Dynamic Equilibrium Economies: A Framework for Comparing Models and Data

D Equilibrium Economies: A Framework for Comparing Models and Data FrancisX. Diebold Lee E. Ohanian JeremyBerkowitz UniversityofPennsylvania UniversityofMinnesota Federal Reserve Board and and NBER UniversityofPennsylvania RevisedMarch 1997 Address correspondence to: FrancisX. Diebold Department of Economics UniversityofPennsylvania 3718 Locust Walk Philadelphia,PA 19104-6297 Abstraet: We propose aconstructive, multivariateframework for assessingagreement between (generallymisspecified)dynamicequilibriummodelsanddata, whichenablesacomplete secondorder comparison ofthe dynamicproperties ofmodelsanddata. We use bootstrap algorithmsto evaluatethe significanceofdeviationsbetween modelsanddata, and we use goodness-of-fit criteriato produce estimators that optimizeeconomically-relevantloss functions. We provide a detailedillustrativeapplicationto modelingtheU.S. cattlecycle. Acknowled~ment~: The Co-Editor andreferees provided helpfulandconstructive input,as did participants at meetingsofthe Econometric Society,ECARE/CEPR, NBER, andnumerous universityseminars. We gratefullyacknowledgeadditionalhelpfrom BillBrown, FabioCanova, TimCogley,Bob Lucas, EllenMcGrattan, DannyQuah, LucreziaReichlin,SherwinRosen, Chris Sims,Tony Smith,JimStock, Mark Watson, andespeciallyLarsHansen, AdrianPagan, andTom Sargent. Allremainingerrors andinaccuraciesare ours. JOS6Lopez provided dedicatedresearch assistanceinthe early stages ofthisproject. We thank theNational ScienceFoundation, the SloanFoundation and theUniversityofPennsylvaniaResearch Foundation for support.

-2- 1. Introduction Dynamicequilibriummodels are now usedroutinelyinmanyfields. Such models,for example,havebeenused to address avariety of macroeconomicissues,includingbusiness-cycle fluctuations,economic growth, and the effects of governmentpolicies.~Additionalprominent fieldsof applicationincludeinternationaleconomics,publiceconomics,industrialorganization, labor economics,and agriculturaleconomics.2 At present, however, manyimportant questionsregarding the empiricalimplementationof dynamicequilibriummodelsremainincompletelyanswered. The questionsfallroughlyinto two methodologicalgroups. The first group involvesissuesrelated to assessingmodeladequacy,and the secondinvolvesissuesrelated to modelestimation. We contribute to anemergingliterature that hasbegunto dealwithboth issues, includingWatson (1993), KingandWatson (1992, 1996), Canova,FinnandPagan (1994), KimandPagan (1994), Pagan (1994), Leeper and Sims(1994), Cogleyand Nason (1995), and Hansen, McGrattan and Sargent (1997). A 1996Journal of Economic Perspectives symposiumfocused on these issues, andtwo important messages emerged:3 (1)dynamicequilibriummodels,We allmodels, areintentionallysimpleabstractions andtherefore shouldnot beconstrued as the true data generatingprocess, and (2) formalmethods shouldbe developed andused to helpus assess the modelsmore thoroughly. In thispaper, we take a step inthat direction. Someparts of our framework are new, whileothers buildon earlierwork ininteresting ways. In manyrespects, our work beginswhere Watson (1993) ends. With aneyetoward future research, Watson notes that “...one ofthe most informativediagnostics ...istheplot ofthemodel 1Among manyothers, see KydlandandPrescott (1982), Hansen (1985), Christian and Eichenbaum(1995), andRotemberg andWoodford (1996) (businesscycles),Lucas (1988), Jones andManuelli(1990), Rebelo (1991), andGreenwood, Hercowitz, and Krusell(1997) (growth), andLucas (1990), Cooleyand Hansen (1992), and Ohanian(1997) (policyeffects). 2Among manyothers, seeBackus, Kehoeand Kydland(1994) (internationaleconomics), Auerbach and Kotlikoff(1987) (publiceconomics), Ericson andPakes (1995) (industrial organization),Rust (1989) (laboreconomics), andRosen, Murphy and Scheinkman(1994) (agriculturaleconomics). 3See Kydlandand Prescott (1996), Sims(1996) andHansen and Heckman (1996).

-3anddata spectra,” and herecommends that inthefuture researchers “presentboth model anddata spectra as aconvenientway ofcomparingtheircomplete set of secondmoments.”4 Our methods, whichare based on comparison ofmodelanddata spectraldensityfunctions,can beused to assessthe performance of amodel (for a givenset ofparameters), to estimatemodelparameters, and to test hypothesesaboutparameters or models. To elaborate, our approach is: A. Frequency-domainandmultivariate. Working inthefrequencydomainenables decomposition ofvariation across frequencies,whichisofien useful,andthe multivariatefocus facilitatessimpleexaminationofcross-variablecorrelations and lead-lagrelationships,at the frequenciesofinterest. B. Based on afullsecond-order comparison of modelanddata dynamics. Thisisin contrast to acommon approach used inthe businesscycleliterature ofcomparing onlyafew variancesand covariancesofdetrended variableshorn the model economy andthe actualeconomy. The spectrumprovides a complete summary of Gaussiantime seriesdynamicsand an approximate summaryofnon-Gaussian time seriesdynamics. C. Based on therealisticassumptionthat au modelsare misspecified. We regard allofthe models we entertain asfalse,inwtich case traditionalstatisticalmethods lose some oftheir appeal. D. Graphicaland constructive. The framework permits oneto assessvisuallyand quickly the dimensionsalong whichamodelperforms well,and thedimensionsalong whichitperforms poorly. E. Based on acommon set oftools that can beused byresearchers withpotentiallyvery differentobjectivesandresearch strategies. The framework can beused to evaluatestrictlycalibratedmodels, anditcan also beused formallyto estimate and test models. F. Designedto facilitatestatisticalinferenceabout objectsestimated from data, includhg spectra, goodness-of-fit measures, modelparameters, andtest statistics. Bootstrap methods playan importantrole inthat regard; we develop anduse a simple nonparametricbootstrap algorithm. G. Mathematicallyconvenient. Under regularityconditions,the spectrum isabounded continuousfunction,whichmakesfor convenientmathematicaldevelopments. 4He also notes that hisfailureto studycross-variablerelationshipsisapotentially important omission.

-4- Allofthe classicalideasof business-cycleanalysisdiscussed,for example,byLucas (1977)have spectralanalogs,rangingtiom univariatepersistence (typicalspectral shape)to multivariateissuesofcomovernent(coherence) andlead-lagrelationships(phase shtits) at business-cyclefrequencies. We highlightthese linksanddraw upon the business-cycleliterature for motivationinthe methodologicalsections2 and 3. The methods we develop,however, are not wed to macroeconomicsin anyway;rather, theycan beused inavariety offields. Therefore, to introduceresearchers indifferentareas to the use of our framework, we applyour methods to a simpleand accessible,yetrich, macroeconomicmodelinsection4. We concludein section5. 2. Assessing Agreement Bet~veenModel and Data Our basicstrategy isto assessmodels bycomparingmodelspectra to data spectra. Our goalisprovision of a graphicalframework that facfitates visualcomparisons ofmodel spectra to intervalestimates ofdata spectra. We compute modelspectra exactly(eitheranalyticallyor numerically);thus, theyhaveno samplinguncertainty. Samplingerror does, however, affectthe sampledata spectra, whichare of coursejust estimates oftrue butunknown (population)data spectra. We exploitwell-establishedprocedures for estimatingspectra, andwe develop anduse bootstrap techniquesto assessthe samplinguncertaintyofestimated spectra.5 2a. Estimatin~St)ectr~ Consider theN-variate linearlyregular covariance stationary stochasticprocess, m Yt= P + B(L) Et= p + ~ Bi et-i Q ift =s E(e~e~)= 0 otherwise, { 5Alternatively,onecould fixthe data spectrum and assess samplingerror inthe model spectrum bysimulatingrepeated realizationsfrom the model. The two approaches are essentially complementary,corresponding to the “Wald”and “Lagrangemultiplier”testingperspectives. See, for example,Gregory and Smith(1991).

-5where E(e~)= O,B. =I, andthe coefficientsare square summable(inthe matrix sense).6 The . autocovariancefunction is r(~) = ~ B1Q B1~tand the spectraldensityfunctionis JL ~=–W A Considernow a genericoff-diagonalelementofF(u), fu(u). In polar form thecrossspectraldensityisfu(ti) = gau(o) exp[iphu(m)],where gau(o) = [re2(fM(m)+) im2(fU(o))]12isthe gainor amplitude,andwhere phkl(~)–– arctan{im(fti(~))/ re(fti(m))}isthephase. Asiswell known, the gaintellshow the amplitudeofylismultipliedincontributingto the amplitudeofy~at frequencyO, andphase measures the lead of y~over ylat frequency~. (Thephase shiftintime unitsisph(w)/~.) We shalloften finditconvenientto examinecoherencerather than gain,where ga~(u) thecoherence isdefinedas coh~l(w) = whichmeasures the squared correlation f~~(~)f~](o)’ between y~andylat tiequency u. Given a samplepath {yl,, ..., yNt}~.l,we estimatethe .Nx1meanvector p with j’ = (y], ..., yN)’.From thispointonward, we assume that allsamplepaths havebeencentered around this samplemean. We estimate:h~lautocovariance functionwith ~(~) = [~~1(~)](k = 1, 1 ..., N, 1= 1,..., N), where ~~l(z) = — ~ ~ Y~,Y9l,+.~, = ~, 51, ..., *(T-1). Weestirnatethe spectraldensitymatrix usingthe Blac&n-Tukqlag-window approach inwhich wereplace the samplespectraldensityfunction, ~(mj) = L ‘~) ~(~)e-ioj’ (~j = ~, j = ~, .,., ~-1~ 2n.=.@-1) with oneinvolvin the “windowed”sampleautocovariance sequence, 4 $ -1) ‘~’ F *(uj) = ~ A(r) ~(~) e-itiz, where 2n .=-~-l) A(T)isa matrix oflag windows. The Blachn-Tukey procedure results ina consistent estimator ifwe adjustthe lagwindowA(%)with samplesizein sucha waythat varianceandbias deche simuhaneously.7We then obtainthe samplecoherence andphase at anyfrequency oj by transforming the appropriate elementsof F “(tij). 2b. AssessingSalnDk~ v~iabilit~ 6In manycases, detrending of some sort willbenecessaryto achievecovariance stationarity. 7Alternatively,ofcourse, onemaysmooth the samplespectraldensityfunctiondirectly. The dualitybetween the two approaches, for appropriate window choices,iswellknown. See Priestley (1981).

-6- .4key issuefor our purposes ishow to ascertainthe samplingvariabilityofthe estimated spectraldensityfunction. To do so, we use an algorithmfor resamplingfrom timeseriesdata, whichwe callthe Choleskyfactor bootstrap.8 The basicideais straightforward. Firstwe compute the Choleskyfactor of the samplecovariancematrix ofthe seriesofinterest. We then exploitthe fact that, up to second order, the seriesof interest can be written astheproduct ofthe Choleskyfactor and seria~yuncorrelated disturbances, whichcan beeasilybootstrapped using parametric or non-parametric procedures.9 An importantfeature ofthisvery simpleapproach is that itcan beused to bootstrap objectsother than the spectraldensityfunction. Later, for example,we willuseit to assessthe uncertaintyina model’sestimatedparameters. First we need somedefinitionsandnotation. ht z~= (ylt, ..., y~~)’,and let II z = (Zl, Z2>““”? z~‘)‘. Then z - (l@p, 2), where 1isanN-dimensionalcolumnvector of ones, and 2 = Toeplitz(r(0), r(l), ..., r(T-1)). By symmetryandpositivedefiniteness,we can write 2 = PP ~,where the unique Choleskyfactor P islowe;t;~~ngular. We estimate 2 by ~ = Toeplitz(fi(0), ~(l), ..., ~(T-1)), where F(r) = ~ AT ~,z, ~~lT1’ T = o, *1, ...?k(T-l); thisensures that Wecan WriteZ = PP , where theuniqueCholesky factor ~ islower triangulm. Now let {A,i.j,};:;,=obe a set ofdecreasing weightsappliedto the successiveoff-diagonalblocksof ~, andcalltheresultingmatrix 2“. Finally,let P *bethe Choleskyfactor of Z*. iid The fact that z - (l@p, PP impfiesthat data generated bydrawing e(i) - (O,1~~) and forming z(i)= pZ + P&(l), gThe Choleskyfactor bootstrap iscloselyrelated to theRamos (1988) bootstrap. We developthe Choleskyfactor bootstrap inthe timedomain,however, whereas Ramosproceeds in the frequencydomain. 9Note that the Choleskyfactor bootstrap willmissnonlineardynamicssuch asGARCH -it isdesignedto capture onlysecond-order dynamics,inidenticalfashionto standard (as opposed to higher-order) spectralanalysis. Users shouldbecautious inemployingour procedure if nonlinearitiesare suspected to be operative, aswould likelybethecase, for example,for highfiequency financialdata. Such nonlinearitiesare not likelyto beas importantfor the lowerfrequencydata typicallyanalyzedinmanyareas of macroeconomics,publicfinance,international economics,industrialorganization,agriculturaleconomics,etc.

-7where pZ = l@p, willhavethe samesecond-order properties as the observed data. In practice we replace theunknown population first and second moments withthe consistent estimates described above. Thus, to perform aparametric bootstrap, we draw e(i)- N(O, 1~~),form ~(i)= ~ * p *&(i)- N(z, ~*), where z = l@~, andthen compute both theestimates F *(i)(uj),j = 1, ,.,, ~-l, i= 1,..., R and confidenceintervals. Alternatively,to perform anonparametric bootstrap, we note that practiwcee~, “awe(i) ~(i) = p -J ~7).In (z – withreplacementfrom P *-’(z – z), form . from whichwe compute F *{i)(oj),j = 1, .,., $-1, i= 1,..., R, and then construct cotidence intervals. In summary,there me severalappealingfeatures ofthe Choleskyfactor bootstrap: (1) it isavery simpleprocedure, (2) itcan beused to bootstrap avarietyof objects, (3) itdoes not involveconditioningon afitted modelandtherefore imposesminimalassumptionson dynamics. Thislast feature maybeattractive forresearchers who choose not to viewthe data through the lensof anassumedparametric model. Alternativebootstrap procedures includethe VAR bootstrap (e.g., Canova,Finnand Pagan, 1994),whichcan bea usefulapproach for those interested infittinga specificparametric modelto thedata. Thus, the Choleskyapproach and the VAR approach can beviewed ascomplementaryprocedures. We hasten to add, however, that the literature on bootstrapping timeseriesin general-and spectra inparticular -- isvery young andvery much unsettled. We stillhave agreat dealto learnabout thecomparativeproperties ofvarious bootstraps, both asymptoticallyandinfinite samples,andthe conditionsrequired for variousproperties to obtain. Presently availableresults ci~er dependingon the specificstatisticbeingbootstrapped, and moreover, onlyscattered firstand second-order asymptoticresults are available,andevenlessisknown about actualfinitesampleperformance. With thisinmind,we present both theoretical andMonte Carlo analysesof theperformance of the Choleskyfactor bootstrap intwo appendicesto thispaper. In appendix 1, we document the bootstrap’s small.-sa.mpleperformance. In a second appendix(availablefrom the authors upon request), we establishfirst-order asymptoticvalidity.

-8- ~10 2c, Constructin~ConfidenceTunne ~ If interest centers on onlyone fl.equency,we simplyuse thebootstrap distributionat that frequencyto construct the usualbootstrap confidenceinterval. That is, we find q:, q~”suchthat P(f*(”)((A)s) q;) = 1-; and P(f ‘(”)(w)> q~-’)= 1-~, where (1-a) isthe desired confidence level,“L”standsfor lower, “U”standsfor upper, the “T”subscriptindicatesthat we tailor the bandto thefinite-samplesizeT, andthe (.) superscript indicatesthat.wetake theprobability under thebootstraP distribution. The (1-a)70 two-sided confidenceintervails [qTLq,-lu]. However, one often wants to assessthe samplingvariabilityof the entire spectraldensity fu?zctionover manyfrequencies(e.g., business-cyclefrequencies,or perhaps allfrequencies)to learn about the broad agreementbetween data andmodel. One approach isto form thepointwise bootstrap confidenceintervalsdescribed above, andthen to “comect the dots.” But obviously,a set of (1-a)70 confidenceintervalsconstructed for each of n ordinates willnot achieve (1-a)y~ joint coverage probability. Rather, theactualconfidencelevelwillbecloser to (1-a)”%, which holdsexactlyifthepointwiseintervalsare independent. Abetter approach isto use the Bonfenoni method to approximatethe desired coverage level,byassigning (1 - a/n)% coverage to each ordi.nate.11Theresulting’’confidencetunnel”hascoverage of at least (1 - a)% and therefore provides aconservativeestimate ofthetunnel.12 A third approach to confidencetunnelconstruction is thesupremummethod of Woodroofe andvan Ness (1967) and Swanepoel andvan Wyk (1986), whichuses an estimate of the (standardized) distributionof sup If*(uj) - f(oj)l, @j= ~, j = 1, ..., $-1, to O<uj<n l“In this section,for notational simphcitywe focus oncoltildencetunnelsfor univariate spectra. As willbeclear, the extensionto cross spectra isimmediate. IIIn the univariatecase, typicallyn =T/2 -1. In the multivariatecase, thequestion arises as to “howwideto cast the net”informingconfidencetunnels. Onemightvieweach elementof the spectraldensitymatrixinisolation,for example,inwhichcase each ofthe respective confidencetunnelswould use n =T/2 -1.. At the other extreme, onecould use n = N2(T/2-1), effectivelyforming atumel for the entirematrix. 12Bonferroni tunnelsachievethe desired coverage onlyfor (1) independentvaluesofthe estimated function across ordinates, whichisclearlyviolatedin spectraldensityestimationas the smoothingrequired for consistencyresults inaveragingacross fi”equencies,and (2)large n, because (1 - a/n)” 2 (1 - a), for anyfiniten.

-9construct aconfidencetumel for thecurve. Specificafly,13 (1.) Calculate f’(~(uj), @j= ~, j = 1, ..., ~-l.. 2 (2) Findc suchthat: –o!, where we evaluatetheprobabilitywithrespect to the t>ootstrapdistribution. (3) Construct the cotildence tunnel, f“(~j) * c~f”(~j), m = ~2,njJ = 1, ..., Z- . j 91 Unlikethe Bonferroni tunnels,the suprernumtunnelsattain asymptoticallycorrect “ coverage rates even with statisticaldependenceamong ordinates. Littleishewn, however, about the compmativefinite-sampleperformance ofthe Bonferroni and supremumtunnels,and the supremumtunnelsmayrequire verylarge samplesfor accurate coverage.14 3. Estimation: Maximizing Agreement Between Model and Data Now we considerestimation,together with therelated issues of goodness-of-fit and hypothesistesting. To makethe discussionas transparent aspossible,we first discussthe univariatecase, andthen weproceed to the multivariatecase. . 3a. Umvariat.~ Estilnationrequires aloss function,or goodness-of-fit measure, for assessingcloseness between modelanddata. A strength of our approach isthat manylossfunctionsmaybe entertained;theparticular lossfunction adopted reflects the user’spreferences.ls In most cases it would seemthat afinction ofthe form cow(e)= jg(fm(u; 6), f “(m))W(0) dm a o willbeadequate. The functiong measures thedivergencebetween f~(m; 0) (modelspectrum) ISThisprocedure is stiar to the one advocated inGallant,Rossi andTauchen (1993). 14See Hannan (1970),p. 294. IsFor an interestingandearlydiscussionof thisandrelated points, seePagan (1994).

-1oand f ‘(u) (estimateofdata spectrum).lb We weightthisdivergenceacross frequenciesbythe 27CJ function w(m). Inpractice, wereplace the integralwith a sumover frequencies oj = — T’ T j = 1, ..., —-1. Quadratic losswith uniformweightingover allfrequencies,for example, n corresponds~tog(a, b) = (a-b)2 and w(m) = 1, yieldingC~W(0)= ~ (f~(mj;(3) - f*(uj))2. The goodness-of-fit measure mayreadilybetransformed into a: estimationcriterion by taking eg=wargmin Cgw(e). e The GaussianML estimator isasymptotically fthisfern> for aparticular andpotenti~lly restrictive choiceof g, f*, and w; itis argma -+~h fm(”j; e) - ;~ f ;“J)6) . . e“ To compute standard errors andinterv! 1esti~’tes for parameters if i#ter#st, a1d to test hypotheses aboutthe elementsof 6~W,we againuse the Choleskyfactor bootstrap. We proceed asfo~ows: (1) At bootstrap replication (i),draw a bootstrap sampleof sizeT usingthe Cholesky factor algorithm. (2) NumericallyminimizeC~$(0)to get $~~. (3) Repeat Rtimes. (4) Compute standard errors, form intervalestimates, implementbiascorrections, or test hypothesesusingthedistributionof ~(i) i= 1,..., R. gw’ Note that, unlikemost implementationsof the bootstrap, ours does not involveconditioningon the model;instead,we generate the bootstrap samplesdirectlyfrom the autocovariancematrix of thedata. Thisisimportant inour environment,inwhichallmodelsare bestregarded asfalse. In closingthis section,letus elaborate on our allowancefor differentialweightingby frequency. There are at leasttwo reasons for entertainingthispossibility. First, use of a loss functionthat weights dtierentially byfrequencymaybe helpfulindealingwith measurement error, whichofien maynot contaminateallfrequenciesequally. Thus, it would seemprudent to 1~Note that the model spectrumiseithercomputableanalyticallyor numerica~yto any desired degree of accuracy. The data spectrum on the other hand,isconsistentlyestimable.

-11downweight those frequenciesthat are assumedto bemore contaminated bymeasurementerror. Second, use of a lossfunctionthat weightsdifferentiallybyfrequencymaybe important in misspecifiedmodels. For example,asdiscussed byHansen andHeckman (1996), model misspecificationmaycontaminate solmefrequenciesmore than others. Examplesofthisinclude potential contaminationat seasonalhequencies, asinthe work ofHansen and Sargent (1993) and Sims(1993). Watson (1993) also advocates the use ofdtierent.ialweightinginparameter estimation,for the samereason, althoughhedoesn’tpursue the matter. As Watson notes, optimizingalossfunction atparticular frequenciescorresponds to constructing an analog estimator alongthe linesofManski(1988). . 3b. Multlvmi~’t~ The multivariateanalogof our earfierloss functionis cGw(e)= /G(Fm(O; 0), F*(w)) 0 W(W) d~, o where ~ denotes component-by-component multiplication. The multivariateanalogof our earlierunivariatequadratic lossfunction,for example,is C~i~,(0)= ~ tr(D‘(uj; 6) D(uj; 6)), 2nj where D(oj; o) = F~(uj; 6) - F *(mj), Qj = — j ~ ~7..,, —T—~e T’ 2 The estimationcriterion functionhasthe sameform asinthe univariatecase, 0 = argefi cGw(e), Gw andthe bootstrap approaches to computing standard errors, confidenceintervals,andhypothesis testingparallelthe univariatecase precisely. Furthermore, asexpected, the multivariateGaussian ML estimator emerges as a specialandpotentiallyrestrictive case;itis Itisworth emphasizinghow allparts of the spectrum contribute to lossin themultivariatecase. Consider,for example,abivariatemodel (variablesx andy)under quadratic loss. Then

-12dxx(~j;0) dxy(uj;0) D(wj; 0) = dYX(~j;6) dn(mj; 6) ‘ [ 1 where ‘y,(@j;e) = fy.xm(oj;e) - $~(@j)= fxym(tij;e) - fx;(mj) = ~(oj; 0). Thus, tr(D/(wj; O)D(Oj;O))= [d~(~j;e) ~ dXY(tij;O)dYX6(o)]j; ~ [~(uj;e) + dxy(oj;6)dyx(w$j;)] = [f . X . xw(tij;0) - fx;(oj)]z + 2[re(fXY@(0j6;)) - re(fX~.(Oj))]z Thisexpression shows clearlyhow the goodness offit of both univariatespectra, aswe~as both thereal and imaginaryparts of the cross spectrum, contribute to loss. 4. Applicatio~~:‘1’heU.S. Cattle Cycle Letus beginby summarizingthe framework for assessingandestimatingdynamic stochasticmodelsdevelopedin sections2 and 3 ofthispaper. We firstperform afill seconciorder comparison ofmodelanddata byvisuallycomparing modelspectra, data spectra, and associated confidencetunnelsabout thedata spectra computed usilg the simpleCholeskyfactor bootstrap. To forma~yassessdi~’ergencebetween modelanddata spectra, andto estimate model

-13parameters, we develop anexplicitlossfunctionthat reflects the specificobjectivesof the investigation. Finally,we assessthe samplingdistributionsofestimatedparameters againusing the Choleskyfactor bootstrap. It iswellknown that cattle stock andconsumption are among themost periodictime seriesineconomics, with acycleofroughlyten yearsinU.S. (“thecattle cycle”). In this section, weprovide adetailedillustrationofthe use of our assessmentandestimationtechniquesby applyingthemto an important modelofthe cattle cycledeveloped byRosen, Murphy, and Scheinkman(RMS, 1994). This simpleyetrich modelallowsus to illustratevery clearlythe applicationof allthe tools inour framework, andmoreover, our findingsprovide new insightinto the RMS modelandits agreement with thedata. 4a. The Data We use amual data on U.S. cattle consumption and stock, 1900-1989.17We plot the seriesinFigures 1and 2, andthe cycleisvisuallyapparent. Moreover, the seriesare clearly trending. FollowingRMS, we remove alineartrend horn each seriesprior to additionalanalysis, allowingfor a break inthe slopeof the trend in 1930.~g We present the estimateddata spectrum inFigure3.19We makeuse -- here and throughout -- of amatrix graphicwith univariatespectraplotted on the maindiagonal,coherence inthe upper-right corner, andphase inthe lower-left corner. Not allfrequenciesme ofequal interest, however. The frequenciesmost relevantto aninvestigationofthe cattle cycle,typically thought to haveaperiod ofroughlyten years, are not those inthe entire [0, m]range, butrather those inasubset that excludesvery low andvery highfrequencies. Thispresents noproblemfor our procedures andinfactprovides a good opportunity to illustratethe ease with whichtheycan betailored to study specificapplications. Thus, for much of our analysis,we concentrate on the 17The data were kindlysuppliedbySherwinRosen and were originallyobtainedfrom Historical Statistics: Colonial Times to 1970 andAgricultural Statistics, publishedbythe U.S. Department of Agriculture. 18The fitted trends are also shown inFigures 1and 2. 19We smooth the sampleautocovariancefunctionusingaBartlett windowwith truncation lag 24.

-14fiequency bandcorresponding to periods of 30 yearsto 4 years,indicatedbythe shadedregion in Figure 3 and subsequentfigures. Four features ofthepoint estimates of the data spectrum stand out. First, the consumption spectrum (and to alesser extent, the stock spectrum) displaysapower concentration atroughly a ten-year cycle. Second, both theconsumption and stock spectra otherwise have Granger’s(1966) typicalspectral shape, with highpower atlow frequencies,and decliningpower throughout the fi-equencyrange. Third, the coherence between consumption and stock is generallyhighandvaries across frequencies,with amatium (about .85) atroughly aten-year cycle. Finally,the phase SM varieswith frequency;withinthe bandofinterest, the maximum (about oneyear) isagainatroughly aten yearcycle.20 In Figure4 wepresent the data spectrum alongwith 90% cotidence tumels computed usingthe conservativeBotierroni techniqueinconjunctionwith the Cholesky-factorbootstrap.21 To facilitateevaluation,weplot the consumption and stock spectra on alogarithmicscale.22All ofthepoint estimatesdisplaysubstantialuncertainty,asmanifestinthe 90Y0confidencetunnels. Such uncertaintyassociated withestimated spectra istypicalof economictime series,althoughit often goes unacknowledged. 4b. The Model We beginwith someaccountingidentities. The headcount of allanimals(yt) isthe sum of the adultbreeding stock (xt), the stock ofcalves(assumedequalto gxt-l), and the stock of yearlings(assumed equalto gxt-z), where g isafertilityparameter. That is, Y~= x~+ gx~-~+ gxt-~. The adultbreedingstock consists of survivingstock fi-omthepreviousperiod (assumed equalto 20Phase shiftismeasured inyearsbywhichconsumptionleadsstock. 21The detrended consumptionand stock data are neverthelesshighlypersistent. We present someMonte Carlo evidenceinAppendix 1that indicatesthat the Choleskyfactor bootstrap performs wellin suchstationary, buthighlypersistent, environments. 22From thispoint onward, we adopt thelog scalefor consumption and stock spectra wheneverconfidencetunnelsare included.

-15- (1-b)xt-l ) and the yearlingsfrom t-1 enteringthe adultherd (gx~-~) lessthe numberthat are marketed (ct), Xt= (1–a)xt-l + gx~.g– Ct” We are concerned with the equilibriumdeterminationofctand yt. Therisk-neutral rancher maximizesthepresent discounted valueof expectedprofits, whichinvolvesequatingthe expected marginalbenefitofmarketing an animalfor consumptionto the expected marginalbenefitof holdingthe animalfor breedtig. First, supposethat therancher markets the animalfor consumption. He receivesnet revenue qt=p~–mt,where ptisprice and ~ isfishing cost. Alternatively,suppose therancher holdsan animalfor breeding. Expected discounted netrevenue isthe sumofexpected discounted revenue from sellingtomorrow plusexpected discounted revenue from marketing its offspring,lessexpected total holdingcosts (~, Et[~(l -b)q~+l‘p3gqt,g-zt]. Total holdingcost equalsthe sumoftimet holdingcosts (h~), discounted holdingcosts of theresultant timet+l calves,and discounted holdingcosts of the resultant timet+2 yearlings. That is, zt=ht+~gyO\+l+~2gy1ht+2(assumingproportional COStSfor calvesandyearlings, yOand YI). In equilibrium,the expected marginalnetrevenue from marketingfor consumptionequals the expected marginaldiscounted netrevenue from holdingfor breeding;that is, Et[qt] = Et[~(l -~)qt+l + ~3gqt+g- ‘tl” We closethe modelby specifyingthe exogenousprocesses {~, ht, d~ asfirst-order autoregressions.23FollowingRMS, we assumethat each ofthe three shockshascommon serialcorrelation parameter p. The model structure impliesthat thereduced-form equationsfor ctand y,can beexpressed interms of a singledisturbance, mt,whichisalinearcombinationof the independentinnovations horn the three AR(1) drivingprocesses. In particular, c~-ARMA(2,1) and yt-ARMA(4,2): (l-klL)(l-pL) C, = -(1 - $IL) Ut 23dtisapreference shock. We havenot discussedthe demand sideof the model,because we do not use itinestimation.

-16- (l-LIL)(l -(J)2L)(l-@gL)(l-pL) y, = (1 + gL + gL2) ~,> where @listhe oneunstableroot and {+2, $3} are the two stableroots of 0’ - (1-8)$2 g = (), and Al isthe one stableroot of gp3a3 + (1-5)pA - 1 = o. The associated univariatespectra are 1(1- @,eio)12 fc(m) I(1 - ~leio)(l - pei”)12 1(1 + geio + ge2io)12 f},(m)= 0: @2ei0)(l - @3ei@)(l- pei0)12 ‘ 1(1 - ~leio)(l andthe cross spectrum is -(1 - @lei@)(l - $2ei”)(l - $3ei@)f (U) fq(m) = (1 + Y“ geio + ge2io) These equationsprovide a fulldescription ofthe modelinthe frequencydomain. o: isa complicatedfunction ofthe structural parameters, includingsomehorn the demand sideof the model. Alloftheparameters ofpresent interest, however, maybe identitle.dfrom the other reduced-form parameters, with the exception ofyOandyl. We therefore treat o; as afree parameter andestimate it subjectto no restrictions. RMS do not estimatethe cattle cyclemodel. Rather, theychoose valuesfor the behavioralparameters andreport that thecalibratedmodelfitsthedata w’ell.In the following section,we explicitlyestimate themodelandcompare our findingsto those ofRMS. 4c. Asse,ssing.Est’wtinc. andTestin~the Model To assess agreement between aparametrized version ofthe modelandthe data, or to

-17estimateparameters formally,it isnecessaryto construct an explicitloss function. We use aloss functionthat explicitlyincorporates the focus inthe cattlecycleliterature on cyclesofroughly 10 years. The loss function,whichmeasuresdivergencebetween modelanddata spectra onlywithin aparticular frequencyband,leadsus to anestimator that we callband-restricted maximum Welihood (Band-ML). We excludefrequenciescorresponding to periods ofmore than 30 years or lessthan 4 years.~ From the standpointof our earlierdiscussionoffrequencydownweighting, thiscorresponds to weightingfrequenciesinthe band ofinterest equally,and givingtiequencies outsidethe band zero weight. In Figure 5 we displaythe modelspectrum evaluatedat theBand-ML parameter estimates. Giventhe objectiveofconstructing a simplemodelthat isconsistent withperiodic behaviorinthese series,a surprisingfindingisthat neitherthe consumptionnor the stock model spectrum hasapeak corresponding to aten-year cycle. Instead, the maindistinguishingfeature of both modelspectra isGranger’s (1966) classicspectral shape. Thissuggests that at the band-ML optirnu~ themodeldoes not easilyproduce cyclicalbehavior. The modelphase shiftalso declinesmonotonically,whichcontrasts somewhatwith thepoint estimateof thephase shift, whichhas alocalpeak atroughly the ten-year cycle. Finally,the modelcoherenceremindsus of yet another of themodel’slimitations: becauseit isdrivenbya singleshock,themodelis singular,whichproduces unitcoherence at allfrequenciesregardless of theparameter configuration. To evaluatedivergencebetween modelanddata, weplot the model spectruminFigure 6, together withthe earlier-discussed90% confidencetunnelsfor the data spectrum, produced with 200replicationsof the non-parametric Choleskyfactor bootstrap.25The diagonalelements provide comparative assessmentsofmodelanddata univariatedynamics,andthe off-diagonal elementsprovide comparative assessmentsofcross-variabledynamics. wGaussian Band-ML isthe maximumlikelihoodanalogofEngle’s(1974) band-spectral linearregression. Band-ML mayofcourse beundertaken for modelsmuchmore complicated than simplelinearregression, suchas thepresent one. 25When constructing the bootstrap confidencetunnel,we applyaBartlett window to the off-diagonalelementsof the covariancematrix,and we use atruncation lag of24.

-18- Figure 6reveals somedivergencebetween modelanddata beyondthe earlier-discussed fact that the model spectrumfailsto displaytheinternalspectralpeaks found inthe data spectrum. First, therate of decay ofthe modelconsumption spectrum appears significantlyslower than that ofthe data spectrum thus, althoughtheconsumptionmodelanddata spectra agree over most of therelevantfrequencyrange, they beginto deviate substantiallyfor cycleswithperiods of4 years or less. Second, andconversely,therate ofdecay ofthe modelstock spectrum appears significantlyfaster than that ofthe data spectrum. The two divergenot onlyat highfrequencies, but also over much oftherelevantfrequencyrange. In particular, the model stock spectrumlies slightlyoutsidethe lowerregion ofthe 9070confidencetunnelfor cyclesof about 20 yearsand less. Third, thephase shiftimpliedbythe modeltends to be significantlylarger than thephase shiftfound inthe data over the frequenciesofinterest. Finally,modelanddata coherence diverge; in spiteofthe fact that the confidencetumel isverywide, the unitmodelcoherence is always outsidethe confidencetunnelfor the data coherence. Let us now discussthe band-MLestimationingreater detail. We estimatemodel parameters usingthe simplexalgorit~ whichisa derivative-freemethod, asimplementedinthe Matlabfmins.mprocedure. Usingpenaltyfunctions,we constrain the discountfactor to be between 0.65 and 1.00,the fertilityrate to bebetween 0.00 and 1.00,the death rate to bebetween 0.00 and 1.00,thepersistenceparameter to bebetween 0.00 and 1.00,and the scaleparameter to bebetween 0.10 and7.00. We start theiterations with theRMS parameter valuesfor the discountrate, fertilityrate, death rate, andpersistence parameter.2GIn our experience,estimation isnumericallystraightforward and stable;the estimatedparameter vector isalwaysintheinterior of the constraint set, convergenceisfast, and alternativestartingvaluesproduce virtuallyidentical estimates. In contrast, the RMS modelhasproven to k quite dficult to estimateusingmore traditionalapproaches. For example,Hansen, McGrattan, and Sargent (1997) findthat standard time-domainML failsto converge unlessthe discountfactor isfixedprior to estimation. Wepresent the Band-ML estimates andthe RMS parameter valuesinTable 1. We have two mainfindings. First, severaloftheparameter valuesobtainedbyband-restricted maximum likelihoodare similarto those chosen byRMS. Inparticular, the estimate of thedeath rate 26RMS did notreport avaluefor the scaleparameter; we start it at 1.7.

-19parameter (.08) isnearlyidenticalto the RMS value(O.10),andthe estimate oftheproducer’s discountfactor (.86) iscloseto the RMS value(0.91). The estimated fertilityparameter (0.67) is lower than but neverthelesscloseto the RMS value(0.85), whichRMS choose based on biologicalconsiderations. Our second mainfindingisthat the bandML estimate ofthepersistence parameter, which isa fundamentalobjectinthe RMS model,dfiers substantiallyfrom the RMS value. RMS choose a fairlypersistent valueof0.6. In contrast, we findthat optimizingthe band-MLloss functionrequires verylittlepersistence inthedrivingprocess (0.2). Thisirnpfiesthat the RMS modelhas a strong internalpropagation mechanism: the modeltakes shocks withrelativelylittle serialcorrelation andtransforms theminto seriesthat displaysubstantialpersistence in equilibrium. Thisdimensionof the RMS modeldiffersfundamentallyfrom standard dynamic equilibriummodelsused inmacroeconomics, internationaleconomics,andpublicfinance. As Watson (1993) and others havenoted, modelsused inthose fieldstypicallyhaveweak internal propagation mechanisms-- theyrequire highlypersistent underlyingshocksto generate arealistic amount of serialcorrelation inthe variablesdeterminedinequilibrium. Thisisconsidered to bea shortcomingofthe models andisthe focus of muchcurrent research. Thus, apotentially important contribution ofthe RMS modelisthat therich nature ofitsdynamicpropagation mechanismsmaybe adapted to helpresearchers inother fieldsconstruct modelswith stronger internalpropagation. In additionto findingtheparameter estimatesthat maximize agreement between model anddata, we can assess their samplinguncertaintywithinour framework. Standard errors are of someuse inthatregard, in spiteofthe fact that the samplingdistributionsneed not beGaussian. We compute themusing 200replicationsofthe Choleskyfactor bootstrap procedure, andwe report theminparentheses belowthe estimatedparameters inTable 1. More generally,our bootstrap procedures allowus to estimatethe entire samplingdistributionsofthe estimated parameters; we report intheminFigure7. The estimated samplingdistributionsof the discount factor, the depreciationrate, andthepersistenceparameter are fairlyconcentrated, whilethe estimated samplingdistributionoffertilityrate ismore dispersed. Our framework also enablesus to examinethejoint distributionofthe estimated parameters. In Table2 wepresent bootstrap estimates ofthe correlations between the estimated

-20parameters. Perhaps the most interestingrelationshipisthe strong negativecorrelation between the discountfactor andthefertilityrate, whichoccurs becausethe discountfactor and thefertility rate enter multiplicativelyinone ofthe cubicequationsthat definethe ARMApolynomials. This impliesthat the loss functiontrades offhighfertilityrates for low discountfactors, and suggests that fixingeither one oftheparameters at the higherRMS valuewould tend to result inan even lowerestimatefor the other. 5. Concluding Remarks We havedescribed aframework for evaluatingdynamiceconomicmodelsthat shouldbe usefil to appliedeconomistsinmanyfields. The framework isflexible-- it can beused by researchers to formallyevaluatepurelycalibratedmodels, anditcan also beused byresearchers interested inestimatingparameters andconductinginference. Moreover, itisgraphicaland constructive, andittakes seriouslyseveralimportant issuesinthe quantitativeanalysisof simple, dynamicequilibriummodels: modelmisspecification,the user’sobjectives,and smallsamplesizes. Its frequency-domainfoundationsprovide usefuldiagnosticsthat nicelycomplementalternative time-domainapproaches, such as Canova,FinnandPagan’s(1994) approach based on estimated VARS. Our analysisof the RMS modelofcattle cyclesNustrated the use of our tools for assessingagreement between modelsanddata atpre-set pmameter values, aswellasfor forma~y estimatingmodels andperforming statisticalinference. In addition,it shednew lighton the characteristics ofthe RMS model, andinparticular, its strong internalpropagation mechanism. Our analysisalsorevealed severaldeficienciesofthe model, not the least ofwhichisitsinabilityto generate internalspectralpeaks inthe model spectra evaluated atthe band-MLestimates. The ultimategoal oftheresearch program of whichthispaper isapart isto facilitate communicationbetween researchers withpotentiallyverydtierent research objectivesand strategies, thereby bringingmodern dynamiceconomictheory into closer andmore frequent contact withdynamiceconomicdata. Aseconomists usericher andmore complicatedmodelsto understand a widervariety ofdata, we hopethat our framework willfinduse indiscerningthe dimensionsalongwhichmodelsare consistent -- andinconsistent-- with data. That information can inturn beused to construct new andimprovedmodels.

-21- References Auerbach,A.J. and Kotlikoff,L.J. (1987),Dynamic FiscalPolicy. Cambridge: Cambridge UniversityPress. Backus, D.K., Kehoe, P.J. and Kydland,F.E. (1994), “Dynamicsofthe Trade Balanceandthe Terms ofTrade: The J-Curve ?,”American Economic Review, 84,84-103. Canova,F., Finn,M., andPagan, A.R. (1994), “EvaluatingaRealBusinessCycleModel,”inC.P. Hargreaves (cd.),Nonstationary Time Series and Cointegration. Oxford: Oxford UniversityPress. Christian, L.J, andEichenbau~ M. (1995), “LiquidityEffects, Monetary Policyandthe BusinessCycle,”Journal of Money, CreditandBanking, 27, 1113-1136. Cogley,T. andNason, J.M. (1995), “OutputDynamicsinRealBusinessCycleModels,” American Economic Review, 85,492-511. Cooley,T.F. and Hansen, G. (1992), “TaxDistortions inaNeoclassicalMonetary Economy,” Journal of Economic Theory, 58,290-316. Engle,R.F. (1974), “BandSpectrum Regression,”International EconomicReview, 15, 1-11. Ericson, R. andPakes, A. (1995), Markov-Perfect Industry Dynamics: AFramework for EmpiricalWork,”Review of Economic Studies, 62,53-82. Gallant,A.R., Rossi, P.E. andTauchen, G. (1993), “NonlinearDynamicStructures,” Econometric, 61,871-908. Granger, C.W.J. (1966), “TheTypicalSpectral Shape of anEconomicVariable,”Econometric, 34, 150-161. Greenwood, J., Hercowitz, Z. and KruseH,P. (1997), “LongRunImplicationsof Equipment-SpecificTechnologicalChange,”American EconomicReview, inpress. Gregory, A.W. and Smith,G.W. (1991), “Calibrationas Testing,”Journal ofBusiness and Economic Statistics, 9,297-303. Hannan,E.J. (1970), Multiple Time Series. New York: JohnWiley. Hansen, G.D. (1985), “IndivisibleLabor andthe BusinessCycle,”Journal ofMonetary Economics, 16,309-327. Hansen, L.P and Heckman, J.J. (1996), “TheEmpiricalFoundationsof Calibration,”Journal of

-22- Economic Perspectives, 10,87-104. Hansen, L.P., McGrattan, E. and Sargent, T.J. (1997), “MechanicsofForming andEstimating DynamicLinearEconomies,“inJ. Rust, D. Kendrick,andH. Amman (eds.), Handbook of Computational Economics, inpress. Hansen, L.P. and Sargent, T.J. (1993), “Seasonalityand ApproximationErrors inRational Expectations Models,”Journal of Econometrics, 55,21-55. Jones, L. and Manuelli,R. (1990), “AConvexModelofEquilibriumGrowth: Theory andPolicy Implications,”Journal of Political Economy, 98, 1008-1038. ~ K. and Pagan, A.R. (1994), “TheEconometric Analysisof CalibratedMacroeconomic Models,”inH. Pesaran andM. Wickens (eds.), Handbook ofApplied Econometrics. Oxford: Blackwell. ~g, R.G. andWatson, M.W. (1992), “Onthe Econometrics of ComparativeDynamics,” Manuscript, Departments of Economics, UniversityofRochester andNorthwestern University. King,R.G. andWatson,’M.W.(1996), “Money,Prices, Interest Rates andthe BusinessCycle,” Review of Economics and Statistics, 78,35-53. Kydland,F. andPrescott, E.C. (1982), “Timeto BufidandAggregate Fluctuations,” Econometric, 50, 1345-1370. Kydland,F.E. andPrescott, E.C. (1996), “TheComputationalExperiment: AnEconometric Tool,” Journal of EconomicPerspectives, 10,69-86. Leeper, E.M. and Sims,C.A. (1994), “Toward aModern Macroeconomic Model Useablefor PolicyAnalysis,”in O. Blanchard and S. Fischer(eds.),NBER Macroeconomics Annual, 81-117. Lucas, R.E. (1977), “UnderstandingBusinessCycles,”Carnegie-Rochester Series on Public Policy, 5,7-29. Lucas, R.E. (1988), “OnThe Mechanicsof EconomicDevelopment,”Journal of Monetary Economics. 22,3-42. Lucas, R.E. (1990), “SupplySideEconomics: An AnalyticalReview,”Manuscript, Department ofEconomics, Universityof Chicago. Manski,C. (1988),Analog Estimation Methods inEconometrics.NewYork: Chapmanand Hall.

-23- Ohanian,L.E. (1997), “TheMacroeconomic Effects ofWar Financeinthe United States: World War II andthe Korean War,”AmericanEconomicReview,inpress. Pagan, A. (1994), “Calibrationand Econometric Research: An Overview,”inA. Pagan (cd.), CalibrationTechniquesandEconometrics,specialissueofJournalofApplied Econometrics,9,S1-S10. Priestly,M.B. (1981), SpectralAnalysisandTimeSeries. London: AcademicPress. Ramos, E. (1988), “ResamplingMethods for Time Series,”TechnicalReport ONR-C-2, Department of Statistics,Harvard University. Rebelo, S. (1991), “LongRun PolicyAnalysisandLong RunGrowth,” JournalofPolitical Economy,99,500-521. Rosen, S., Murphy, K.M. and Scheinkman,J.A. (1994), “CattleCycles,”JournalofPolitical Economy,102,468-492. Rotemberg, J. andWoodford, M. (1996), “RealBusinessCycleModelsand Forecastable Movementsin Output,”AmericanEconomicReview,86,71-89. Rust, J. (1989), “ADynamicPrograrnrningModel ofRetirementBehavior,” inD. Wise (cd.), The EconomicsofAging. Chicago: Universityof ChicagoPress. Sims,C.A. (1993), “RationalExpectations Modelingwith SeasonallyAdjustedData,”Journalof Econometrics,55,9-19. Sti, C.A. (1996), “Macroeconomicsand Methodology,”JournalofEconomicPerspectives,10, 105-120. Swanepoel,J.W.H. andvanWyk,J.W.J. (1986), “TheBootstrap Appliedto Power Spectral DensityFunction Estimation,”Biometrika,73, 135-141. Watson, M.W. (1993), “MeasuresofFitfor CalibratedModels,”JournalofPoliticalEconomy, 101, 1011-1041. Woodroofe, M.B. andvan Ness, J.W. (1967), “TheMaximumDeviationof SampleSpectral Densities,”AnnalsofMathematicalStatistics,38, 1559-1569.

Tabie 1 ParameterEstimates Band-Restricted Maximum Likelihood R;stimation o . Estmtlon or CalibrationMethod Band-ML .86 .67 .08 .21 2.10 (.03) (.09) (.03) (.10)(.37) RMs .909 .85 .10 .60 NA (NA) (NA) (NA) (NA) (NA) Notesto Table: ~ isthediscount factor, g isthe isthe fertilityrate, 5 isthe death rate, and p isthe persistenceparameter. Band-MLdenotes band-restricted rnaxirnumlikelihoodestimation,withthe frequencyband used for estimationcorresponding to periods horn 30 to 4 years. Standard errors, based on 200 bootstrap replications, appear in parentheses. RMS denotes the Rosen-Murphy- Scheinkmancalibratedparameters. (Theyhavenostandarderrors, becausetheywere not estimated.) Table 2 Estimated Parameter CorI*elations Band-Restricted Maximum Likelihood Estimation -.73 1.00 g 6 .49 -.37 1.00 P -.19 .10 .06 1.00 Notes to Table: ~ is the discount factor, g is the fertilityrate, 5 is the death rate, and p is the persistence parameter. Estimated parameter correlations are based on 200 bootstrap replications. The fi”equencybandused for esthnation corresponds to periods fi-om30 to 4 years.

—. k’lgure1 U.S. Cattle Consumption, 1900=1990 Actual and Estimated Trend 50 1 lo~ -- -- -- 00 10 20 30 40 50 60 -/u 8(.J 9U Yem Notes to Figure: We show cattle consumption (solid line) and the estimated kinked-lineartrend (dashedline). Figure 2 U.S. Cattle Stock, 1900-1990 Actual and Estimated Trend 140, 40*, ,,,,,,,,,,,,,,,,,,,,,,,,, ,,,,,,,,,,,,,,,,,,,,,,, ,,, -~q 00 10 20 30 40 50 60 70 go 90 Notes to Figure: We show cattle stock (solidline)and the estimated ktiked-linear trend (dashed line).

Figul*e3 Estimated Spectral Densit~’Matrix U.S. Cattle Consumption and Stock 1.2 1.0 0.2 0.0 . — 0 JL n Frequency Frequency 5 700 4 600 500 3 200 o 100 -1 0 o n Frequency Frequency Notesto Figure: We detrenclalldata usingthe kinked-linearmethod. We showthepoint estimate ofeachelementofthespeetraldensitymatyix,The fre~~uencybandindicatedbyverticaldashed lines correspondsto cycleswithperiodsof30to 4 yearsandisthe bandofprimaryrelevancefor studying cattlecycles.

Figure 4 Estimated Spectral Densit}~Matrix and Confidence Tunnels U.S. Cattle Consumption and Stock o o n Frequency Frequency 15 10 -5 -10 0 o n n Frequency Frequency Notesto Figure: We detrenclafldata usingthekinked-linearmethod. We showthepoint estimate togetherwitha9070 confidencetunnelforeachelementofthespeetrd densitymatrix. The frequency bandindicatedbyverticaldashedlinescorresponds to cycleswithperiods of 30 to 4 years andisthe band ofprimaryrelevancefor studyingcattlecycles.

Figure 5 Model Spectrum Evaluated at Band-ML Estimates U.S. Cattle Consumption and Stock 1.2 1.0 ------------------- 0.8 0.6 0.4 0.2 0.0 Tc n Frequency Frequency 700 600 500 400 300 200 100 0 o o n n Frequency Frequency Notesto Figure: Wedetrendafldatausingthektiked-linear method. We show the model spectruln evaluated at the band-restricted maximumWelihood parameter values, for each element of the spectraldensitymatrix. Thefrequencybandindicatedbyverticaldashed linescorresponds to cycles withperiods of30 to 4 years andisthe bandofprimaryrelevancefor studyingcattle cycles.

I?igure6 Model Spectra, and Data Spectra Confidence Tunnels [J.S. Cattle Consumption and Stock 6 1.2 1.0 4 0.8 E *2 & m o 0.2 -2 0.0 Frequency Frequency 15 10 -5 -lo Frequency Frequency Notesto Figure: Wedetrendafldatausingthekinked-linearmethod. We show the 90% confidence tllnnel for the data spectrum together with the model spectrum evaluated at the band-restricted maxilnumlkelihoodparametervalues,foreachelementofthespectraldensitymatrix. The frequency bandindicatedbyverticaldashedlinescorresponds to cycleswithperiods of 30 to 4 years andisthe bandof prilnaryrelevancefor studyingcattle cycles.

l~igure7 Bootstrap Estimates of Sampling Distributions 140 140 a 120 120. 1 100 100- 80 Discount s 80- 60 8 60- Fertility 40 40. Rate - 20 20- 0 0.00 0.25 0.s0 0.75 I.QO 0.00 0.25 0.50 0.75 1.00 ParameterValue ParameterValue 140 140 1 120 120 1 100 100 80 Persistence 60 Parameter 40 20 20 0 0 - :- 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 ParameterValue ParameterValue Notes to Figure: Estimated samplingdistributionsare based on 200 bootstrap repficat.ions.

Appendix 1 Finite-Sample Properties of the Cholesky Factor Bootstrap In thisappendix,we describetheresults of aMonte Carlo comparison of the tite-sample properties ofthe Choleskyfactor bootstrap and conventionalasymptotic. The experimentis smallbynecessity,as Monte Carlo evaluationofbootstrap procedures isextremelyburdensome computationally,but we believethat it shedssomeinterestinglighton thefinite-sample performance of the bootstrap. We use a data-generating process withreahstic dynamics,givenby Yt = 1.335yt-1 - .401y~-z+ et, T = 1, e.., 100, whichcorresponds to Rudebusch’s(1993) estimatefor detrended log GNP andisrepresentative ofthedynamicsof a typicaldetrended macroeconomic series. We examinethe empiricalcoverage ofthe nominal80%and 90%intervalsconstructed usingthe Choleskyfactor bootstrap andconventionalasymptotic. We examinetwo bootstrap intervals,parametric (Gaussian) and nonparametric. At each of 1.000Monte Carlo replications, we applythe Choleskyfactor bootstrap with 2000 bootstrap replications. At each bootstrap replicationwe estimate the spectraldensityat frequencies n/6 and n/2. In TableAl, we present the empiricalcoverage rates for bootstrap and asymptotic confidenceintervalsfor three innovationdistributions. First, we set et- iidN(O,l). At frequency n/6, the actualcoverage ofallthree intervalsexceedsnominalcoverage. However, both the parametric andnonparametricbootstrap coverage rates are muchcloser to nominalcoverage than those ofthe asymptoticapproximation. At frequency n/2, the asymptoticintervalssimilarly deliverexcessivelyhighcoverage rates buttheparametric bootstrap intervalinparticular (andto a lesserextent the nonparametric) displaynearlyexact coverage. Second, we set et to aconditionallyGaussianGARCH(l,l). As expected, the nonparametric bootstrap outperforms theparametric bootstrap inthiscase. However, neitherthe nonparametric bootstrap nor the asymptoticapproximationappeardefinitivelybestinterms of actualcoverage. Finally,the innovationisiid X2(2),normalizedto havezero mean andunitvariance. As withiidN(O,1)innovations,we findthat the asymptoticapproximationtends to giverise to excessivelywideconfidenceintervals. At anominalcoverage levelof 9070,both bootstraps delivermore accurate coverage rates. At thenominal80%level,ody theparametric bootstrap dominatesthe asymptoticinterval. References Rudebusch,G.D. (1993), “TheUncertain Unit Root inRealGNP,” Anlerican Ec>onon~Riecview, 83,264-272.

Table Al EmpiricalCoverage Bootstrap and Asymptotic Confidence Intervals Parametric Nonparametric Nominal Bootstrap Bootstrap Asymptotic Coverage Interval Interval Interval . ausslanInnovat”o s f(:/;) .80 .827 .831 .912 .90 .913 .910 .974 f(n/2) .80 .795 .780 .827 .90 .904 .901 .980 . . . Cond-y Gauw (1.1) Innovations f(m/6) .80 .696 .718 .767 .90 .808 .838 .845 f(n/2) .80 .770 .818 .789 .90 .863 .905 .924 Standardized Chi-Sauar~ Innovations f(n/6) .80 .843 .862 .913 .90 .916 .933 .963 f(n/2) .80 .798 .852 .824 .90 .901 .939 .979 Notes to Table: For each innovationdistribution,we generate data from an AR(2) with parameters 1.335and -.401, with samplesizeT=1OO.We perform 2000 bootstrap iterationsin each of 1000Monte Carlo trials.