differsjrzj

EstimationofDSGEmodelswhenthedataarepersistent$YuriyGorodnichenkoa,,SerenaNgbaDepartmentofEconomics,UniversityofCalifornia,Berkeley,NBER,andIZA,USAbDepartmentofEconomics,ColumbiaUniversity,USAarticleinfoArticlehistory:Received9October2007Receivedinrevisedform17February2010Accepted19February2010Availableonline6March2010JELclassication:C32C5E32Keywords:PersistentdataFiltersTrendsUnitrootSpuriousestimatesBusinesscyclesabstractDynamicstochasticgeneralequilibrium(DSGE)modelsareoftensolvedandestimatedunderspecicassumptionsastowhethertheexogenousvariablesaredifferenceortrendstationary.
However,evenmilddeparturesofthedatageneratingprocessfromtheseassumptionscanseverelybiastheestimatesofthemodelparameters.
Thispaperproposesnewestimatorsthatdonotrequireresearcherstotakeastandonwhethershockshavepermanentortransitoryeffects.
Theseprocedureshavetwokeyfeatures.
First,thesamelterisappliedtoboththedataandthemodelvariables.
Second,thelteredvariablesarestationarywhenevaluatedatthetrueparametervector.
Theestimatorsareapproximatelynormallydistributednotonlywhentheshocksaremildlypersistent,butalsowhentheyhavenearorexactunitroots.
Simulationsshowthattheserobustestimatorsperformwellespeciallywhentheshocksarehighlypersistentyetstationary.
Insuchcases,lineardetrendingandrstdifferencingareshowntoyieldbiasedorimpreciseestimates.
&2010ElsevierB.
V.
Allrightsreserved.
1.
IntroductionDynamicstochasticgeneralequilibrium(DSGE)modelsarenowacceptedastheprimaryframeworkformacroeconomicanalysis.
Untilrecently,counterfactualexperimentswereconductedbyassigningtheparametersofthemodelswithvaluesthatarelooselycalibratedtothedata.
Morerecently,seriouseffortshavebeenmadetoestimatethemodelparametersusingclassicalandBayesianmethods.
Thispermitsresearcherstoassesshowwellthemodelstthedatabothinandoutofsamples.
Formalestimationalsopermitserrorsarisingfromsamplingormodeluncertaintytobeexplicitlyaccountedforincounterfactualpolicysimulations.
Arguably,DSGEmodelsarenowtakenmoreseriouslyasatoolforpolicyanalysisbecauseofsuchseriouseconometricinvestigations.
AnyattempttoestimateDSGEmodelsmustconfrontthefactthatmacroeconomicdataarehighlypersistent.
ThisfactoftenrequiresresearcherstotakeastandonthespecicationofthetrendsinDSGEmodels.
Specically,totakethemodeltothedata,aresearcherneedstousesampleanalogsofthedeviationsfromsteadystatesand,indoingso,mustdecidehowtodetrendthevariablesinthemodelandinthedata.
Table1isanon-exhaustivelistingofhowtrendsaretreatedinContentslistsavailableatScienceDirectjournalhomepage:www.
elsevier.
com/locate/jmeJournalofMonetaryEconomicsARTICLEINPRESS0304-3932/$-seefrontmatter&2010ElsevierB.
V.
Allrightsreserved.
doi:10.
1016/j.
jmoneco.
2010.
02.
008$ThispaperwaspresentedatBrownUniversity,theUniversityofMichigan,the2007NBERSummerInstitute,PrincetonUniversity,UCBerkeley,andtheNewYorkAreaMacroConference.
WethankMarcGiannone,TimCogley,AnnaMikusheva,andtwoanonymousreferees,theAssociateEditorandtheEditorformanyhelpfulcomments.
ThesecondauthoracknowledgesnancialsupportfromtheNationalScienceFoundation(SES0549978).
Correspondingauthor.
E-mailaddress:ygorodni@econ.
berkeley.
edu(Y.
Gorodnichenko).
JournalofMonetaryEconomics57(2010)325–340somenotablepapers.
1Somestudiesassumestochastictrendsforthemodelanduserstdifferenceddatainestimation.
Anumberofstudiesspecifydeterministictrendsforthemodelanduselinearlydetrendeddatainestimation.
StudiesthatapplytheHodrick–Prescott(HP)ltertothedatadifferinwhattrendsarespeciedforthemodel.
Someassumesimplelineartrends,whileothersassumeunitrootprocesses.
Table1demonstratesthatavarietyoftrendshavebeenspeciedforthemodelandavarietyofdetrendingmethodshavebeenusedinestimation.
Theproblemforresearchersisthatitisnoteasytoascertainwhetherhighlypersistentdataaretrendstationaryordifferencestationaryinnitesamples.
Whilemanyhavestudiedtheimplicationsforestimationandinferenceofinappropriatedetrendinginlinearmodels,2muchlessisknownabouttheeffectsofdetrendinginestimationofnon-linearmodels.
FromsimulationevidenceofDoorn(2006)foraninventorymodel,itseemsthatHPlteringcansignicantlybiastheestimateddynamicparameters.
Whilethelocal-to-unitframeworkisavailabletohelpresearchersunderstandthepropertiesoftheestimatedautoregressiverootwhenthedataarestronglypersistent,itisuncleartowhatextenttheframeworkcanbeusedinnon-linearestimationeveninthesingleequationcase.
WhatmakesestimationofDSGEmodelsdistinctisthattheyconsistofasystemofequationsandmisspecicationinoneequationcanaffectestimatesinotherequations.
Thispaperdevelopsrobustestimationproceduresthatdonotrequireresearcherstotakeastandonwhethershocksinthemodelhaveanexactoranearunitroot,andyetobtainconsistentestimatesofthemodelparameters.
Allrobustprocedureshavetwocharacteristics.
First,thesametransformation(lter)isappliedtoboththedataandthemodelvariables.
Second,thelteredvariablesarestationarywhenevaluatedatthetrueparametervector.
TheestimatorshavetheclassicalpropertiesofbeingTpconsistentandasymptoticallynormalforallvaluesofthelargestautoregressiveroot.
Ourpointofdepartureisthattherathercommonpracticeofapplyingdifferentlterstothemodelvariablesandthedatacanhaveundesirableconsequences.
Aswillbeshownlater,estimatesofparametersgoverningthepropagationandamplicationmechanismsinthemodelcanbeseverelydistortedwhenthetrendspeciedforthemodelisnotconsistentwiththeoneappliedtothedata.
Weinsistonestimatorsthatapplythesametransformationtoboththemodelandthedata.
This,however,maystillleadtobiasedestimatesifthelterdoesnotremovethetrendsactuallypresentinthedata.
Accordingly,oneneedstoworkwithltersthatcanremovebothdeterministicandstochastictrendswithouttheresearcherARTICLEINPRESSTable1Summaryofselectedwork.
PaperEquationsForcingvariableModellterDatalterEstimatorKydlandandPrescott(1982)SystemARMA(1,1)LTHPCalibrationAltug(1989)SystemI(1)FD1FD1MLEChristianoandEichenbaum(1992)SystemI(1)ztHPGMMBurnsideetal.
(1993)SystemAR(1)LTHPGMMBurnsideandEichenbaum(1996)SystemI(1)ztHPGMMMcGrattanetal.
(1997)SystemVAR(2)LTLT,HPMLEFuhrer(1997)EquationNotspeciedNotspeciedHP,LT,QTGMMClaridaetal.
(2000)EquationAR(1)NotspeciedLT,HP,CBOGMMKim(2000)SystemAR(1)LTLTMLEIreland(2001)SystemAR(1)LTLTMLESmetsandWouters(2003)SystemAR(1)LTHPBayesianDib(2003)SystemAR(1)LTLTMLEFuhrerandRudebusch(2004)EquationNotspeciedNotspeciedHP,CBO,QTMLE,GMMLubikandSchorfheide(2004)SystemAR(1)LTHP,LTBayesianAltigetal.
(2004)SystemARI(1,1)FD1FD1GMMIreland(2004)SystemI(1)FD1FD1MLEBouakezetal.
(2005)SystemAR(1)LTLTMLEChristianoetal.
(2005)SystemNotspeciedNotspeciedVARGMMDelNegroetal.
(2007)SystemARI(1,1)FD1FD1BayesianFaia(2007)SystemAR(1)LTHPCalibrationSmetsandWouters(2007)SystemAR(1)FDFDBayesianNote:CBOdenotesactualseriesminustheCongressBudgetOfce'smeasureofpotentialoutput.
I(1)andARI(1,1)denoteforcingvariableswithstochastictrends.
ARandARMAdenotetrendstationaryforcingvariables.
VARdenoteslteringwithavectorautoregressionwhichcanaccommodatetrendanddifferencestationaryprocesses.
FDisrstdifferencing,FD1isrstdifferencingwiththerestrictionthattheforcingvariablehasaunitroot(e.
g.
,rz1),LTisprojectiononlineartimetrend,QTisprojectiononquadratictimetrend,HPisHodrick–Prescottlter,ztisdetrendingbytheleveloftechnology.
Thesecondcolumnshowswhetherapaperestimatesasystemofequations(''system'')orasinglestructuralequation(''equation'').
1AsofJune2009,thesepaperswerecitedalmost2500timesattheWebofScience(formerSocialScienceCitationIndex)andalmost8000timesatGoogleScholar.
2Forexample,NelsonandKang(1981)showedthatlineardetrendingaunitrootprocesscangeneratespuriouscycles.
CogleyandNason(1995a)foundthatimproperlteringcanalterthepersistenceandthevolatilityoftheserieswhilespuriouscorrelationsintheltereddatawasdocumentedinHarveyandJaeger(1993).
Singleton(1988)andChristianoanddenHaan(1996)discussedhowinappropriatelteringcanaffectestimationandinferenceinlinearmodels.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340326takingastandbeforesolvingandestimatingthemodel.
Theideaofapplyingrobustlterstoboththemodelandthedataisnotnew.
ChristianoanddenHaan(1996)aswellasBurnside(1998)appliedtheHPltertoboththemodelandthedata,buttheyhadtoresorttoestimationbysimulationstogetaroundthelargestatevectorthattheHPlterinduces.
ThelterstobeconsideredhavethesamedesirablefeatureastheHPinthattheyadapttothetrendsinthedata.
However,theycanbeimplementedwithsimplemodicationstothestatespacesystemwhilekeepingthedimensionofthestatevectorsmall.
Specically,fourtransformationsareconsidered:(i)quasi-differencing,(ii)unconstrainedrstdifferencing,(iii)hybriddifferencing,and(iv)theHPlter.
AlllterscanbeusedinGMMestimationbutnoteverymethodcanbeimplementedinthelikelihoodframework.
Importantly,onecanusestandardasymptoticinferenceasthenitesampledistributionoftheestimatorsarewellapproximatedbythenormaldistributionnotonlywhenthelargeautoregressiverootisfarfromone,butalsowhenitisnearorontheunitcircle.
TheprocedurescanbeappliedtoDSGEmodelswhosesolutioncanbeshowntoexistandisunique,andcanbesolvedusingvariationsofthemethoddiscussedinBlanchardandKahn(1980)andSims(2002).
AsdiscussedinIskrev(2010)andKomunjerandNg(2009),DSGEmodelsaresusceptibletoidenticationfailure,inwhichcase,consistentestimationofparametersisnotpossibleirrespectiveofthetreatmentoftrends.
Inviewofthisconsiderationandtoxideas,thispaperusesasimplestochasticgrowthmodelwhosepropertiesarewellunderstood.
Themodel,whichwillbepresentedinSection2,willalsobeusedtoperformbaselinesimulationexperiments.
ThenewestimatorsarepresentedinSections3and4.
DiscussionoftherelatedliteratureisinSection4.
Sections5and6usesimulationstoshowthattherobustapproachesperformwellespeciallywhentheshocksarehighlypersistentyetstationary.
Theseresultsalsoholdupinlargermodelsthoughsomeltersaremoresensitivetothenumberofshocksthanothers.
Incontrast,lineardetrendingandrstdifferencingoftenleadtoseverelybiasedestimates.
ImplementationissuesarediscussedinSection7.
Section8concludes.
2.
PreliminariesConsidertheonesectorstochasticgrowthmodel.
TheproblemfacingthecentralplannerismaxEtX1t0btlogCtyLtsubjecttofeasibilityandtechnologicalconstraintsYtCtItKat1ZtLt1aKt1dKt1ItZtexpgtexpuzt;uztrzuzt1ezt;jrzjr1Letbmtbct;bkt;bltctgt;ktgt;ltmtmzwhereYtisoutput,Ctisconsumption,Ktiscapital,Ltislaborinput,Ztistheleveloftechnology,eztisaninnovationintechnology.
Notethatrzisallowedtobeontheunitcircle.
Letlowercaselettersdenotethenaturallogarithmofthevariables,e.
g.
ct=logCt.
Letc*tbesuchthatctc*tisstationary;k*tandz*taresimilarlydened.
Byassumption,laborLtisstationaryforallrzr1andthusl*t=0.
Collecttheobservedmodelvariablesintothevectormt=(ct,kt,lt)anddenotethetrendcomponentofthemodelvariablesbym*t=(c*t,k*t,l*t).
Ingeneral,howm*tisdened,howthemodelislinearizedandestimatedwilldependonwhetherrzo1orrz1.
SolvingthesystemofexpectationalequationsyieldsthereducedformbmtPbmt1Buztuztrzuzt1ezt1AsallrootsofPareassumedtobestrictlylessthanone,non-stationaritycanonlyarisebecauserzisontheunitcircle.
Notethatwhenrz1,themodelneedstobelinearizedandsolvedwithmtutgt;utgt;0zt;zt;0.
Despitethefactthatthepermanentshockuztisnowapartofmt*,(1)isstillthereducedformrepresentationforthelevelsofthelinearlydetrendedvariables.
Inotherwords,thereducedformrepresentationforbmtiscontinuousinrzeventhoughhowonearrivesatthisrepresentationwilldependonrz.
Hence,withoutlossofgenerality,therepresentation(1)willbealwaysusedinsubsequentdiscussionsforallvaluesofrz.
Notethatbydenition,bmtisthelinearlydetrendedcomponentofthemodelvariablesmt.
Inotherwords,bmtisamodelconcept.
Hereafter,letdtdenotethedataanalogofmt.
Forthestochasticgrowthmodel,dt=(ct,kt,lt)arethedataseries.
Letbdtbeobtainedbyremovingdeterministictrendsfromdt.
Thenbdtisthedataanalogofbmt.
3.
RobustestimatorsThissectionpresentsrobustmethodsthatdonotrequiretheresearchertotakeastandonthepropertiesoftrendsinthedata.
Thestochasticgrowthmodelisusedtoillustratetheintuitionbehindtheproposedmethods.
ARTICLEINPRESSY.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340327ManymethodshavebeenusedtoestimateDSGEmodels.
3Ourfocuswillbeamethodofmoments(MM)estimatorthatminimizesthedistancebetweenthesecondmomentsofdataandthesecondmomentsimpliedbythemodel,asinChristianoanddenHaan(1996)andChristianoandEichenbaum(1992).
AdaptationtolikelihoodbasedestimationwillbediscussedinSection7.
LetYdenotetheunknownstructuralparametersofthemodelandpartitionYY;rz.
ThegenericalMMestimatorcanbesummarizedasfollows.
Step1appliesalter(ifnecessary)todtandcomputesbOdj,theestimatedcovariancematrixofthelteredseriesatlagj.
CollectthedatamomentsinthevectorbodvechbOd00vecbOd10.
.
.
vecbOdM00Step2solvestherationalexpectationsmodelforaguessofY.
ComputeOmj,themodelimpliedautocovariancesofthelteredbmtanalyticallyorbysimulation.
CollectthemodelmomentsinthevectoromvechOm00vecOm10.
.
.
vecOmM00Step3estimatesthestructuralparametersasbYargminYJbodomYJThechoiceofmomentsinMMcanbeimportantforidentication(seee.
g.
CanovaandSala,2009).
Theunconditionalautocovariancesareusedinthispaperbutmatchingimpulseresponsescanalsobeconsidered.
AlthoughMMissomewhatlesswidelyusedthanmaximumlikelihoodestimatorsintheDSGEliterature,itdoesnotrequireparametricspecicationoftheerrorprocessesanditiseasytoimplement.
Aswillbediscussedlater,themoreimportantreasonforusingMMispracticalasitcanbeusedwithmanypopularlters.
Therobustapproachesconsideredherealwaysapplythesameltertothemodelandthedatasothatthelteredvariablesarestationarywhenevaluatedatthetrueparameterofrz,whichcanbeoneorclosetoone.
ThestatisticalpropertiesofbYwilldependonrzandtheltersused.
Thenextfoursubsectionsconsiderfourlters.
Section4thenexploreswhichofthesehavebetternitesamplesproperties.
Propertiesofestimatorsthatdonothavethesefeatureswillalsobecomparedlater.
3.
1.
TheQDestimatorLetDrz1rzLbethequasi-differencing(QD)operatorandletDrzbmt1rzLbmt.
Multiplyingbothsidesof(1)byDrzandusinguztrzuzt1eztgivesDrzbmtPDrzbmt1Bezt2Notethattheerrorterminthequasi-differencedmodelisani.
i.
d.
innovation.
AsDrzbmtisstationaryforallrzr1,itsmomentsarewelldened.
Incontrast,themomentsofbmtarenotwelldenedwhenrz1.
ThismotivatesestimationofYasfollows.
First,initializerz.
Second,quasi-differencebdtwithrztoobtainDrzbdt.
ComputebOdDrzjcovDrzbdt;Drzbdtj;thesampleautocovariancematrixofthequasi-differenceddataatlagj0;M.
DenebUdDrzjbOdDrzjbOdDrz0andletbodDrzvecbUdDrz10vecbUdDrzM00.
Third,foragivenrzandY,solveforthereducedform(1).
ApplyDrztobmtandcomputeOmDrzj;j1;M,themodelimpliedautocovariancematricesofthequasi-differencedvariables.
LetUmDrzjOmDrzjOmDrz0.
DeneomDrzrzvecUmDrz10vecUmDrzM00.
Fourth,ndthestructuralparametersbYQDargminYJbodDrzrzomDrzYJ.
TheQDestimatorisbasedonthedifferencebetweenthemodelandthesampleautocovariancesofthelteredvariables,normalizedbytherespectivevariancematrixODrz0.
TheQDdiffersfromastandardcovarianceestimatorinoneimportantrespect.
Theparameterrznowaffectsboththemomentsofthemodelandthedatasincethelatterarecomputedforthedataquasi-differencedatrz.
AsrzandYareestimatedsimultaneously,thelterisdatadependentratherthanxed.
Thecrucialfeatureisthatthequasi-transformeddataarestationarywhenevaluatedatthetruerz,whichsubsequentlypermitsapplicationofacentrallimittheorem.
ThenormalizationofthelaggedautocovariancesbythevarianceamountstousingthemomentscovDrzbdt;DrzbdtDrzbdtjcovDrzbmt;DrzbmtDrzbmtjforestimation.
Thej-thdifferenceofDrzbdtisalwaysstationaryandensuresthattheasymptoticdistributioniswellbehaved.
Finally,observethatsincethemodelissolvedinlevelsandthetransformedvariablesareusedonlytocomputemoments,allequilibriumrelationshipsbetweenvariablesarepreserved.
ARTICLEINPRESS3ThisincludeslikelihoodandBayesianbasedmethods(e.
g.
Fernandez-VillaverdeandRubio-Ramirez,2007;Ireland,1997),two-stepminimumdistanceapproach(e.
g.
,Sbordone,2006),aswellassimulationestimation(e.
g.
,Altigetal.
,2004).
Ruge-Murcia(2007)providesareviewofthesemethods.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–3403283.
2.
TheFDestimatorIfDrzbmtisstationarywhenrzr1,thedatavectorisalsostationarywhenquasi-differencedatrz1.
Denotetherst-differencing(FD)operatorbyD1Landconsiderthefollowingestimationprocedure.
First,computebOdDjcovDbdt;Dbdtj,thesampleautocovariancematrixoftherstdifferenceddataatlagj1;M.
DenebodDvechbOdD10vecbOdDM00.
Second,foragivenY,solveforthereducedform(1).
ComputeOmDj,themodelimpliedautocovariancematricesoftherst-differencedvariablesDbmt.
DeneomDvechOmD00vecOmDM00.
Third,ndthestructuralparametersbYFDargminYJbodDomDYJ.
Tobeclear,theautocovariancesarecomputedfortherstdifferenceddataandthemodelvariables,butrzisafreeparameterwhichisestimated.
NotethattheQDandFDestimatorsareequivalentwhenrz1.
ThekeydifferencebetweenFDandQDisthatFDisaxedlterwhiletheQDisadatadependentlter.
3.
3.
ThehybridestimatorOnedrawbackoftheFDestimatoristhatwhenrzisfarfromunity,over-differencinginducesanon-invertiblemoving-averagecomponent.
Theestimatesobtainedbymatchingasmallnumberoflaggedautocovariancesmaybeinefcient.
TheQDestimatordoesnothavethisproblem,butbOdDrzjisquadraticinrz.
Aswillbeexplainedbelow,thisiswhybOdDrzjwasnormalizedbybOdDrz0.
Theseconsiderationssuggestahybrid(HD)estimator.
First,transformtheobserveddatatoobtainDrzbdt(asinQD)andDbdt(asinFD).
Second,computebOdQD;DjcovDrzbdt;Dbdtj.
DenebodQD;DvecbOdQD;D00vecbOdQD;DM00.
Third,foragivenY,solveforthereducedform(1),andcomputethemodelimpliedautocovariancesbetweenthequasi-differencedandtherstdifferencedvariables.
DeneomQD;DvecOmQD;D00vecOmQD;DM00.
Fourth,ndthestructuralparametersbYHDargminYJbodQD;DrzomQD;DYJ.
NoticethatbOdQD;Djisnowlinearinrz,unlikebOdDrzj.
3.
4.
TheHPestimatorLinearlterssuchastheHPandthebandpasscanalsoremovedeterministicandstochastictrends,seeBaxterandKing(1999)andKingandRebelo(1993).
TheHPdetrendedseriesisdenedasHPLdtl1L21L121l1L21L12dtTheestimatorcanbeconstructedasfollows.
First,computetheautocovariancematricesoftheHP-ltereddatabOdHP0bOdHPM.
DenebodHPvechbOdHP00vecbOdHPM00.
Second,foragivenguessofY,solveforthereducedform(1),andcomputeOmj,theautocovariancesofbmt.
ApplytheFouriertransformtoobtainthespectrumforbmtatfrequencies2ps=T,s0;T1.
MultiplythespectrumbythegainoftheHPlter.
InverseFouriertransformtoobtainOHPj,theautocovariancesoftheHP(L)bmt.
DeneomHPvechOmHP00vecOmHPM00.
Third,ndthestructuralparametersbYHPargminY:bodHPomHPY:.
ThisapproachissimilartoBurnside(1998)whoalsorstappliestheHPltertoboththemodelandthedataseries,andthenusessimulationstocomputemodel-impliedmoments.
LiketheFD,rzdoesnotenterthelterbutboththeltereddataandthelteredmodelvariablesarestationaryforallrzr1.
NotethatHPlteringinvolvesestimationofmanymoreautocovariancesthantheotherestimatorsconsideredabove.
4.
PropertiesoftheestimatorsLetbodjgenericallydenotethej-thsamplemomentsofthelteredvariableswhileomjYdenotethemodelmomentbasedonthesamelter.
DenegjYbodjomjYandletgYg0Y;g1YgMY.
ThentheMMestimatorbYminYJgYJisanon-linearGMMestimatorusinganidentityweightingmatrix.
Thissub-optimalweightingmatrixisusedbecausewhentherearefewershocksthanvariablesinthesystem,stochasticsingularitywillinducecollinearityinthevariablesresultinginamatrixofcovariancesthatwouldbesingular.
Evenifthereareasmanyshocksasendogenousvariables,AbowdandCard(1989),AltonjiandSegal(1996)andothersndthatanidentitymatrixperformsbetterthantheoptimalweightingmatrixinthecontextofestimatingcovariancestructures.
Theoptimalweightingmatrix,whichcontainshighordermoments,tendstocorrelatewiththemomentsandthiscorrelationunderminestheperformanceoftheestimator.
LetGYbethematrixofderivativesofgYwithrespecttoY.
Instandardcovariancestructureestimation,theparametersenterthemodelmomentsomYbutnotthesamplebod,sothatifbodaremomentsofstationaryvariables,ARTICLEINPRESSY.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340329thenunderregularityconditionssuchasstatedinNeweyandMcFadden(1994),theconventionalresultthatbYisconsistentobtains.
Furthermore,TpbYY0!
dAN0;SwhereAG00G01G00,TpgY0!
dN0;S,andG0istheprobabilitylimitofGYevaluatedatYY0.
ThisdistributiontheoryappliestotheFDandtheHPbecausethesetwoltersdonotdependonunknownparametersandthelteredvariablesarealwaysstationary.
FortheHDestimator,boddependsonrzbutitsrstderivativedoesnot,sothataquadraticexpansionoftheobjectivefunctioncanstillbeusedtoderivetheasymptoticdistributionoftheestimator.
AlthoughGYfortheHDhasarandomlimitwhenrz1,bodisavectorofcovariancesofstationaryvariableswhenevaluatedatthetruevalueofrz.
Thus,the'standardized'HDestimator(orthetstatistic)remainsasymptoticallynormal.
TounderstandthepropertiesoftheQDestimator,anexplanationforwhythelaggedautocovariancesarenormalizedbythevarianceisnecessary.
Suppose~gjYbOdDrzjOmDrzjwasusedinsteadofgjYbodDrzjomDrzjwhereodDrzjbOdDrzjbOdDrz0,andomDrzislikewisedened.
MinimizingJ~gYJoverYyieldsanestimator,say,QD0.
TheproblemhereisthatbOdDrzjisacross-productofdataquasi-differencedatrz,andisthusquadraticinrz.
ThequadraticexpansionofJ~gYJaroundY0containstermsthatarenotnegligiblewhenrzisone.
Assuch,thesampleobjectivefunctioncannotbeshowntoconvergeuniformlytothepopulationobjectivefunction.
Gorodnichenkoetal.
(2009)showinasimplersettingthattheQD0estimatorforrzisconsistentbutithasaconvergencerateofT3/4andisnotasymptoticallynormal.
TheQDestimatorismotivatedbythefactthattheoffendingterminthequadraticexpansionofbOdDrzjiscollinearwithbOdDrz0whenrz1.
Proposition1.
ConsideraDSGEmodelwhosereducedformisgivenby(1)andallrootsofPlessthanone.
LetYbetheunknownparametersofthemodelandletbYQDbetheQDestimatorofY.
ThenTpbYQDY0!
dN0;AvarbYQD.
Asketchoftheargumentisgiveninthesupplementarymaterialforthebaselinemodelwhoseclosed-formsolutionisknown.
Bysubtractingthevariancefromeachlaggedautocovariance,thequadratictermsintheexpansionoftheobjectivefunctionareasymptoticallynegligible.
ThisleadstotheratherunexpectedpropertythatbYisasymptoticallynormalevenwhenrz1.
Fromapracticalperspective,theprimaryadvantageoftherobustestimatorsisthatwhenproperlystudentized,theestimatorsarenormallydistributedwhetherrzo1orrz1,whichgreatlyfacilitatesinference.
Sinceallestimatorsareconsistentandasymptoticallynormal,itremainstoconsiderwhichestimatorismoreefcientinnitesamples.
4.
1.
RelatedliteratureThereisasmallliteratureonestimationofnon-lineardynamicmodelswhenthedataarehighlypersistent.
Cogley(2001)considersseveralestimatorsandndsthatusingcointegrationrelationshipsintheunconditionalEulerequationsworksquitewell.
OurmethodissimilartoCogleys(2001)inthatneitherrequirestheresearchertotakeastandonthepropertiesofthetrendfunctionandyetthemomentsusedinGMMestimationarealwaysstationary.
However,thereareimportantdifferences.
First,quasi-differencingcaneasilyhandlemultipleI(1)orhighlypersistentshocks.
Incontrast,cointegrationrelationshipscanbeusedonlyforcertaintypesofshocks.
Forexample,iftheshocktodisutilityoflaborsupplyisanI(1)process,thereisnocointegrationvectortonullifyatrendinhours.
Second,cointegrationofteninvolvesestimatingidentitiesandthereforetheresearcherhastoaddanerrorterm(typicallymeasurementerror)toavoidsingularity.
Wedonotestimatespecicequationsandhencedonotneedtoaugmentthemodelwithadditional,atheoreticalshocks.
Finally,somestructuralparameterssuchasadjustmentcostscannotbeidentiedbycointegrationrelationsbecausetheyarezerobyconstructioninthesteadystate.
Incontrast,theestimatorsproposedhereutilizeshort-rundynamicsinthedatatoestimatetheparametersgoverningtheshort-rundynamicsofthemodel.
FukacandPagan(2006)considerhowthetreatmentoftrendsmightaffectestimationofDSGEmodels,buttheiranalysisisconnedtoasingleequation.
TheyproposetousetheBeveridge–Nelsondecompositiontoestimateandremovethepermanentcomponentsinthedata.
ThisassumesthattherestrictionsimpliedbyBeveridge–Nelsontrendareconsistentwiththedata.
Canova(2008)explicitlytreatsthelatenttrendsasunobservedcomponentsandestimatesthetrendsandcyclesdirectly.
Whilethisallowsthedatatoselectthetrendendogenously,theprocedurecanbeimprecisewhentherandomwalkcomponentissmall.
CanovaandFerroni(2008)considermanyltersandtreateachasthetruecyclicalcomponentmeasuredwitherror.
Theyareprimarilyconcernedwiththeconsequencesofdatalteringtakingthemodelspecicationasgiven.
Thispapertakestheviewthatthetrendsspeciedforthemodelshouldbeconsistentwiththefactsthatwesoughttoexplain.
Assuch,itshouldnotbetakenasgiven.
ARTICLEINPRESSY.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–3403305.
Simulations:baselinemodelThissectionusesthestochasticgrowthmodeltoconductMonteCarloexperiments.
Dataaregeneratedwitheitherdeterministictrendsrzo1orstochastictrendsrz1usingthemodelequationsforbmt.
Themodelvariablesarethenrescaledbacktolevelformandtreatedasobserveddatadt=(ct,kt,yt,lt),whicharetakenasgiveninestimation.
Thevarianceandtherstorder(M=1)auto-andcrossvarianceofthefourvariablesareusedasmoments.
Alternativechoicesofobservedvariables,suchasexcludingthecapitalstockseries,yieldverysimilarresults.
Themodeliscalibratedasfollows:capitalintensitya0:33;disutilityoflabory1;discountfactorb0:99;depreciationrated0:1;grossgrowthrateintechnologyg0:005.
Thereisonlyoneshockinthisbaselinemodel.
Thus,thestandarddeviationofeztissettosz1withoutlossofgenerality.
Theadmissiblerangeoftheestimatesofais[0.
01,0.
99].
Thepersistenceparameterrztakesvalues(0.
95,0.
99,1).
Todecouplethetreatmentoftrendsfromtheidenticationissues,theparametervectorYa;r;sisestimatedwhileb;d;yisassumedknown.
4Ineachofthe2000replicationsforeachparameterset,serieswithT=200observationsarecreated.
Othersamplesizesarealsoconsidered.
Inallsimulationsandforallestimators,thestartingvaluesinoptimizationroutinesareequaltothetrueparametervalues.
ThemodelissolvedusingtheAndersonandMoore(1985)algorithm.
Mildlyexplosiveestimatesareallowedbecauseotherwisesolutionsforbrzwillbetruncatedtotherightatonemakingthedistributionofbrzhighlyskewed.
Onlyparametervaluesconsistentwithauniquerationalexpectationsequilibriumareallowed.
5Table2reportssimulationresultsforthebaselinegrowthmodel.
Thepersistenceoftechnologyshocksisgivenintheleftcolumn.
Therstandsecondrowsindicatewhichlterisappliedtoboththedataandthemodelvariables.
Columns(1)–(4)reportresultsforthefourestimators.
Byandlarge,allfourltersyieldestimateswhichareveryclosetothetruevalues.
Noticethatwhilerzisalwayspreciselyestimated,thevarianceoftheestimatesvariessubstantiallyacrosslters.
TheQDestimateshavetheloweststandarddeviationwhiletheHPestimatesaretwotovetimesmorevariablethantheQD.
TheHDismoreprecisethantheFDbutislessprecisethantheQD.
Thispatternisrecurrentinallsimulations.
Fig.
1showstherootmeansquarederror(RMSE)fordifferentestimatorsandsamplesizes.
TheQDestimatorperformsthebestwhiletheFDtendstohavethelargestRMSEinalmostallcases.
Insmallsamples,theHPtendstoleadtolargeRMSE.
However,inlargersamples,theHPapproachestheHDwhichisonlyslightlyinferiortotheQD.
ARTICLEINPRESSTable2Neoclassicalgrowthmodel.
rzDatalterQDHDFDHPLTFD1HPHPModellerQDHDFDHPLTFD1LTzt(1)(2)(3)(4)(5)(6)(7)(8)Estimateofa0.
95Mean0.
3180.
3330.
3670.
3500.
4800.
4000.
6750.
990St.
dev.
0.
0520.
0610.
1100.
1030.
1200.
0830.
0220.
99mean0.
3080.
3240.
3720.
3600.
8100.
3770.
7890.
990St.
dev.
0.
0530.
0660.
1150.
1200.
2010.
1090.
0241.
00Mean0.
3040.
3120.
3490.
3510.
9050.
3570.
8170.
990St.
dev.
0.
0540.
0610.
1050.
1150.
1830.
1130.
022Estimateofrz0.
95Mean0.
9490.
9490.
9500.
9500.
9141.
0000.
5411.
000St.
dev.
0.
0060.
0140.
0170.
0150.
0420.
0490.
99mean0.
9890.
9900.
9910.
9910.
8641.
0000.
4851.
000St.
dev.
0.
0020.
0050.
0070.
0160.
0940.
0411.
00Mean0.
9991.
0000.
9981.
0000.
6941.
0000.
4611.
000St.
dev.
0.
0010.
0030.
0050.
0110.
1230.
039Estimateofsz0.
95Mean0.
9811.
0211.
1571.
0761.
1351.
3341.
9490.
046St.
dev.
0.
1230.
1870.
4410.
2910.
2830.
2850.
1670.
0060.
99Mean0.
9621.
0011.
1541.
1074.
3481.
1852.
9120.
042St.
dev.
0.
1110.
1700.
3670.
3032.
1690.
3470.
3970.
0061.
00Mean0.
9550.
9741.
0731.
08719.
8031.
1073.
2890.
041St.
dev.
0.
1080.
1450.
2950.
51310.
6810.
3410.
4780.
005Note:Thenumberofsimulationsis2000.
SamplesizeisT=200.
LTislineardetrending,HPisHodrick–Prescottlter,FDisrstdifferencing,FD1isrstdifferencingwiththerestrictionthatrz1,QDisquasi-differencing,HDishybriddifferencing,ztisdetrendingbytheleveloftechnology.
4Theaveragegrowthrategisestimatedinthepreliminarywhentheseriesisprojectedonalineartimetrend.
5Arationalexpectationssolutionissaidtobestableifthenumberofunstableeigenvaluesofthesystemequalsthenumberofforwardlookingvariables.
Stabilityinthiscontextreferstotheinternaldynamicsofthesystem.
Thisisdistinctfromcovariancestationarityofthetimeseriesdata,whichobtainswhenrzo1.
Itispossibleforrztobemildlyexplosiveandyetthesystemhasastable,uniquerationalexpectationsequilibrium.
Onlyatinyfractionofsimulationswasdiscardedduetonon-uniquenessoftherationalexpectationsequilibrium.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340331ARTICLEINPRESSFig.
1.
Rootmeansquarederrors.
Note:Thisgureshowstherootmeansquarederrors(RMSE)forfourestimatorswhichapplythesametransformation(QD,HP,FD,HD)todataandmodelseries.
RMSEareshownforthreeparameterestimates:a,elasticityofoutputwithrespecttocapital;rz,persistenceoftechnologyshocks;sz,standarddeviationoftechnologyshocks.
Fig.
2.
KerneldensityofsimulatedTpbaa.
Note:ThisgureshowsthekerneldensityofTpbaaforfourestimatorswhichapplythesametransformation(QD,HP,FD,HD)todataandmodelseries.
KerneldensitiesareshownforthreesamplesizesT=150,300,and2000.
Parameteraistheelasticityofoutputwithrespecttocapital.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340332ARTICLEINPRESSFig.
3.
KerneldensityofsimulatedTpbrzrz.
Note:ThisgureshowsthekerneldensityofTpbrzrzforfourestimatorswhichapplythesametransformation(QD,HP,FD,HD)todataandmodelseries.
KerneldensitiesareshownforthreesamplesizesT=150,300,and2000.
Parameterrzisthepersistenceoftechnologyshocks.
Fig.
4.
KerneldensityofsimulatedTpbszsz.
Note:ThisgureshowsthekerneldensityofTpbszszforfourestimatorswhichapplythesametransformation(QD,HP,FD,HD)todataandmodelseries.
KerneldensitiesareshownforthreesamplesizesT=150,300,and2000.
Parameterszisthestandarddeviationoftechnologyshocks.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340333Figs.
2–4presentthekerneldensityofthenormalizedestimator(i.
e.
TpbYY)forsamplesizesofT=150and300.
ResultsarealsoreportedforT=2000tostudytheasymptoticpropertiesoftheestimators.
Approximatenormalityofbrzwhenrzisclosetoone,istotallyunexpected,giventhattheliteratureonintegratedregressorspreparedustoexpectsuperconsistentestimatorswithDickey–Fullertypedistributionsthatareskewed.
Instead,alldensitiesarebell-shapedandsymmetricforallrzr1withnoapparentdiscontinuityasweincreaserztoone.
Thenormalapproximationisnotperfectinsmallsamples,suggestingthatsomesizedistortionwilloccurifoneusesthetstatisticforinference.
Inunreportedresults,t-statisticsconstructedusingNewey–WeststandarderrorshaverejectionratesgreaterthanthenominalsizeforallestimatorsexcepttheHP,whichcanbeundersized.
Forexample,therejectionrateoftheQDestimatorforthetwo-sidedt-testofrzatthetruevalueof1is0.
055whenT=200whilefortestingaatthetruevalueof0.
33,therejectionrateis0.
21.
Thisislargerthanthenominalsizeof0.
05.
Asthesamplesizeincreases,theactualsizegetscloser(andeventuallyconverges)tothenominalrates.
Forexample,atT=1000forQD,thetwo-sidedt-testofrz1hasarejectionrateof0.
05,whilethettestfora0:33is0.
10.
TheQDandHDgenerallyhavebettersizethantheFDandtheHP.
Thenitesamplesizedistortionseemstobeageneralproblemwithcovariancestructureestimatorsandnotspecictotheconsideredestimators.
BurnsideandEichenbaum(1996)reportedsimilarresultsincovariancestructureestimationwithmanyoveridentifyingrestrictions,alsousingtheNewey–Westestimatorofthevarianceofmoments.
5.
1.
VariationstothebaselinemodelInresponsetothendinginCogleyandNason(1995b)thatthebasicrealbusinesscyclemodelhasweakinternalpropagation,researchersoftenaugmentthebasicmodeltostrengthenthepropagationandtobettertthedataatARTICLEINPRESSTable3Augmentedversionsoftheneoclassicalgrowthmodel.
rzDatalterQDHDFDHPLTFD1HPHPModellerQDHDFDHPLTFD1LTzt(1)(2)(3)(4)(5)(6)(7)(8)PanelA:seriallycorrelatedgrowthrateintechnologyEstimateofk00.
95Mean0.
0100.
0010.
0010.
0190.
1800.
1000.
2240.
369St.
dev.
0.
0630.
0580.
0500.
1600.
1650.
0350.
0380.
1060.
99Mean0.
0140.
0030.
0020.
0160.
4980.
0210.
2550.
429St.
dev.
0.
0440.
0460.
0410.
1550.
0880.
0300.
0380.
0701.
00Mean0.
0140.
0030.
0030.
0200.
6000.
0020.
2560.
446St.
dev.
0.
0380.
0390.
0350.
1610.
0290.
0280.
0380.
058PanelB:habitformationinconsumptionEstimateoff00.
95Mean0.
0200.
0080.
0060.
0140.
4100.
0860.
1930.
679St.
dev.
0.
0750.
0730.
0710.
2550.
3390.
0660.
3820.
0740.
99Mean0.
0190.
0090.
0080.
0230.
6470.
0180.
4950.
637St.
dev.
0.
0700.
0740.
0690.
1680.
2410.
0830.
4000.
0701.
00Mean0.
0180.
0230.
0110.
0250.
7020.
0110.
6030.
622St.
dev.
0.
0670.
0870.
0760.
1560.
1740.
0950.
3730.
068PanelC:preferenceshocksqtEstimateofa0:33sq0:50.
95Mean0.
3440.
3350.
3610.
3290.
4650.
2990.
5910.
337St.
dev.
0.
0400.
0250.
0800.
0750.
1260.
0430.
0370.
1371.
00Mean0.
3530.
3420.
3520.
3350.
5080.
3520.
6650.
485St.
dev.
0.
0520.
0340.
0670.
0720.
3570.
0630.
0500.
265sq1:00.
95Mean0.
3390.
3410.
3490.
3310.
4310.
3160.
5040.
344St.
dev.
0.
0230.
0300.
0490.
0600.
1030.
0220.
0310.
0221.
00Mean0.
3470.
3470.
3570.
3400.
6110.
3410.
5290.
364St.
dev.
0.
0280.
0340.
0500.
0550.
2570.
0250.
0360.
023sq1:50.
95Mean0.
3380.
3430.
3490.
3330.
3990.
3260.
4690.
378St.
dev.
0.
0210.
0300.
0420.
0530.
0780.
0170.
0200.
0241.
00Mean0.
3410.
3460.
3530.
3380.
5150.
3380.
4770.
391St.
dev.
0.
0200.
0290.
0380.
0490.
2030.
0170.
0210.
023Note:PanelsAandB:rzandkorfareestimated;a0:33andsz1arexed.
PanelC:veparametersareestimateda;rz;rq;sz;sq.
Thenumberofsimulationsis2000.
SamplesizeisT=200.
LTislineardetrending,HPisHodrick–Prescottlter,FDisrstdifferencing,FD1isrstdifferencingwiththerestrictionthatrz1,QDisquasi-differencing,HDishybriddifferencing,ztisdetrendingbytheleveloftechnology.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340334businesscyclefrequencies.
Oneconsiderationistointroduceserialcorrelationinthegrowthrateofshockstotechnologybyassumingutrzkut1krzut2ezt.
Thisspecicationgeneratesserialcorrelationofkinthegrowthrateoftechnologywhenrz%1.
Thebaselinemodelcorrespondstok0.
Whendataaresimulatedwithk0andkisestimatedfreely,theQD,HD,FD,andHPcorrectlyndthatk0(Table3,PanelA).
Habitinconsumptionisanotherpopularwaytointroducegreaterpersistenceinbusinesscyclemodels.
Considertheutilityfunction:lnCtfCt1yLtwherefmeasuresthedegreeofhabitinconsumption.
Dataaregeneratedwithfsettozero.
Whenfisfreelyestimatedalongwithotherparameters,therobustestimatorsagainndbftobenumericallysmallandnotstatisticallydifferentfromzeroforallvaluesofrz(Table3,PanelB).
AthirdvariationtothebaselinemodelisapreferenceshockQtsuchthattheutilityislnCtyLt=QtwhereqtlnQtrqqt1eqtandeqt$iid0;s2q.
Inthesimulations,rq0:8(sothatthepreferenceshockisstationary)andsq0:5;1:0;1:5.
Toconservespace,PanelCinTable3reportsonlyestimatesfora.
Consistentwiththeresultsthusfar,theHPestimateshavethelargestvariabilityalthoughthedifferencewithotherestimatorsisnotaslargeasitwasinthebaselinemodel.
Notethatassqincreases,thedifferenceacrossmethodsshrinkswhiletheprecisionforallestimatorsimproves.
ArecurrentresultisthattheHPestimateshavethelargestvariabilityandiscomputationallymostintensive.
Burnside(1998)reportsthattheHPlterremovesvariationpotentiallyinformativeaboutthestructuralparametersbutthattheHPlteredmodelanddataseriesstillhavesufcientvariabilitytodiscriminatecompetingtheoriesofbusinesscycles.
OnepossibilityfortheresultsreportedhereisthattheHPltersoutmorelowfrequencyvariationthanotherlters,andtheparametersfandkareidentiedfromthesefrequencies.
AnotherpossibilityisthattheHPimplicitlyusesmanymoreestimatedautocovariances(recallthattheinverseFouriertransformisappliedtomanyautocovariances).
Thisextensiveuseofsampleautocovariancescanalsointroducevariabilitytotheestimator.
6.
Non-robustestimatorsandamodelwithmultiplerigiditiesThissectionreportsresultsforthenon-robustestimatorstoillustratehowtreatmentoftrendscanbringaboutmisleadingconclusionsaboutthepropagatingmechanismofshocks.
Inadditiontothebasicstochasticgrowthmodel,theestimatorsarealsocomparedforamodelwithmanymoreendogenousvariables.
6.
1.
AlternativedetrendingproceduresUptothispoint,theconsideredapproachesapplythesametransformationtothedataandthemodelvariables.
Muchhasbeenwrittenabouttheeffectsoflteringonbusinesscyclefacts.
KingandRebelo(1993)andCanova(1998)showedthattheHPltereddataarequalitativelydifferentfromtherawdata.
Canova(1998)showedthatthestylizedfactsofbusinesscyclesaresensitivetothelterusedtoremovethetrendingcomponents.
GregoryandSmith(1996)usedacalibratedbusinesscyclemodeltoinvestigatewhattypeoftrendcanproduceacyclicalcomponentinthedatathatissimilartothecyclicalcomponentinthemodel.
AlthoughtheseauthorsdidnotestimateaDSGEmodelonltereddata,theyhintedthattheparameterestimatescanbeadverselyaffectedbyltering.
Toinvestigatetheconsequencesofusingdifferentand/orinappropriatelters,fourcombinationsareconsidered:(A)theautocovariancesarecomputedforlinearlydetrendedmodelanddataseries;(B)theautocovariancesarecomputedfortherstdifferencedmodelanddataserieswithimposedrz1;(C)thesampleautocovariancesarecomputedforHPltereddatabutthemodelautocovariancesarecomputedforthelinearlydetrendedvariables;(D)thesampleautocovariancesarecomputedforHPltereddatawhilethemodelautocovariancesarecomputedforseriesnormalizedbytheleveloftechnology,i.
e.
,mtztwhereztistheleveloftechnology.
Eachcombinationhasbeenusedintheliterature(seee.
g.
Table1).
(A)and(B)areaimedtoshowtheeffectsofimposingincorrectassumptionsabouttrends.
(C)and(D)illustratetheconsequenceswhendifferenttrendsareappliedtothemodelandthedata.
6TheresultsforthebasicstochasticgrowthmodelarereportedinTable2.
For(A),whichisreportedincolumn(5),theparameterestimatesareslightlybiasedwhenrz0:95.
Asrzincreases,theestimatesarestronglybiased.
Thisshowsthatwhenrzisclosetounityyetstationary,assumingtrendstationaritystillyieldsimpreciseestimates.
Atrz1,themeanofbrzis0.
694(insteadof1),themeanofbaisapproximately0.
905(insteadof0.
33),themeanofbszis19.
8(insteadof1).
Thecaseofrzr1isempiricallyrelevantbecausemacroeconomicdataarehighlypersistentandwellapproximatedbyunitrootprocesses.
Theseresultsshowthatlineardetrendingofnearlyintegrateddatainnon-linearestimationcanleadtobiasedestimatesofthestructuralparameters,reminiscentoftheunivariatendingofNelsonandKang(1981)thatprojectingaserieswithaunitrootontimetrendcanleadtospuriouscycles.
Turningto(B)incolumn(6)ofTable2,theestimatesarefairlyprecisewhenrzisindeedequaltoone,butasrzdepartsfromone,theestimatesgetincreasinglybiased.
Henceimposingastochastictrendwhenthedatageneratingprocessistrendstationarycanleadtoseriouslydistortedestimates.
Resultsforcombination(C)arereportedincolumn(7)ofTable2.
ARTICLEINPRESS6Asageneralobservation,thestartingvaluesareveryimportantfornon-robustmethodsastheoptimizationroutinescangetstuckinlocaloptima.
Withtherobustestimators,theconvergedestimatesdonotchangeastheoptimizationstartsfromvaluesotherthanthetrueparameters,thoughthesearchforglobalminimumwasoftenlong.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340335Theestimatesofrzaredownwardbiasedwhilebaandbszareupwardbiased.
Takenatfacevalue,theseestimatessuggestasignicantroleforcapitalasamechanismforpropagatingshocksinthemodel.
Resultsfor(D)arereportedincolumn(8)ofTable2.
Here,theestimatesofaoftenhittheboundaryofthepermissibleparameterspacewhileestimatesofszareclosetozero.
Thereasonisthatwhenzthasaunitroot,shockstomtztaretransitoryandconsumptionadjustsquicklytothepermanenttechnologyshock.
ButtheHPltereddataareseriallycorrelated.
Thus,theestimatorisforcedtoproduceparametervaluesthatcangeneratestrongserialcorrelationinthemodelvariables.
Resultsfor(C)and(D)areconsistentwiththendingsofCogleyandNason(1995a),KingandRebelo(1993)andHarveyandJaeger(1993).
ThesepaperssuggestthattheHPlterchangesnotonlythepersistenceoftheseriesbutalsotherelativevolatilityandserialcorrelationoftheseries.
Thistranslatesintobiasedestimatesofallparametersbecausetheestimatorisforcedtomatchtheserialcorrelationoftheltereddata.
Clearly,largeestimatedvaluesofawillalerttheresearcherthatthemodelislikelymisspecied.
Supposetheresearcherallowsforseriallycorrelatedshocksintechnologygrowthbyestimatingkfreely.
PanelAinTable3showsthatthenon-robustmethodsnowyieldestimatesofaaround0.
4–0.
5,whichseemmoreplausiblethanwhenkwasassumedzero.
However,theseestimatesareachievedbyhavingbkstronglynegativeandstatisticallysignicantwhenthetruevalueofkiszero.
Supposenowtheresearchermodiesthemodelbyallowingforhabitsinconsumption.
Evidently,theestimatedhabitformationparameterfissensitivetowhichnon-robustestimatorisused.
Inparticular,(A)hasastrongdownwardbias,while(B)producesanegativebiasinbfwhenrzdepartsfromone.
Ontheotherhand,(C)and(D)haveastrongupwardbias.
Witheithermodication,thetofthemisspeciedmodelsimprovesrelativetothecorrectlyspeciedmodel.
However,thesemodicationsshouldnothavebeenundertakenastheydonotexistinthedatageneratingprocess.
Theseexamplesindicatehowthetreatmentoftrendscanmisleadtheresearchertoaugmentcorrectlyspeciedmodelswithspuriouspropagationmechanismstomatchthemomentsofthedata.
ResultsforthemodelwithanadditionallaborsupplyshockarereportedinTable3,PanelC.
Theestimatescontinuetobebiasedalthoughthebiasestendtobesmallerthaninthebaselinemodelwithasinglepersistentshock.
Ingeneral,aARTICLEINPRESSTable4SmetsandWouters(2007)model.
rzDatalterQDHDFDHPLTFD1HPHPModellerQDHDFDHPLTFD1LTzt(1)(2)(3)(4)(5)(6)(7)(8)Estimateofpersistenceintechnologyshocksrz0.
95Mean0.
9650.
9670.
9620.
9450.
8641.
0000.
1001.
000St.
dev.
0.
0380.
0370.
0440.
1370.
1420.
1570.
99Mean0.
9860.
9840.
9860.
9670.
8361.
0000.
1141.
000St.
dev.
0.
0270.
0270.
0280.
1230.
2270.
0901.
00Mean0.
9900.
9890.
9930.
9710.
7441.
0000.
1231.
000St.
dev.
0.
0270.
0260.
0250.
1230.
3050.
075Estimateofinvestmentadjustmentcostf5:480.
95Mean5.
0575.
3815.
2275.
0663.
9324.
7004.
4479.
818St.
dev.
2.
2362.
5482.
3063.
3541.
9172.
4870.
2650.
6090.
99Mean5.
4325.
5635.
3735.
0955.
5955.
2364.
3669.
662St.
dev.
2.
3212.
4632.
4043.
0122.
6472.
7940.
2570.
5881.
00Mean5.
8636.
2536.
0145.
6176.
1736.
0494.
3779.
541St.
dev.
2.
3752.
7752.
7813.
2792.
9833.
0460.
2300.
548Estimateofhabitformationl0:710.
95Mean0.
7250.
7300.
7490.
7530.
7300.
8643.
9320.
673St.
dev.
0.
0570.
0630.
0620.
0490.
0630.
1421.
9170.
1340.
99Mean0.
6990.
7180.
7190.
7180.
5430.
7440.
9080.
941St.
dev.
0.
0560.
0530.
0620.
1340.
1770.
0530.
0330.
0061.
00Mean0.
6860.
7110.
7160.
7090.
4700.
7310.
9120.
940St.
dev.
0.
0560.
0550.
0640.
1450.
2610.
0570.
0280.
005Estimateofwageadjustmentprobabilityxw0:730.
95Mean0.
7040.
7300.
7340.
6860.
6570.
7590.
4840.
220St.
dev.
0.
0730.
0630.
0750.
1170.
1050.
0770.
0850.
0190.
99Mean0.
6860.
7040.
7090.
6590.
5300.
7180.
4580.
213St.
dev.
0.
0810.
0650.
0790.
1250.
2140.
0840.
0780.
0161.
00Mean0.
6730.
6970.
7000.
6410.
4570.
7000.
4440.
210St.
dev.
0.
0920.
0680.
0830.
1380.
2620.
0910.
0720.
015Note:Thenumberofsimulationsis2000.
SamplesizeisT=150.
LTislineardetrending,HPisHodrick–Prescottlter,FDisrstdifferencing,FD1isrstdifferencingwiththerestrictionthatrz1,QDisquasi-differencing,HDishybriddifferencing,ztisdetrendingbytheleveloftechnology.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340336smallerrzandalargersqleadtosmallerbiases.
Insomecases,onendsbsq4bsz,sothattheresearchermaybetemptedtoconcludethatpreferenceshockshavelargervolatilitythanshockstotechnologywhiletheoppositeistrue.
6.
2.
TheSmetsandWoutersmodelAlthoughthebaselinemodelisanilluminatinglaboratorytoevaluatehowtheestimatorsperform,itisoverlysimplistic.
Toassessthepropertiesoftheestimatorsinamorerealisticsetting,considerthemodelofSmetsandWouters(2007)(henceforthSW).
TreatingSW'sestimatesforthepost-1982sampleasthetrueparametervalues,seriesofsizeT=150aregeneratedandtheestimatorsareappliedtothegeneratedseries.
Toseparateidenticationissuesfromissuesrelatedtothetreatmentoftrends,onlyfourparametersareestimated:persistenceoftechnologyshocksrzwhosetruevaluevariesacrosssimulations;investmentadjustmentcostfwhosetruevalueis5.
48;externalhabitformationinconsumptionlwhosetruevalueis0.
71;andCalvo'sprobabilityofwageadjustmentxwwhosetruevalueis0.
73.
TheresultsarereportedinTable4.
Allrobustmethodsyieldpreciseestimatesoftheparameters.
AlthoughtheHPcontinuestobelessprecise,thedifferencewiththeotherthreerobustestimatorsissmallerthaninthebaselinemodel.
Asimilarfeaturewasobservedwhenthetwo-shockandone-shockneoclassicalgrowthmodelswerecompared.
Thesedifferencesbetweenthebaselineandthemorecomplicatedmodelscanoccurforseveralreasons.
First,inlargermodelswithmanyotherstructuralshocks,technologyshockexplainsonlyafractionofvariationinkeymacroeconomicvariables.
TheHPestimatormaysimplyneedmoreshockstoidentifytheparameters.
Second,biggermodelsimposemanymorecrossequationrestrictionsthatmayimprovetheefciencyofsomeestimatorsmorethanothers.
Thegeneralobservation,however,isthattheproposedrobustestimatorsperformreasonablywellforallvaluesofrzinsimpleandmorecomplexmodels.
Incontrast,thenon-robustestimators(A)through(D)havedramaticbiasesinallfourparametersbeingestimatedwhen(i)thelterusedforthemodelandthedataaredifferent,when(ii)theassumedtrendsaredifferentfromtrendsinthedatageneratingprocess,orwhen(iii)thedataarestationarybuthighlypersistent.
Obviously,theimpulseresponses(andotheranalysesrelatedtotheroleofrigiditiesinamplicationandpropagationofshocksinbusinesscyclemodels)basedonthesebiasedestimatesofthestructuralparameterswillbemisleading.
Asanillustration,Fig.
5highlightsthedifferencebetweenthetrueresponseofkeymacroeconomicvariablestoatechnologyshockintheSWmodelandtheresponsesbasedonparameterestimatesfromapproaches(A)through(D).
Forinstance,considertheresponseofARTICLEINPRESS0510152000.
511.
52Output051015200.
500.
511.
52Consumption0510152010123Investment051015200.
500.
511.
5Wages0510152010.
500.
5Employment051015200.
150.
10.
0500.
05InflationTrue(LT,LT)(FD1,FD1)(HP,LT)(HP,zt)Fig.
5.
EstimatedimpulseresponsesfunctionstoatechnologyshockinSmetsandWouters(2007)model.
Note:ThisgureplotsimpulseresponsefunctionsbasedonparameterestimatesobtainedfromestimatorsapplyingdifferentlterstomodelanddataserieswhenthedataaregeneratedbytheSmetsandWouters(2007)model.
LTisprojectiononalineartimetrend.
FD1isrstdifferencing.
HPistheHodrick–Prescottlter.
Theshockisa1percentincreaseintheleveloftechnology.
Persistenceoftechnologyshockisrz0:99.
Seesupplementalmaterialforotherimpulseresponses.
Y.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340337consumption.
Estimatesfromapproaches(A)and(C)implygrosslyunderstatedresponses.
Estimatesfromapproach(D)suggestaconsiderablymoredelayedconsumptionresponsethanthetrueone.
Theconsumptionresponseimpliedbyapproach(B)isqualitativelysimilartothetrueresponse,buttheresponsesarenoticeablydifferentquantitativelyespeciallywhenrzisfurtherawayfromone.
7.
ExtensionsandimplementationissuesThissectiondiscussesseveralpracticalissuesandextensionspertainingtotherobustestimators.
7.
1.
MultipleshocksThereducedformsolution(1)canbeeasilygeneralizedtoothermodelsandtakestheformbmtPbmt1Bututrut1Set3whereutisnowavectorofexogenousforcingvariables,etisavectorofinnovationsinut,andthematricesP,B,S,rareofconformablesizes.
SupposethereareJunivariateshockprocesses,eachcharacterizedby1rjLujtejt;j1;JwheresomeJ*oftherjmaybeontheunitcircle.
DeneDrLYJj11rjLNowthequasi-differencingoperatoristheproductoftheJ*polynomialsinlagoperator.
Oncethemodelissolvedtoarriveat(3),onecancomputemomentsforDrLbmt.
Whethernone,one,ormoreshocksarepermanent,theautocovariancesofthetransformedvariablesarewelldened.
Forexample,ifoneknowsthatshockstotastesdissipatequicklywhiletechnologyshocksztarehighlypersistent,onecanstilluse1rzLasDr.
7.
2.
LikelihoodestimationAslikelihoodandBayesianestimationiscommonlyusedintheDSGEliterature,onemaywonderhowtheideasconsideredinthispapercanbeimplementedinlikelihoodbasedestimation.
Supposeonecanwritethemodelinastatespaceformwhichinvolvesusingthemeasurementequationstoestablishastrictcorrespondencebetweenthedetrendedseriesinthemodelandinthedata.
Thenonecanderivethelikelihoodwhichmakesmaximumlikelihood(MLE)andBayesianestimationpossible.
Asanexample,considerthemodelgivenin(3).
ThemeasurementequationcorrespondingtotheFDestimatorisxtHstCC0st4wherextisthevectoroflteredvariable,Cistheselectionmatrix,ands0tbmt;bmt1;utisthestatevector.
Thecorrespondingtransitionequationisbmtbmt1ut264375P0BrI0000r264375bmt1bmt2ut1264375BS0S264375etorstPst1Bet5withet$i:i:d:0;S.
Themeasuredvariablextisstationaryirrespectiveofwhetherbmthasstochasticordeterministictrends.
FortheQD0estimator,HCrC0.
Aswithallquasi-differencingestimators,thetreatmentofinitialconditionisimportantespeciallywhenthereisstrongpersistence.
Insimulationswiththerstobservationheldxed,theMLEversionoftheFDgivespreciseestimates,butthetstatisticsarelesswellapproximatedbythenormaldistributioncomparedtoMM-FD(seesupplementarymaterial).
Fortheotherthreeestimators,theextensiontoMLEiseithernotpossibleornotpractical.
ForMLE-HP,onewouldneedtowriteouttheentiredatadensityoftheHPltereddata,andtheJacobiantransformationfromtheunlteredtoltereddatainvolvesaninnitedimensionalmatrix.
FortheQDestimator,recallthattheautocovariancesarenormalizedbythevariance.
Byanalogy,MLE-QDwouldrequiremodifyingthescorevector.
Althoughsuchmodicationispossibleintheory,itisnotstraightforwardtoimplement.
FortheHDestimator,theMLEimplementationiscumbersomebecauseHDexploitsARTICLEINPRESSY.
Gorodnichenko,S.
Ng/JournalofMonetaryEconomics57(2010)325–340338covariancesofvariablescomputedwithdifferentlters.
ThedifferencebetweentheMMandMLEreallyboilsdowntoachoiceofmoments,andtheMMismorestraightforwardtoimplement.
7.
3.
ComputationMomentsofthelteredmodelvariablescanbecomputedanalyticallyorbyusingsimulations.
Weusetheanalyticalmomentswheneverpossiblesinceittendstobemuchfasterthansimulationsanditdoesnothavesimulationerrors.
Althoughthereareavarietyofmethodsforanalyticalcalculations,amethodthatisespeciallyattractiveforlargemodelsistocombinethemeasurementequationxt=HstandthestateequationstPst1Bettoobtainxtst"#0HP0Pxt1st1"#HBBetLetw0tx0t;s0tsothatwtD0wt1D1etThevariancematrixOw0Ewtw0tcannowbecomputedbyiteratingtheequationOiw0D0Oi1w0D00D1SD016untilconvergence.
TheautocovariancematricescanthenbecomputedasOwjDk0Ow0.
Sinceoneisonlyinterestedincomputingthemomentsofvariablesinthemeasurementvectorxt,onecaniterateEq.
(6)untiltheblockthatcorrespondstoxtconverges,i.
e.
JOix0Oi1x0Joe.
TocomputethemomentsoftheHPltereddata,observethattheHPlteredseriescanalternativelybeobtainedasfollows:HPLdtHPLDdtl1L1L121l1L21L12DdtInpractice,usingHP+(L)andtheautocovariancesforDdtandDbmttendstogivemorestableresultswhenrzisclosetoone.
ItispossibletospeedupestimationbasedonHPlteredseriesbyusingasmallernumberofleadsandlagsatthecostoflargerapproximationerrors.
7Finally,anoteonthetreatmentofstationaryvariablesisinorder.
Recallthatinthestochasticgrowthmodel,mtgt;gt;0whenjrzjo1andmtutgt;utgt;0whenjrzj1,wherethethirdcomponentofm*tisthetrendforlaborsupply,lt.
Sincelthasnodeterministicorstochastictrendcomponent,theautocovariancesarecomputedforltandnotblt,thoughtheresultsdonotchangemateriallyifthelteredserieswereused.
Ingeneral,ifthej-thcomponentofm*tiszero,itisunderstoodthattheautocovariancesarecomputedforthelevelofthevariablebothinthemodelandinthedata.
Analternativeistodealwiththesenon-trendingvariablesthroughthemeasurementequation.
Thensomevariablescanbequasi-differencedorrst-differenced,whileothersrequirenotransformation.
8.
ConcludingremarksArealisticsituationencounteredwithestimationofDSGEmodelisthat(a)thedataaretrending;(b)deviationsfromthetrendarepersistent;(c)theresearcherdoesnotknowwhetherthedatageneratingprocessisdifferenceortrendstationary.
ThispapershowsthatthetreatmentoftrendscansignicantlyaffecttheparameterestimatesofDSGEmodelsandproposeseveralrobustapproachesthatproducepreciseestimateswithouttheresearcherhavingtotakeastandoftrendspecication.
Thekeyistoapplythesameltertothedataandthemodelvariablestoyieldwell-denedmomentsfortheestimationofthestructuralparameters.
Severallterscanbeusedinmethodsofmomentsestimation.
Theseestimatorshaveapproximatelynormalnitesampledistributions.
Undoubtedly,theestimatorsrequirefurtherscrutinyandcanbeimprovedinvariousdimensions.
8Thepresentanalysisisarststepinthesparseliteratureonnon-linearestimationwhenthedataarehighlypersistent.
AppendixA.
SupplementarydataSupplementarydataassociatedwiththisarticlecanbefoundintheonlineversionatdoi:10.
1016/j.
jmoneco.
2010.
02.
008.
ARTICLEINPRESS7Asimulationprocedurecanalsobeconsidered.
ForeachY,themodelisusedtogeneratej=1,y,RsamplesofsizeTandthemomentsarecomputed.
AveragingoverjgivesomHP.
Thisprocedureiscomputationallymoreintensiveandtheresultsaresimilartotheoneconsideredhere.
8Forexample,onecanusethebootstrapdevelopedforcovariancestructuresinHorowitz(1998)tocorrectforsmall-samplebiases.
Onemightalsoconsideramodel-basedinsteadofadata-basedweightingmatrixcomputed.
Finally,onemayusesimulationbasedestimators,seeCoibionandGorodnichenko(2010)foranexample.
Y.
Gorodnichenko,S.
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