A common trends model identification, estimation a

ACOMMONTRENDSMODEL:IDENTIFICATION,ESTIMATIONANDINFERENCEANDERSWARNEAbstract.Commontrendsmodelsprovideausefultoolforstudyinggrowthandbusi-nesscyclephenomenainajointframework(seeKing,Plosser,StockandWatson(1991)).Inthispaperwestudytheproblemofhowtoestimateandanalyseacommonstochastictrendsmodelforanndimensionaltimeserieswhichiscointegratedoforder(1,1)withrncointegrationvectors.Identicationofk=nrpermanent(trend)andrtransi-toryinnovationsisdiscussedintermsofimpulseresponsesandvariancedecompositions.Finally,wederiveanalyticalexpressionsoftheasymptoticdistributionsforestimatesofthesefunctions,therebymakingformalhypothesistestingandinferencepossiblewithinthisframework.Keywords:Cointegration,commontrends,impulseresponsefunction,permanentandtransitoryshocks,variancedecomposition.JELClassificationNumbers:C32,C51.1.IntroductionInmanymodelsonmacroeconomicuctuationsthedichotomybetweengrowthandcycleshasplayedanimportantrole.Traditionally,growthhasoftenbeentreatedasindependentoffactorsthatresultincyclicaluctuations(seeKing,PlosserandRe-belo(1988a)).Incontrast,stochasticgrowthmodels(seee.g.King,PlosserandRebelo(1988b),andKing,Plosser,StockandWatson(1987))allowgrowthshockstoinuencetheshortrunuctuations.Acommonfeatureofthesemodelsisthatthenumberofgrowthdisturbancesisratherlowrelativetothenumberofvariables.Theprevailingviewinthetheoreticalliteratureseemstobethatmacroeconomicuc-tuationsarisefromshockstofundamentalvariablessuchaseconomicpolicy,preferences,andtechnology.Theseshocksarethenpropagatedthroughtheeconomyandresultinsystematicpatternsofpersistenceandcomovementsamongmacroeconomicaggregates.Consequently,itshouldbeofinteresttoanalyseasimpletimeseriesmodelwhichmakesDate:October1993.IhavebenettedfromdiscussionswithMichaelBergman,NilsGottfries,NielsHaldrup,DennisHoman,SvendHylleberg,TorJacobson,SrenJohansen,KatarinaJuselius,SuneKarlsson,ErikMellander,Lars{ErikOller,andAndersVredin.ThispaperisbasedonachapterofmyPhDthesis.Responsibilityforerrorsandobscuritiesrestswithmealone.FinancialsupportfromBankforskningsinstitutetandHumanistisk{SamhallsvetenskapligaForskningsradetisgratefullyacknowledged.12ANDERSWARNEitpossibletoexamineconnectionsbetweengrowthrelatedshocksandtransientuctua-tions.Suchamodelwillthenbynecessityincorporatestochasticratherthandeterminis-tictrends.Furthermore,toconsiderthenotionofafewimportantgrowthdisturbances,therewillingeneralbefewerstochastictrendsthantimeseries.InpapersbyKing,Plosser,Stock,andWatson(1987,1991)andStockandWatson(1988)theconnectionbetweencointegrationandcommonstochastictrendswasrstex-aminedinsomedetail.Thebasicideaisthatthereisareducednumberoflinearstochastictrendsfeedingthesystem.Thisimpliesthatthereexistscertainlinearcombinationsofthelevelsserieswhichensurethatthetrendsaverageout,i.e.theresidualsfromthelinearcombinationsarewidesensestationarystochasticprocesses.King,Plosser,Stock,andWatson(1987)investigateacommontrendsmodelforveU.S.macroeconomictimeseries(output,consumption,investments,thepricelevel,andthemoneystock)andmodelgrowthbytwostochastictrends,anominalandarealtrend.Withvetimese-riesandtwoindependentstochastictrends,commonsense(oralgebra)suggeststhatwecanconstructthreeindependentvectorswhicheliminatethetrends,i.e.therearethreecointegratingvectorswhichdescribeasteadystateinsuchasystem.Ashortcomingoftheirpaperisthatthedescriptionoftheestimationandcomputationstrategytheymakeuseofissomewhatlimited.Forexample,aninversionalgorithmneededtoobtainestimatesof,e.g.impulseresponsefunctionsandforecasterrorvariancedecompositionsisonlymentioned.Moreimportantly,asymptoticpropertiesofthesefunctionsarenotconsidered.Apurposeofthispaperistomathematicallyestablishhowonemayestimatetheparametersinacommonstochastictrendsmodelwhenthetimeseriesofinterestarecointegratedoforder(1,1)(seeBlanchardandQuah(1989),Park(1990),andShapiroandWatson(1988)forapproacheswhicharerelatedtotheoneIshallexaminehere;orGonzaloandGranger(1992)forafactormodelapproachtocommontrends).Fur-thermore,Ishallshowhowonemayperformdynamicanalysiswithinthisframeworkwhentheinnovationstothesystemareeitherpermanentortransitory,i.e.whenthere-sponsesinatleastonevariabletoaninnovationareorarenotpersistent.Inparticular,thecalculationofimpulseresponsefunctionsandforecasterrorvariancedecompositionswillbelookedintoinsomedetail.Finally,Ishallderiveasymptoticdistributionsofestimatesofthesefunctionsinthepresentsetting.Here,thetheoryisbasedonBaillieACOMMONTRENDSMODEL3(1981,1987),Lutkepohl(1988,1989,1990),LutkepohlandPoskitt(1990),LutkepohlandReimers(1992),andSchmidt(1973,1974),althoughtheparticularinnovationsIexaminecomplicatetheanalysissomewhat.Thepaperisorganizedasfollows.Insection2,Idiscusssomerepresentationswhichareequivalentforcointegratedtimeseries.Thereitisshownthatarestrictedvectorautoregressiverepresentationforcointegratedtimeseriesexistsunderfamiliarcircum-stances.Sincethisrepresentationisinvertible,itiswellsuitedforcalculatingallotherparametersofinterest(seealsoWarne(1990)).Section3isconcernedwiththeWoldmovingaverageparametersand