can be understood more generally as the reduced form process for more elaborate distortions, such as stickywages. The production function F(k ,Z l ) is constant returns to scale, where Z is labor-augmenting t t t t technological progress. Firms are static and maximize profits F(k ,Z l ) − w l − r k . Pert t t t t t t capita output equals production, y = F(k ,Z l ), and the economy’s resource constraint is y = t t t t t c +(1+γ)k −(1−δ)k . Theexogenousdriversfollowlinearstochasticprocesses: t t+1 t logZ = µ +logZ +σ εZ t z t−1 z t logτ = (1−ρ )τ¯ +ρ logτ +σ εl l,t+1 l l l l,t−1 l t whereεZ andεl areiidstandard-normalrandomvariables. Theyarethetechnologyshock,respect t 5

tivelylaborshock. ρ measuresthepersistenceofthetransitorylabortax. Thescalefactorsσ and l z σ determine their relative importance in the model. (µ is the drift in log-technology and τ¯ is the l z t averagetaxrate.) ThecalibrationisidenticaltothebaselinemodelofCEV,whichusesparametervaluesfamiliar from the business cycle literature. Utility is specified as u(c,l) = logc+ψlog(1−l) (consistent with balanced growth) and the production function is Cobb-Douglas F(k,l) = kθl1−θ with a capital share of θ = 0.33. The labor preference parameter is set to ψ = 2.5. On an annualized basis, the calibration sets the depreciation rate to 6%, the rate of time preferences to 2% and population growth to 1%.1 Following CEV, the transitory shock is calibrated as an AR(1) with persistence ρ = 0.986. This calibration is identical to the values used by CKM except for their l valuesof φ = 1.6andρ = 0.95. l Themodeleconomyiscalibratedoverdifferentratiosinthevarianceoftransitorytopermanent shocks,σ2/σ2,whichtranslateintodifferentassumptionsabouttheshareofoutputfluctuationsexl z plainedbytechnologyshocks.2 Asabenchmark,maximum-likelihoodestimatesofCEVobtained from fitting the model to U.S. post-war data imply that around two-thirds of the bandpass-filtered varianceinoutputcanbeattributedtotechnologyshocks.3 Thebandpassfilteremployedthroughout this paper considers only fluctuations with durations between two-and-a-half and eight years, whichisconsistentwiththeNBERdefinitionsof BurnsandMitchell(1946). Data is simulated for samples of length T = 180, corresponding to 45 years of quarterly data; identical to the simulations of CKM and CEV. Following CEV and CKM, bivariate VARs are estimated using simulated data of the (log) growth rate of labor productivity and hours worked; (cid:183) (cid:184) T X = ∆log(y /l ) logl .4 For each simulated sample, the lag length of the VAR(p) is chot t t t 1Thedriftintechnologyissetto0.4%andtheaverage“labortax”issetto24.2%perquarter. 2CKMextensivelydocumenthowdifferentratiosinthevarianceoftransitorytopermanentshocks,σ2/σ2,affect l z theperformanceofstandardVARsbothinpopulationandinsmallsample. McGrattan(2005)showsthatinthelimit, σ /σ → 0,afiniteorderVAR(evenap = 1)inproductivitygrowthandhoursrecoversthetrueresponses—though l z thetruesystemdoesnothaveafinite-orderVARrepresentation. Inthisspecialcasethemodelreducestoastandard, one-shockRBCmodel. 3Tobeprecise,CEVestimateσ =0.00562,σ =0.00953correspondingtoatechnologyshareof67.5%. l z 4Inadditiontothis“LSVAR”specification,CKMrunalsoVARswithquasi-differencedhours. Thisreplacesthe secondVARelementl with(1−αL)l (α∈{0;0.999}). Dependingonα,thiscapturespopular(butalsocontested) t t 6

sen by minimizing the Schwartz Information Criterion (SIC), typically picking small values close to one.5 When computing population moments, a VAR(1) is used. For each calibration, 1,000 samplesaresimulated. Whenlookingatdatasimulatedfromthismodel,twostatisticsareofparticularinterestforthis paper: How do hours worked respond to a technology shock? What is the share of fluctuations due to technology shocks? These questions are typically asked by empirical researchers trying to evaluate predictions from business cycle models with SVARs, such as Gali (1999) or Christiano, Eichenbaum,andVigfusson(2004). 3 Long-Run Identification in VARs The linearized solution to the model described in the previous section is only one example from a wider class of linear dynamic models to which the SVAR methods discussed here can be applied. None of the issues discussed in this section will be specific to the model described above. An economicmodelfromthisclassissupposedtospecifyaVARrepresentationforastationaryvector ofobservablevariables6 X : t X = B(L)X +e (1) t t−1 t (cid:80) where B(L) is a polynomial in the lag-operator L,B(L) = ∞ B Lk−1 whose roots lie all k=1 k outsidetheunitcircleandtheinnovationsare iid,e ∼ iid(0,Ω). t Inprinciple,themodelprescribesaninfinite orderVAR.WhenB = 0fork > pthisneststhe k case of a finite order VAR. But as noted by Cooley and Dywer (1998), many interesting models specifications: Ontheonehandthe“LSVAR”withhoursinlevelsandα = 0andontheotherhandthe“QDSVAR” withα = 0.999,whichapproximatesaVARwithdifferencedhourswithoutintroducingaunitMAroot. ThequasidifferencingisdiscussedinmoredetailbyCKMaswellasMarcet(2005),GaliandRabanal(2004)andChristiano, Eichenbaum,andVigfusson(2003). 5Resultsareinsensitivetousingotherinformationcriteria,suchastheAkaikecriterion(AIC).Ingeneral,AICis knownforpickinghighervaluesofpcomparedtoSIC.Forthislabeconomy,AIChasbeenfoundtopicklaglengths ofuptop=6withanaverageofp=2. 6Fornotationalconvenience,butwithoutlossofgenerality,X representsthedemeanedvariables,whichisequivt alenttoincludingaconstantinaVARusingtheoriginaldata. 7

have only infinite order VAR representations. In the remainder of this paper the true VAR representation is always assumed to be of infinite order. The linearized solution to the model described inSection2hassuchaninfiniteorderVARrepresentation;detailsareshowninAppendix B. For the identification of structural shocks, there has to be an invertible one-to-one mapping from innovations e to the structural shocks ε driving the underlying model—such as technology, t t monetarypolicyerrors,exogenousgovernmentspendingetc.: e =A ε (2) t 0 t whereA issquareand|A | 6= 0. Ferna`ndez-Villaverdeetal.(2007)deriveconditionswhenalin- 0 0 eardynamicmodelhasaninvertibleVARrepresentation.7 (ThesearesummarizedinAppendixB.) Thispaperconsidersonlycaseswheretheseconditionsaresatisfied,thoughpossiblyonlyinaninfinite order VAR representation. The same applies to the situations studied by CKM, CEV as well as Erceg, Guerrieri, and Gust (2005). Excluding the complications arising from non-invertibilities allowstofocusonproblemsowingtosmallsamplebiasandthefiniteorderapproximationsofthe VAR.8 Itwillbehandytointroducethenotation (cid:88)∞ C(L) ≡ (I −B(L)L)−1 = C Lk where C = I (3) k 0 k=0 forthenon-structuralmovingaverage(VMA)coefficientsofX = C(L)e . Thestructuralmoving t t averagerepresentationfor X isthenX = A(L)ε withA(L) = C(L)A . t t t 0 In the spirit of CEV and CKM, only one of the structural shocks will be identified. For concreteness, let it be the first one, denoted εz, and call it “technology shock”. Think of the first t element of X as being a growth rate (a difference in logs), like the change in labor-productivity t 7Fernandez-Villaverde, Rubio-Ramirez, and Sargent (2005) give examples of interesting models where the conditions are satisfied and where not. For all calibrations considered, the model of Section 2 satisfies the condition of Ferna`ndez-Villaverdeetal.(2007). 8SeeforexampleGiannoneandReichlin(2006)onthenon-invertibilityproblem. 8

(Gali1999)oroutputgrowth(BlanchardandQuah1989). Theidentifyingassumptionisthenthat only the technology shock has a permanent effect on the level of the first element of X . This t (cid:80) restrictsthematrixoflong-runcoefficients, A(1) = ∞ A : i=0 i   a¯ 0 ... 0  11  A(1) = C(1)A =   and a¯ > 0 (4) 0 11 # # ... # ThisrestrictionholdsexactlyinthelinearizedsolutiontothemodeldescribedinSection 2. AkeyobjectforimplementingthisconstraintisthespectraldensityofX . Thespectraldensity t (cid:80) atfrequencyω isdefinedasS (ω) ≡ ∞ E(X XT )e−iωk = C(e−iω)ΩC(e−iω)T whereiis X k=−∞ t t−k theimaginaryunit.9 A(1)factorsthespectraldensityof X atfrequencyzero: t A(1)A(1)T = C(1)ΩC(1)T = S (0) (5) X OnewaytocomputethefirstcolumnofA isbyrecoveringA(1)fromtheCholeskydecomposition 0 ofS (0). (Thisistheuniquelowertriangularfactorizationofapositivedefinitematrix.10): X A(1) = chol{S (0)} X CEV show that the restriction in (4) uniquely pins down the first column of A and the Cholesky 0 factorizationisonepossibleimplementation.11 The long-run coefficients can then be mapped into the matrix of impact responses using the 9Throughoutthispaper,transposesarecomplexconjugate. 10The spectral density S (0) = C(1)ΩC(1)T is strictly positive definite when the variance covariance matrix X of the forecast errors, Ω, is nonsingular. S (0) inherits positive definiteness from Ω since C(1) is nonsingular. X I−B(1)=C(1)−1existsbecauseoftheassumedstationarityoftheVARprocess. 11Undertherestrictionsstatedin(4),A(1)cangenerallybedescribedas (cid:183) (cid:184) 1 0 A(1)=chol{S (0)} forsomeW suchthatWWT =I X 0 W In the lab economy described later, the VAR will be bivariate and the forecast errors e are a linear combination of t onlytwoshocks. Knowingthetechnologyshockwillthenalsoidentifythesecondshockuptoitssign,|W|=1. 9

VARdynamicsencodedinthepolynomialoflagcoefficients B(L): A = (I −B(1))A(1) (6) 0 3.1 OLS Implementation with Finite-Order VAR Since the VAR innovations in (1) are assumed to be white noise, they satisfy the OLS normal equations EX eT = 0 (∀k). And in principle, the coefficients B could be estimated from least t−k t k squares projections of X on its infinite past. In practice, an empirical implementation can only t workwithafinitelaglength. HenceforthB(L)OLS denotesalagpolynomialoffiniteorderp < ∞: (cid:88)p B(L)OLS ≡ BOLSLk−1 k k=1 vOLS ≡ X −B(L)OLSX (7) t t t−1 ΩOLS ≡ E [vOLS(vOLS)T] v t t wherethenormalequationsareimposedforalllags k ≤ p EX (vOLS)T = 0 (8) t−k t TheassociatedVMAisC(L)OLS ≡ (I−B(L)OLSL)−1. OnlystableVARsareconsidered,formally thisrequiresallrootsof C(L)OLS tobeoutsidetheunit-circle. The standard procedure is to assume uncorrelated residuals, vOLS. Following Blanchard and t Quah (1989), the long run restriction (4) is implemented based on an estimate of the spectral density at frequency zero constructed from the OLS estimates. Impact coefficients are computed 10

bypluggingtheseestimatesinto(6): (cid:161) (cid:162) S (0)OLS = C(1)OLS ΩOLS C(1)OLS T (9) X v (cid:161) (cid:162) (cid:169) (cid:170) AOLS = I −B(1)OLS chol S (0)OLS (10) 0 X Using a finite-order VAR when the data has been generated from an infinite order process induces a truncation bias into the estimates. In this case, the OLS assumption of uncorrelated forecast errors vOLS is violated, which is an example of what Cooley and Dywer criticized as t an “auxiliary” (but not innocuous) assumption. This truncation bias arises even when the true population moments of the data generating process were known. Applied to data generated from a business cycle model the truncation bias in SVARs can be substantial, as shown by Cooley and Dywer(1998),Erceg,Guerrieri,andGust(2005),Ravenna(2007)orCKM.Thetruncationbiasis alsosizablefordatafromthemodeldescribedSection 2aswillbeseeninFigure 3below. 3.2 CEV: Combining OLS with Spectral Estimate CEV propose an alternative estimator for the matrix of impact coefficients. This new estimator uses a mixture of the OLS estimates of B(1) and a non-parametric estimator for S (0). The X procedure is motivated by observing that OLS projections construct B(L)OLS not necessarily with regard to B(1) but in order to minimize the forecast error variance ΩOLS. Following Sims (1972), v the least-squares objective seeks OLS coefficients which minimize the average distance between themselvesandthetrueB(e−iω),weightedbythespectraldensityofX ,whichmayormaynotbe t largeatthezerofrequency:12 min ΩOLS = Ω + v v BOLS,...,BOLS 1 p (cid:90) π (cid:161) (cid:162) (cid:161) (cid:162) B(e−iω)−B(e−iω)OLS S (ω) B(e−iω)−B(e−iω)OLS T dω (11) X −π 12This can be derived by applying the definition of spectral density to vOLS = v +[B(L)−B(L)OLS]X and t t t−1 computingthevarianceΩOLSastheintegraloverthespectraldensity. v 11

Accordingly, OLS will try to set B(1)OLS close to B(1) only if the data’s spectrum is high at the zero frequency and S (0)OLS need not be the best possible estimate for the spectral density at X frequencyzero. Instead of using S (0)OLS, CEV employ a spectral estimator of S (0) to construct A(1). x X In Christiano, Eichenbaum, and Vigfusson (2006a), they consider two estimators, one based on Newey and West (1987) and the other on Andrews and Monahan (1992). Both are based on truncated sums of autocovariance matrices. To ensure positive definiteness, these are weighted by a Bartlettkernel. WhereNewey-Westsumsoverthe(sample)autocovariancesof X , t (cid:181) (cid:182) (cid:88)b |k| (cid:163) (cid:164) S (0)NW = 1− E X XT (12) X b+1 t t−k k=−b Andrews-Monahan uses first the VAR to prewhiten the data and then sums over the residual autocovariances: (cid:161) (cid:162) S (0)AM = C(1)OLSSNW(0) C(1)OLS T (13) X v (cid:181) (cid:182) (cid:88)b |k| (cid:163) (cid:164) where S (0)NW = 1− E vOLS(vOLS)T (14) v b+1 t t−k k=−b b is a truncation parameter, also known as “bandwidth”, which will be discussed in more detail below.13 TheAndrews-MonahanestimatorneststheOLScasewhen b = 0. The new CEV estimator computes the long-run coefficients from the non-parametric density estimate. CombinedwiththeOLSlagcoefficients,CEVobtaintheirimpactcoefficientsas (cid:161) (cid:162) ACEV-AM = I −B(1)OLS A(1)AM (15) 0 (cid:169) (cid:170) where A(1)AM = chol S (0)AM (16) X Impulse responses are A(L)CEV-AM = C(L)OLSACEV-AM. Using the Newey-West estimator, impact 0 (cid:163) (cid:164) 13Aselsewhereinthissection,estimatorshavebeenwrittenintermsofpopulationmoments,E vOLS(vOLS)T . In t t−k empiricalapplications, thepopulationmomentsarereplacedbysamplemoments. ForsomevariableZ , thesample (cid:80) t momentisE Z ≡1/T T Z . T t t=1 t 12

(cid:161) (cid:162) (cid:169) (cid:170) coefficientsarecomputedasACEV-NW = I −B(1)OLS chol S (0)NW . Forbrevity,theremain- 0 X der of this section will mostly refer to the Andrews-Monahan estimator, with similar arguments holdingfortheNewey-Westestimator. Section 4presentssimulationsusingbothestimators. The bandwidth choice b is critical in estimating spectra, akin to choosing the lag order of a VAR. Bandwidth choice has been shown to be more important using other weighting schemes than the Bartlett kernel (Newey and West 1994).14 For a consistent estimator, b can grow with the sample size but at a smaller rate. CEV use a fixed and fairly large value of b = 150 in a sample of 180observations.15 Theoretically,theprewhiteningofAndrews-Monahanisappealingsinceitremovesspikesfrom the spectral density of X which make spectral estimation difficult (Priestley 1981, Chapter 7). t Andrews and Monahan (1992) and Newey and West (1994) find the prewhitening to fare better in Monte Carlo studies than the original Newey-West estimator. Christiano, Eichenbaum, and Vigfusson (2006a) find no clearly superior choice between the two and Christiano, Eichenbaum, andVigfusson(2006b)proceedtouseonlytheNewey-Westestimator. To minimize the mean-squared error (MSE) of spectral estimates, the bandwidth selection schemes of Andrews (1991) and Newey and West (1994) can be used. However, constructing an MSE optimal estimator of the spectrum does not necessarily translate into an MSE optimal estimate of coefficients like ACEV-AM or ACEV-NW. Their MSE depends not solely on the MSE of 0 0 S (0)AM but—amongstothers—onbiasandstandarderrorofthespectruminwayswhicharespe- X cific to the data generating process.16 Hence, the bandwidth selection scheme of Newey and West (1994) may serve as a useful starting point for bandwidth choice, but it is not necessarily optimal forthepurposeofestimatingimpulseresponseorvarianceshares. 14AlternativeweightingschemesareforexamplediscussedbyPriestley(1981)andPhillips,Sun,andJin(2006). 15The discussion of CEV suggests that this choice is supposed to be compatible with consistency—essentially promising that this bandwidth choice would barely grow as longer data samples become available. Watson (2006) regardsitasapracticallyuntruncatedandinconsistentestimator. 16Tobespecific,ACEV-AM andACEV-NW arefunctionsofaspectralestimateandOLSestimatesoftheVAR.Anal- 0 0 ogouslyto argumentsemployedby Sun, Phillips, and Jin (2008)inthe contextofconstructing confidence intervals, theMSEsofACEV-AMandACEV-NWcouldbeapproximatedtoasecondorder—holdingtheOLSestimatesfixed—bya 0 0 linearcombinationofbiasandvarianceofthespectralestimatewhoseweightsdependonthetruevaluesoftheVAR. Furthermore,theMSEsofACEV-AM andACEV-NW willalsodependonthecovariancebetweenS (0)AM,respectively 0 0 X S (0)NW,andtheOLSestimates. X 13

ThesimulationsreportedbelowuseboththeoptimalbandwidthselectionschemeofNeweyand West (1994) and the large bandwidth choice of CEV. The former tend to pick fairly small bandwidths. Forthevariouscalibrationsofthemodeleconomyconsideredhere,theoptimalbandwidth of the Newey-West estimator is typically close to ten, and the average for the Andrews-Monahan estimator is about four. In simulations not reported here, spectral estimates with intermediate bandwidth choices displayed performance characteristics which were intermediate between what isshownhereforthesetwochoiceshere. 3.3 Conceptual Problems with the CEV Procedure The CEV procedure is motivated by dissatisfaction with B(1)OLS. In conventional SVAR implementations, this estimate is needed for two purposes: First, to construct the long run responses A(1) as in (9), and second in order to map A(1) back into impact responses A as in (10). CEV 0 replace B(1)OLS with a spectral estimate in the first step, but not in the second. This creates a non-negligibleinconsistencyinrepresentingtheoveralldynamicsoftheVAR. By plugging a spectral estimate into their SVAR computations, CEV have weakened the OLS assumption of uncorrelated residuals without fully accounting for its consequences. As a result, the impact coefficients of CEV will in general not reproduce the forecast error variance of the VAR,whichisattheheartofvariancecomputations. Theseandotherconsequencesareillustrated here. Thenextsub-sectionshowshowaspectralfactorizationcouldbeusedtoincorporatespectral estimatesintoaVARmodelwhileretaininganinternallyconsistentmodelofthedata. The spectral estimates embody information about correlation in the VAR residuals vOLS. As t can be seen from (14), the Andrews-Monahan estimator is constructed from autocovariances of the VAR residuals vOLS. Obviously, b > 0 expresses a concern about serially correlated residut als. The Newey-West estimator SNW(ω) also embodies concerns about serially correlated VAR X residuals since it implies a spectrum for the VAR residuals, which is generally not constant across 14

frequencies.17 Under the premise that the true model has only an infinite-order representation, it is indeed very plausible that the residuals from a VAR(p) will be correlated. In the spirit of Andrews and Monahan (1992), the VAR could then be viewed as having merely prewhitened the data. But typically, researchers fit the lag length of their VARs until the point where estimated residuals do not display any significant correlation. Employing a spectral estimate like (14) beyond this point impliesabeliefthatthereisstillusefulinformationtobegleanedfromtheestimatedresiduals—or in other words, it implies a distrust against the lag-selection criteria being chosen for the VAR. By allowing for residual dynamics poses, a researcher risks of overfitting the data, which may still reduce bias in the estimated spectra, but at the expense of a higher standard error.18 Against this backdrop, the assertion of Christiano, Eichenbaum, and Vigfusson (2006a) that the impulse responses computed from their procedure have “smaller bias, smaller means square error” appear even more striking—and as will be seen, these properties do neither extend to the wider set of calibrationsstudiedbelownortootherSVARstatisticslikevariancedecompositions. A researcher adopting the CEV strategy wants S (0)AM 6= S (0)OLS and thus S (0)NW 6= X X v ΩOLS.19 Asadirectimplication,theimpactcoefficientsofCEVdonotreproducetheforecasterror v variance of the VAR, ACEV-AM(ACEV-AM)T 6= ΩOLS. When computing the total variance of the data 0 0 v by summing over the conditional variations implied by the SVAR, the CEV procedure would not matchtheunconditionalvarianceofthedataeither.20 Thismismatchintheunconditionalvariances (cid:163) (cid:164) 17S (ω)NWgeneralizes(12)tothecaseofnon-zerofrequencieswithΓ ≡E X XT : X k t t−k (cid:181) (cid:182) (cid:88)b |k| (cid:161) (cid:162) S (ω)NW =Γ + 1− Γ e−iωk+(Γ )Teiωk X 0 b+1 k k k=1 andtheimpliedspectrumofVARresidualsis(I −B(e−iω)OLSe−iω))S (ω)NW(I−B(e−iω)OLSe−iω))T X 18Inthesimulationsreportedbelow,laglengthischosenseparatelyforeachsamplebasedonthegenerallyconservativeSchwartzInformationCriterion,typicallychoosingalaglengthofoneintheapplicationpresentedhere. 19Thisfollowsfromcomparing(15)and(16)with(9)and(10). 20Theimpulse-responsesA(L)CEV-AM =C(L)OLSACEV-AMimplythefollowingvariancemeasure 0 (cid:88)∞ (cid:88)∞ VarX CEV-AM = COLSACEV-AM(ACEV-AM)T(COLS)T 6= COLSΩOLS(COLS)T =VarX t k 0 0 k k v k t k=0 k=0 Thelaststepholdsbecauseofthenormalequations(8)regardlessofwhethervOLSisiidornot. t 15

of VAR what is implied by the impulse responses of CEV occurs both in population as well as in smallsample. AsimilarargumentappliestotheNewey-WestvariantoftheCEVprocedure,where aresearcherseeksS (0)NW 6= S (0)OLS andthus(I−B(1)OLS)S (0)NW(I−B(1)OLS)T 6= ΩOLS. X X X v [Figure1abouthere.] ForthemodeleconomydescribedinSection2,Figure1illustratesthemismatchinthevariance ofoutputgrowth. Insmallsample,thevariancesimpliedbytheCEVprocedureareonlyabouthalf asbigastheOLSsamplemoments. AscanbeseenintherightpanelofFigure 1,thisoccursboth when using the Newey-West or the Andrews-Monahan variant of the procedure and regardless of the share of fluctuations explained by technology shocks. As depicted in the left panel of the figure, the mismatch is qualitatively different, but also sizable when applying the procedure to the population moments of the model while using a lag length of p = 1 and spectral bandwidth of b = 150. The CEV procedure is motivated by concerns about the ability of OLS estimate to correctly capture the low-frequency dynamics of the data. But implicitly, differences between spectra estimated from OLS and the non-parametric methods are not attributed to misspecified dynamics, but rather to the VAR’s forecast error variance. However, the accuracy of estimating ΩOLS has so v far not been doubted. In fact, getting a good estimate for forecast error variance is precisely the objective of OLS projections—see (11) above. Still, the CEV procedure deviates from previous contributions to the SVAR literature where identification is defined as a search over the space of matricesA ˆ satisfyingA ˆ A ˆT = ΩOLS.21 0 0 0 v Finally, a researcher might want to re-construct structural shocks based on (2) as εCEV-AM = t (cid:161) (cid:162) (cid:161) (cid:162) ACEV-AM −1 vOLS andcomparethemagainstεOLS = AOLS −1 vOLS. Shewillbetroublednoticing 0 t t 0 t thattheestimatedtechnologyshocksareperfectlycorrelated:22 a¯OLS a¯OLS (εz)CEV-AM = 11 ·(εz)OLS and (εz)CEV-NW = 11 ·(εz)OLS t a¯AM t t a¯NW t 11 11 21SeeforexampleFaust(1998),CanovaanddeNicolo(2003)orUhlig(2005). 22Recallfrom(4)thata¯ isthetopelementofA(1). 11 16

This holds for any pair of matrices A1 and A2 constructed from (4) and (6) using B(1)OLS and 0 0 (cid:161) (cid:162) a A(1) satisfying the zero restrictions (4). Under those conditions the top rows of A(1)OLS −1 , (cid:161) (cid:162) (cid:161) (cid:162) A(1)AM −1 and A(1)NW −1 areidenticaluptoascaling.23 Since CEV were only concerned with impulses-responses and A , the problem does not show 0 up in their analysis. The construction of estimated shocks is however often used by researchers, for instance in order to plot historical decompositions or when identifying several shocks (see for exampleAltigetal.(2004)). 3.4 Correct Identification with Spectral Factorization To overcome the problems with the CEV procedure discussed above, it is necessary to parse out dynamicsofvOLS impliedbythespectralestimates. Alsowhenthetruemodelhasaninfiniteorder t VARrepresentation,OLSprojectionsofX onafinitenumberoflagsarewelldefinedinthesense t of satisfying the projection equations (8) for k ≤ p, but the residuals vOLS are not iid. In general, t theresidualsfollowamovingaveragerepresentation: vOLS = e +D e +D e +D e +... = D(L)e t t 1 t−1 2 t−2 3 t−3 t withspectraldensity S (ω) = D(e−ω)ΩD(e−ω)T (17) v where D(L) = (I −B(L)OLSL)C(L) (18) [Figure2abouthere.] CKM and CEV discuss a truncation bias which is hard to detect based on VAR lag-length selection procedures. In terms of the moving average D(L), their results can be read as finding D ≈ 0 but D(1) 6= I. This can also be illustrated in the model economy described in Section 2. i (cid:80) Figure 2 plots the population values of the cumulated sums K D when p = 1 for different k=0 k 23BothCEVandOLSuseB(1)OLSincomputingA−1 =A(1)−1(I−B(1))−1andexceptforthetopleftelement, 0 the first row of A(1)−1 is full of zeros. Applying a standard result for inverting partitioned matrices (Magnus and Neudecker1988,p. 11),thelongrunrestriction(4)placesthesamezerorestrictionsonA(1)−1 asitdoesonA(1). ThisappliesequallytoA(1)CEV-AM,A(1)CEV-NWandA(1)OLS. Finally,thetopleftelementofA(1)−1equals1/a¯ . 11 17

calibrations of the share of fluctuations in output explained by technology shocks. (Results are similarforothervaluesofp.) Ateachlag,theincrementsaresmallandclosetozero,butsumming overmanylagsleadsto D(1) 6= I. Many moving average representations can be consistent with a given spectrum. But only one ofthemisinvertible. Aswillbeshownnext,D(L)isinvertibleandcanbeuniquelyidentifiedwith aspectralfactorizationof S (ω). v Proposition 1 (Invertibility of D(L)). When the underlying model has a fundamental VAR representation as in (1), and the OLS-VAR is stable, the moving average polynomial D(L) defined in (18)hasallitsrootsoutsidetheunit-circle. Proof. The proof is straightforward since (I − B(L))−1 = C(L) = (I − B(L)OLS)−1D(L) has all roots outside the unit circle and the same has been assumed for the VMA of the VAR(p), C(L)OLS = (I −B(L)OLSL)−1. It is straightforward to recover D(L) from S (ω) with a spectral factorization. The “canoniv cal spectral factorization” is a classic theorem in linear quadratic control, assuring existence and uniquenessofaninvertibleD(L)andapositivedefinite Ωconsistentwith(17). Theorem 1 (Spectral Factorization, (Hannan 1970)). Suppose a variable v has autocovariances t (cid:163) (cid:164) Γ ≡ E v vT = (Γ )T andaspectraldensity k t t−k −k (cid:88)q S (ω) ≡ Γ e−ikω ∀ ω ∈ [−π,π] v k k=−q which is non-singular at each frequency (|S (ω)| 6= 0 ∀ω), as well as being non-zero at the zero v frequency,S (0) 6= 0. Thereisauniquefactorizationof S (ω)into v v S (ω) = D(e−ikω)ΩD(e−ikω)T v (cid:80) where Ω is positive definite and D(z) is a q’th order polynomial D(z) = I + q D zk which k=1 k hasallitsrootsoutsidetheunitcircle. 18

The theorem factors a spectrum constructed from a finite number of autocovariances into a finite-order MA. For an empirical application, a finite q has of course to be chosen. But when applying the spectral factorization to the population objects of the true model (1), it remains to considerthatD(L)isingeneralanMA(∞). However,sincetheprocessesforX andvOLS arestat t tionary, their autocovariances and MA-coefficients vanish for large lags (Hamilton 1994, Chapter 3.A). Analogous to Sims (1972), a spectral factorization with an arbitrarily large but finite q can arbitrarily well approximate the true spectrum and true D(L). Alternatively the true D(L) can be thoughtofasbeingthelimitofapplyingTheorem 1toaneverincreasingsequenceof q’s. For a correct identification of the structural shocks, the true impact coefficients (6) can be writteninterms B(L)OLS andD(L)as A(1) = chol{(I −B(1)OLS)−1S (0)(I −B(1)OLS)−T} (19) v A = D(1)−1(I −B(1)OLS)A(1) (20) 0 CEV construct A(1)AM according to (19) while using the spectral estimate S (0)NW. But they v ignoretheresidualdynamicscapturedbyD(1)in(20)whenmappingA(1)AM backintotheimpact coefficients. As illustrated in Figure 2, D(1) is typically not a diagonal matrix in the model economy,farfromequaltotheidentitymatrix. IgnoringtheresidualcapturedbyD(L)isthesourceof thevariancemisrepresentationdiscussedintheprevioussubsection. TocombineVARcoefficientsandspectralestimatesinaninternallyconsistentfashion,aspectralfactorizationmustbeused. ThespectralfactorizationofS (ω)NWyieldsauniqueandinvertible v MA(b),denotedD(L)SF-AM,andaninnovationsvariancematrixΩSF-AM. Thesuperscript“SF-AM” indicatesthatthesearecalculatedfromtheresidualspectrumemployedbytheAndrews-Monahan estimatorS (ω)AM. Impactcoefficientsarethen X (cid:161) (cid:162) ASF-AM = D(1)SF-AM −1 (I −B(1)OLS)A(1)AM 0 (cid:161) (cid:162) = D(1)SF-AM −1 ACEV-AM 0 19

SayedandKailath(2001)surveyanumberofdifferentalgorithmsforperformingspectralfactorizations. The computations reported here use a reliable and efficient algorithm from Li (2005), based on a state space representation of the moving average process of vOLS. (Details are given in t Appendix A.) In contrast to the CEV procedure, the spectral factorization is consistent with the variance of thedata,insampleaswellasinpopulation. Proposition 2. By construction, estimates of ASF-AM and D(L)SF-AM factor the spectral density 0 (cid:82) S (ω)NW and thus reproduce the variance of the VAR residuals ΩOLS = π S (ω)NWdω. As a v v −π v corollary,thispreservesalsotheunconditionalvarianceof X . t Proof. Thespectralestimateis (cid:181) (cid:182) (cid:88)b |k| (cid:161) (cid:162) S (ω)NW = Γ + 1− Γ e−iωk +(Γ )Teiωk v 0 k k b+1 k=1 (cid:82) and the result follows from π e−iωkdω = 0, regardless of whether population moments Γ = −π k (cid:80) EvOLS(vOLS)T or sample moments Γ ˆ = 1 T vOLS(vOLS)T are used. The corollary follows t t−k k T t=k t t−k (cid:80) fromthenormalequationsoftheVAR,whichenforce VarX = ∞ COLSΩOLS(COLS)T. t k=0 k v l A spectral factorization can also be applied directly to the Newey-West estimate of the data’s spectrum, S ωNW, yielding coefficients for the VMA of X , C(L)SF-NW and innovation variance X t ΩSF-NW. Following(6),impactcoefficientsandimpulseresponsescanthenbecomputedas (cid:161) (cid:162) ASF-NW = C(1)SF-NW −1 A(1)NW 0 A(L)SF-NW = C(L)SF-NWASF-NW 0 TheseimpulseresponsesdonotinvolveanyVARcoefficients. AnalogouslytoProposition2,their constructionpreservesthevarianceofthedata. 20

4 SVARs applied to Data from Lab Economy The previous section described several schemes for imposing the long-run restriction (4) on the data. The conventional method, going back to Blanchard and Quah (1989), uses OLS estimates ofaVAR.TherecentlyproposedprocedureofCEVcombinesthiswithanon-parametricestimate of the spectral density at frequency zero. This procedure has been criticized above for its lack of internal consistency. Finally, this paper proposed a new method, combining OLS estimates and spectral estimators in an internally consistent way. This method relies on a spectral factorization (“SF”)touncoverthedynamicsimpliedbythenon-parametricspectralestimators. These procedures are applied here to data simulated from the model economy described in Section2. ThesamedatageneratingprocesshasalsobeenusedbyCEVandCKM.24 FortheCEV and SF methods, there are two variants depending on whether the spectral estimators of Newey andWest(1987)orAndrewsandMonahan(1992)areused. Thissectionreportsresultsforboth. Mimicking conditions faced by empirical researchers, “small” samples with 180 observations are simulated. In small sample, two distinct issues arise. First, there is truncation bias in VARs and spectral estimators arising from the need to specify a finite lag length p, respectively a finite bandwidth b. As discussed in Section 2, lag length is determined individually for each draw with an information criterion and spectral bandwidth is fixed at 150. In addition, alternative results are reported using the bandwidth selection procedure of Newey and West (1994) for Newey-West spectra. (SeeSection 3.2forfurtherdiscussionofbandwidthselection.) Second, there is the small sample bias in estimated parameters known from Hurwicz (1950).25 To isolate the pure truncation effects from the Hurwicz bias, the identification procedures are not only applied to simulated data, but also to VARs and spectral estimates constructed from the model’struepopulationmoments.26 24AsdiscussedinSection2,thecalibrationsemployedhereareidenticaltothesettingofCEV—exceptforconsideringawiderrangeofthetechnologyshareinoutputfluctuations. Asdiscussedaboveaswell,theCKMexperiments differslightlyintheirchoiceofψandρ . l 25Thisbiasisparticularlyacutethesmallerthesampleandthehigherthepersistenceofthedata. Itispertinentin thisexample,sincecalibratingthemodeltomatchsalientfeaturesofU.S.datarequiresahighdegreeofpersistencein thenon-technologyshock,ρ . l 26Inthecaseofthespectralestimators,thismeansevaluating(12)and(13)attruepopulationmoments,insteadof 21

The procedures are evaluated in terms of of their capability to uncover two statistics typically of interest to applied researchers. Following CKM and CEV, estimates of the response of hours to atechnologyshock arecomputed. Forbrevity,the discussionis limitedonimpact coefficients A , 0 sinceallmethodscomputeimpulseresponsesfromC(L)OLSandtheirestimatesofA (exceptwhen 0 factorizingS (ω)NW). Inaddition,theshareoffluctuationsinoutputandhoursduetotechnology X shocksisestimated. Asitistypicalinthebusinesscycleliterature,thesesharesarecomputedafter filtering out any fluctuations which do not correspond to cycles with a duration between two-anda-half and eight years.27 Two criteria are reported to assess the goodness of estimates: Bias and Root Mean Square Error (RMSE), both expressed as percentages relative to the true value known fromthemodel.28 The results show that all procedures are subject to substantial truncation and small biases and none works like a panacea. Different methods display different strengths and weaknesses. The claims by Christiano, Eichenbaum, and Vigfusson (2006a) of “smaller bias, smaller means square error”associatedwiththeirproceduredoneithergeneralizetoawiderrangeofmodelcalibrations nordotheyextendfromtheestimationofimpactresponsestovarianceshares. [Figure3abouthere.] Effects from the truncation and the small sample bias can offset each other. This is the case whenestimatingtheimpactoftechnologyonhours. TheleftcolumninFigure3showshowimpact responsesareoverestimatedinpopulationwhereasthesimulatedbiasshowninthemiddlecolumn ofthefigureislower(morenegative). Thissimulatedbiasdisplaysthetotaleffectfromtruncation and Hurwicz bias. The OLS method has the largest population bias and it is only partially offset by the Hurwicz bias. The two spectral methods suffer from substantially smaller truncation bias, sampleautocovariances,whilekeepingtheBartlettweightsandthetruncationatthechosenbandwidth(here:b=150, respectivelyb=15whencomparingagainstsimulationsusingthebandwidthselectionprocedureofNeweyandWest (1994)). ThecomputationofVARsfrompopulationmomentsisequallystraightforward,anddetailsaredescribedin AppendixB. 27ThevariancecomputationsareexplainedinAppendixC. 28Denotingtheestimatedparameterasθanditsestimateasθˆ,relativebiasiscomputedasE(θˆ−θ)/θ·100%. The (cid:113) (cid:113) RMSEisdefinedasRMSE= E(θˆ−θ)2 = (Eθˆ−θ)2+Varθˆanditisconvertedintoapercentageerrorusing RMSE/θ·100%. Inbothcases,expectationsarecomputedfromthearithmeticaverageover1,000simulations. 22

and depending on the simulated importance of technology shocks, the total bias can be either negative or positive. Coincidentally, the upwards bias in SF-AM and SF-NW is exactly offset aroundtechnologysharesofabouttwothirds,correspondingtotherangeofMLEestimatesofCEV andCKMforU.S.data. (SimilarlyforCEV-AM,butnotCEV-NW.)However,resultsaredifferent for other calibrations of the technology share, which cautions strongly against extrapolating from aparticularresulttodifferentdatasetsanddifferentapplications. Unlessthetrueshareoftechnologyshocksisverylarge,theRMSEofestimatedimpactcoefficients are very large, often surpassing more than 100% of the true value. Interestingly, the RMSE do not differ much across the different methods, as can be seen in the right-most column of Figure 3. If anything, SF-NW is outperforming CEV-NW on bias, at the expense of a worse RMSE. ThisislikelyduetoanoverfittingoftheresidualdynamicsbySF-NW. [Figure4abouthere.] TurningtotheestimatedvariancesharesofoutputandhoursshowninFigure4,therelativeperformanceofthevariousmethodslooksquitedifferent. Thepanelsinthetoprowofthefigureshow bias and RMSE for variance decompositions of output, the bottom row for variance decompositions of hours. For this figure, spectral densities have been estimated with the Andrews-Monahan estimator. Results are broadly similar when using the Newey-West estimator (see Figure A.1 in theseparateappendixwithadditionalresults.) Strikingly, for technology shares in output, bias and RMSE are very similar when using either OLS or CEV. The mismatch in total variance discussed in Section 3.3, does not seem to distort the computations of relative variance measures in this case. But, the two methods differ when decomposingthevarianceofhours. BiasandRMSEinthevariancedecompositionofhoursarean order of magnitude larger than for output, cautioning very strongly against neglecting small sample issues when comparing SVAR estimates against model predictions. Moreover, the variance decompositionsofhoursprovideausefulcounterexampleagainstdisregardingOLSmethodsaltogether, since OLS dominates the spectral methods both in terms of simulated bias and RMSE for allcalibrationsconsideredhere. Allinall,theseresultsunderlinehowtruncationandHurwiczbias 23

interactwiththedifferentmethodsinwayswhicharehardtoanticipateforanempiricalresearcher whodoesnotknowthetruedynamicsofthedata. The results presented in Figures 3 and 4 are based on a large and fixed spectral bandwidth of b = 150. A separate appendix with additional results shows that the results are similar for Newey-West spectra when their bandwidth has been chosen by the automatic bandwidth selection procedure of Newey and West (1994). Compared to the case of a large and fixed bandwidth, only two differences stand out. Estimating technology shares from a direct factorization of the Newey West spectrum perform worse compared to the large bandwidth case, both in terms of bias and RMSE,unlesstechnologyaccountsforlessthantwothirdsofbusinesscyclefluctuationsinoutput (see Figure A.1 in the appendix). Furthermore, the RMSE of impact coefficients estimated with CEV-NWisalmostflatataroundtwothirdsofthetruevalue,independentlyofthetruetechnology share (Figure A.2). Applying the automatic bandwidth selection for the residual spectra of the Andrews-Monahan estimator yields bandwidths close to zero, such that the results are mostly indistinguishablefromtheOLSestimates(Figure A.3). 5 Conclusions In finite sample, truncation bias and Hurwicz bias pose fundamental problems when identifying structuralshocksfromrestrictionsonthelong-runbehaviorofthedata. Theseissuesarepresentin thetimedomainwhenworkingwithaVAR,aswellasinthefrequencydomainwhenworkingwith spectral estimators. Basically, the same estimates of the data’s autocovariances are employed for constructing non-parametric estimates of the spectrum as well as for computing OLS coefficients. In both cases, truncation bias arises since there are only as many sample autocovariances as there are data points. And due to the Hurwicz bias, variance estimates tend to be biased downwards the smaller the sample and the larger the persistence of the data—again affecting both OLS estimates ofVARcoefficientsaswellasnon-parametricestimatesofthespectraldensity. Thus, spectral estimates offer no panacea against the truncation and small sample problems 24

known from OLS. At best, by allowingfor additional dynamics, they might improveupon OLS in termsofbias,butbyoverfittingthedata,thiscomesattheexpenseofincreasingRMSE. The performance of different estimators appears to be very specific to the underlying model anditscalibration,makingithardtopredict,whichprocedurewoulddowellinfutureapplications using new data. Even for a given calibration, when a method performs better in terms of one model statistic, say impact coefficients, this does not necessarily translate into better performance for another statistic, like a variance share. Going forward, it would be more suitable to compare SVARestimates(fromanyprocedure),againstthesmallsample predictions,notthetruemoments, of a specific model as in Cogley and Nason (1995), Kehoe (2006), Dupaigne, Feve, and Matheron (2007)andDupaigneandFeve(2009). 25

Appendix A Spectral Factorization Method Spectralfactorizationhasalongtraditioninthefieldsoflinearquadraticcontrol,robustestimation and control as surveyed for example by Whittle (1996).29 Theorem 1 has been adapted from Hannan (1970, p. 66). The original theorem allows for unit roots in D(L). The version stated abovehasbeenslightlystrengthenedbyexcludingthecaseofzeropowerinthespectraldensityat zero-frequency,toensuretheinvertibilityoftheMA(b).30 In the context of this paper, S (ω) will be the spectral density of vOLS = D(L)e where v t t Ee eT = Ω = A AT. We will be using non-parametric estimates of S (ω) based on weighted t t 0 0 v sumsofthe sample autocovariancefunctionasdescribedinSection 3.2.31 Theorem 1 requires S (ω) to be non-singular. This can be understood as requiring that the v autocovariancesneedtodecaysufficientlyfastinrelationtothenumberofMAlags. Forexample, inthescalarcaseandwithq = 1,thefirst-orderautocorrelationtobematchedwithaMA(1)cannot belargerthan0.5inabsolutevalue.32 Algorithms for implementing the factorization go back to Whittle (1963) and have recently been surveyed by Sayed and Kailath (2001). The simulations reported here use the algorithm of Li (2005), which is based on a state space representation of v and performed very reliably.33 The t remainderofthisappendixdescribesthealgorithminmoredetail. Suppose v follows an MA(q) as above. To represent it in a state space system, define the state t 29ForareferenceinthecontextofeconomicsseeHansenandSargent(2007,2005). 30SupposethatS(0)6=0. SinceΩispositivedefinite,itfollowsthatD(1) 6=0. AllrootsofD(z)arethusoutside theunitcircleandD(L)isaninvertibleMA(b). 31TheΓ fromTheorem1areasmoothedversionofthesampleautocovariancesincetheyarethecoefficientsofan k inverseFouriertransformoftheNewey-Westestimateofthespectraldensity. 32Givenacovarianceγ andfirst-orderautocovarianceγ ,thespectrumequalss(ω)=γ ·(1+2γ cos(ω)). And 0 1 0 1 |s(ω)|6=0requires|γ /γ |<0.5. 1 0 33ThepaperofLialsoshowshowtoreducethenumberofiterationsbystackingtheMA(q)intofirstorderform, however this comes at the cost of inverting larger matrices in the Riccati iterations which proved to be numerically lessstableinthesimulationscomputedforthispaper. 26

(cid:40) (cid:41) (cid:183) (cid:184) (cid:175) T (cid:175) vector s t = E v t v t+1 ... v t+q−1 (cid:175) (cid:175) vt−1 where vt−1 is the entire history of realizations ofv uptotime t−1. Lithenconstructsthefollowingstatespacesystem t s = As +De t+1 t t v = Cs +e t t t     0 I 0 ... 0  m m m m    D 1 0 0 I 0 ... 0     m m m m m (cid:183) (cid:184)     D A =    . . . ... . . .    D =     . . . 2    C = I m 0 m ... 0 m     0 ... 0 I   m m m D q 0 ... 0 0 m m m whereI and0 arethem×midentitymatrix,respectivelythe n×nzeromatrix. m m What is needed is a mapping from the autocovariances of v , Γ , to the state space objects. t k The objects of interest are the matrix D containing the stacked MA coefficients D as well as the i variance Ω = Ee eT of the innovations process. To obtain this mapping, it is useful to stack the t t (cid:183) (cid:184) T . autocovariancesintoamatrix M = ΓT ΓT . . Γt 1 2 q Li(2005,Theorem2)showsthatthevariance-covariancematrixofthestatesΨ ≡ Es sT solves t t theRiccatiequationΨ = AΨAT+(M−AΨCT)(Γ −CΨCT)−1(M−AΨCT)T andthattheMA(q) 0 coefficients can be recovered as D = (M −AΨCT)(Γ −CΨCT)−1 and Ω = Γ −CΨCT. As 0 0 shown by Li (2005), the above Riccati equation can be solved recursively, starting from Ψ(0) = 0 and iterating over Ψ(n+1) = AΨ(n)AT + (M − AΨ(n)CT)(Γ − CΨ(n)CT)−1(M − AΨ(n)CT)T 0 sinceΨ = lim Ψ(n) andΨ(n+1) ≥ Ψ(n). n→∞ Attheendofeachfactorizationcomputedforthispaper,ithasbeenverifiedthatthefactorization produces an invertible MA(q) polynomial, which matches the original spectral density. In all simulations,thishelduptomachineaccuracy. 27

B VARs Implied by Lab Economy Thissectionoutlineshowtoderivethefollowing: First,valuesfromthelabeconomyfortrueVAR objects like A , A(1), B(1), and the autocovariances of X . Second, population coefficients of 0 t finite-orderVARsimpliedbythelabeconomy.34 The linearized solution to the lab economy described in Section 2 yields a state space model forlaborproductivitygrowthandhours   ∆log(y /l )  t t  X =   = CZ with Z = AZ +Bε (21) t t t t−1 t logl t Statevectorandshockvectorare: (cid:183) (cid:184) (cid:183) (cid:184) T T Z = k ˆ εz τz k ˆ εz τz ε = εz εl (22) t t t l,t t−1 t−1 l,t−1 t t t where k ˆ is the log-deviation of detrended capital from its steady state, τ and εz are the labor t l,t t wedgeandthegrowthrateintechnology. (Z includesalsolaggedvariablesduetothepresenceof t laborproductivity growth inX .) t The computationof the matrices A, B and C isstraightforward,please see CKMfor a detailed presentation. TrueVARobjects The decomposition in section 4 uses the following objects of the true process: A , A(1), B(1) 0 as well as the autocovariances of X . Their computation from the state space is straightforward t since true impulse responses and spectrum are given by A(L) = C(I −AL)−1B and S (ω) = X A(e−iω)A(e−iω)T. The impact coefficients A = CB are apparent from (21). Recalling equation 0 34For this specific two-shock economy, details can also be found in McGrattan (2005). For general state space models details can be found in Fernandez-Villaverde, Rubio-Ramirez, and Sargent (2005). To simplify the VAR notation,X hasbeendemeanedpriortotheanalysis. t 28

(2),thisalsopinsdownthecovariancematrixoftheforecasterrors Ω = CBBCT. In order to map forecast errors into structural shocks, A must obviously be square and invert- 0 ible. Furthermore,Fernandez-Villaverde,Rubio-Ramirez,andSargent(2005)showthatinvertibility requires the eigenvalues of A−B(CB)−1CA to be strictly less than one in modulus, which is satisfiedforallcalibrationsconsideredhere. The non-structural moving average representation of X is X = A(L)A−1e = C(L)e . t t 0 t t From (3), the coefficients of the non-structural VAR(∞) representation of the model can be obtainedbyinvertingthismovingaverage,yielding B(L)L = I −C(L)−1. The autocovariances EX XT can be directly computed from the state space model. The t t−k covariance matrix of the states EZ ZT ≡ Ω is obtained as the solution to a discrete Lyapunov t t equation: Ω = AΩ AT +BBT andtheautocovariancesof X areEX XT = CAkΩ CT. t t t−k VAR(p)coefficientsinpopulation Finite-order VAR(p) can be computed as projections of X on a finite number of its past values, t X ...X . In line with the notation of the main text, population coefficients of a VAR(p) are t−1 t−p denotedwithasuperscript“OLS”. X = B(L)OLSX +vOLS t t−1 t (cid:80) Thecoefficientsofthelagpolynomial B(L)OLS = p−1BOLSLi solvetheOLSnormalequations i=0 i (cid:195) (cid:33) (cid:88)p−1 E X − BOLSX XT = 0 ∀j = 1...p t i t−1−i t−j i=0 which are evaluated using the autocovariance matrices of X whose computations are described t (cid:161) (cid:162)(cid:161) (cid:162) in the preceding paragraph. For instance if p = 1, BOLS = EX XT EX XT −1 . Detailed 1 t t−1 t t formulas for higher VARs can be found in Fernandez-Villaverde, Rubio-Ramirez, and Sargent (2005). 29

Chari,Kehoe,andMcGrattan(2005,Proposition1)showthattheVARrepresentationofX in t themodelisofinfiniteorderandresidualsfromaVAR(p)willnotbemartingales. Byconstruction, the projection residuals vOLS are orthogonal to X , ..., X , but they are not orthogonal to the t t−1 t−p complete history of X . The moving average representation of the forecast errors vOLS = D(L)e t t t (cid:161) (cid:162) iseasilyconstructedfrom D(L) = I −B(L)OLSL (I −B(L)L)−1. Varianceequation Even though the VAR(p) residuals vOLS are not iid, the usual variance equation is still applicable. t For notational convenience, take the case of a VAR(1), X = BOLSX + vOLS. The normal t 1 t−1 t equationsimply VarX = BOLS (VarX ) (BOLS)T +ΩOLS (23) t 1 t 1 v (cid:88)∞ = (BOLS)k ΩOLS ((BOLS)k)T 1 v 1 k=0 (cid:88)∞ = COLS ΩOLS (COLS)T k v k k=0 The second line is obtained by recursive substitution of VarX and the third line follows from the t constructionofmoving-averagecoefficientsofaVAR(1),COLS = (BOLS)k. Theargumentiseasily k 1 extendedtoVARswithhigherlagordersbyusingtheircompanionform. C Bandpass-Filtered Variance Share from SVARs This appendix describes how to compute the share of bandpass-filtered fluctuations attributed to ˆ ˆ technology shocks from a set of SVAR parameters, B(L) and A . The bandpass filter employed 0 here considers only cycles with a duration between two-and-a-half and eight years. Denoting the bandpass-filtered level of (log) output y˜, its variance can be easily computed from the transfer t function (cid:183) (cid:184) (cid:179) (cid:180) −1 T (ω) = (1−e−iω)−1 1 I −B ˆ (e−iω)e−iω A ˆ y 0 30

Depending on the identification scheme, A ˆ corresponds to A (true value) or AOLS, ACEV-AM, 0 0 0 0 ACEV-NW, ASF-AM or ASF-NW and B ˆ (L) corresponds to the true (infinite order) polynomial B(L) 0 0 0 or its finite-order counterpart B(L)OLS. When using the spectral factorization of S (ω)NW, the X inverseofI −B ˆ (e−iω)e−iω isreplacedby CSF-NW(e−iω). Usingω = 2π andω = 2π thebandpass-filteredvarianceis 8·12 2.5·12 (cid:90) ω Vary˜ = T (ω)T (ω)Tdω t y y ω andtheshareoffluctuationsattributedtotechnologyshocksistheratio(Vary˜|εz)/(Vary˜),where t t t Vary˜|εz conditionsonlyonfluctuationsattributedtotechnologyshocks. t t   (cid:90) ω 1 0   Vary˜|εz = T (ω) T (ω)Tdω t t y y ω 0 0 Similar computations yield the variance shares for hours, when using the transfer function (cid:183) (cid:184) (cid:179) (cid:180) −1 T (ω) = 0 1 I −B ˆ (e−iω)e−iω A ˆ . l 0 31

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ADDITIONAL RESULTS ARE SPECTRAL ESTIMATORS USEFUL FOR IMPLEMENTING LONG-RUN RESTRICTIONS IN SVARS? List of Figures A.1 TechnologySharesinFluctuationsofOutputandHours(Newey-West) . . . . . . . ii A.2 SimulationswithAutomaticBandwidthSelection(Newey-West) . . . . . . . . . . iii A.3 SimulationswithAutomaticBandwidthSelection(Andrews-Monahan) . . . . . . iv i

ADDITIONALRESULTS ii )tseW-yeweN(sruoHdnatuptuOfosnoitautculFniserahSygolonhceT :1.AerugiF )tuptuO( ESMR detalumiS )tuptuO( saiB detalumiS )tuptuO( saiB noitalupoP 051 051 051 SLO VEC 001 001 FS 001 05 05 05 0 0 0 05− 05− 05− 33 76 001 33 76 001 33 76 001 erahS ygolonhceT erahS ygolonhceT erahS ygolonhceT )sruoH( ESMR detalumiS )sruoH( saiB detalumiS )sruoH( saiB noitalupoP 0002 0002 0002 0051 0051 0051 0001 0001 0001 005 005 005 0 0 0 33 76 001 33 76 001 33 76 001 erahS ygolonhceT erahS ygolonhceT erahS ygolonhceT .sruohrofwormottob,tuptuofonoitisopmocedecnairavrofESMRdnasaibstroperworpoT .erahsygolonhceteurts’ledomehtevitalerstniopegatnecreP :etoN fo egatnecrep eht si sixa-x eht no ”erahs ygolonhceT“ .051 = b fo htdiwdnab dexfi a htiw FS dna VEC rof desu ytisned lartceps eht fo srotamitse tseW-yeweN gnitareneg atad eht ni )sraey thgie dna flah-a-dna-owt neewteb snoitarud htiw selcyc( seicneuqerf elcyc ssenisub ta skcohs ygolonhcet ot eud ytilibairav tuptuo .ssecorp

ADDITIONALRESULTS iii )tseW-yeweN(noitceleShtdiwdnaBcitamotuAhtiwsnoitalumiS :2.AerugiF ESMR detalumiS : A saiB detalumiS : A saiB noitalupoP : A 0 0 0 004 004 004 SLO VEC 003 003 003 FS 002 002 002 001 001 001 0 0 0 001− 001− 33 76 001 33 76 001 33 76 001 erahS ygolonhceT erahS ygolonhceT erahS ygolonhceT ESMR detalumiS :erahS saiB detalumiS :erahS saiB noitalupoP :erahS 002 051 051 001 001 051 05 05 001 0 0 05 05− 05− 0 001− 001− 33 76 001 33 76 001 33 76 001 erahS ygolonhceT erahS ygolonhceT erahS ygolonhceT citamotua htiw rotamitse tseW-yeweN eht gnisu nehw )wor mottob( erahs ygolonhcet dna )wor pot( ygolonhcet ot sruoh fo sesnopser tcapmi detamitsE :etoN elcycssenisubtaskcohsygolonhcetoteudytilibairavtuptuofoegatnecrepehtsisixa-xehtno”erahsygolonhceT“ .)4991,tseWdnayeweN(noitceleshtdiwdnab .ssecorpgnitarenegatadehtni)sraeythgiednaflah-a-dna-owtneewtebsnoitarudhtiwselcyc(seicneuqerf

ADDITIONALRESULTS iv )nahanoM-swerdnA(noitceleShtdiwdnaBcitamotuAhtiwsnoitalumiS :3.AerugiF ESMR detalumiS : A saiB detalumiS : A saiB noitalupoP : A 0 0 0 004 004 004 SLO VEC 003 003 003 FS 002 002 002 001 001 001 0 0 0 001− 001− 33 76 001 33 76 001 33 76 001 erahS ygolonhceT erahS ygolonhceT erahS ygolonhceT ESMR detalumiS :erahS saiB detalumiS :erahS saiB noitalupoP :erahS 002 051 051 001 001 051 05 05 001 0 0 05 05− 05− 0 001− 001− 33 76 001 33 76 001 33 76 001 erahS ygolonhceT erahS ygolonhceT erahS ygolonhceT citamotuahtiwrotamitsenahanoM-swerdnAehtgnisunehw)wormottob(erahsygolonhcetdna)worpot(ygolonhcetotsruohfosesnopsertcapmidetamitsE:etoN elcycssenisubtaskcohsygolonhcetoteudytilibairavtuptuofoegatnecrepehtsisixa-xehtno”erahsygolonhceT“ .)4991,tseWdnayeweN(noitceleshtdiwdnab .ssecorpgnitarenegatadehtni)sraeythgiednaflah-a-dna-owtneewtebsnoitarudhtiwselcyc(seicneuqerf