REGIME SWITCHING GARCH MODELS

COREDISCUSSIONPAPER2006/11REGIMESWITCHINGGARCHMODELSLucBauwens1,AriePreminger,2andJeroenV.K.Rombouts3June2005,correctedJuly17,2006AbstractWedevelopunivariateregime-switchingGARCH(RS-GARCH)modelswhereinthecon-ditionalvarianceswitchesintimefromoneGARCHprocesstoanother.Theswitchingisgovernedbyatime-varyingprobability,speci¯edasafunctionofpastinformation.Weprovidesu±cientconditionsforgeometricergodicityandexistenceofmoments.Becauseofpathdependence,maximumlikelihoodestimationisnotfeasible.Byenlargingthepa-rameterspacetoincludethestatevariables,BayesianestimationusingaGibbssamplingalgorithmisfeasible.WeapplythismodelusingtheNASDAQdailyreturnseries.Keywords:GARCH,regime-switching,Bayesianinference.JELClassi¯cation:C11,C22,C52.1COREandDepartmentofEconomics,Universit¶ecatholiquedeLouvain.2CORE,Universit¶ecatholiquedeLouvain.3HECMontr¶ealandCIRANO,3000CoteSainteCatherine,H3T2A7Montr¶eal(QC),Canada.TheauthorsthankRichardBaillie,EricRenault,SharonRubin,andtheparticipantsoftheInternationalConferenceonHighFrequencyFinanceinKonstanz2006,fortheirhelpfulcomments.AriePremingeracknowl-edgesthe¯nancialsupportprovidedthroughtheEuropeanCommunity'sHumanPotentialProgrammeundercontractHPRN-CT-2002-00232,MicrostructureofFinancialMarketsinEurope,andtheErnstFoundation.ThistextpresentsresearchresultsoftheBelgianProgramonInteruniversityPolesofAttractioninitiatedbytheBelgianState,PrimeMinister'sO±ce,SciencePolicyProgramming.Thescienti¯cresponsibilityisassumedbytheauthors.1IntroductionOverthepasttwodecadestherehasbeenalargeamountoftheoreticalandempiricalresearchonmodellingvolatilityin¯nancialmarkets.Sincevolatilityiscommonlyusedasameasureofriskassociatedwith¯nancialreturns,itisimportanttoportfoliomanagers,optiontradersandmarketmakersamongothers.Further,portfoliooptimization,derivativepricingandriskmanagement,suchasValue-at-Risk(VaR),usevolatilityestimatesasinputs.Sofarintheliterature,themostwidespreadapproachtomodelingvolatilityconsistsoftheGARCHmodelofBollerslev(1986)anditsnumerousextensionsthatcanaccountforthevolatilityclusteringandexcesskurtosisfoundinthedata(seee.g.BollerslevandWooldridge(1992)foranoverviewoftheGARCHliterature).Theaccumulatedevidencefromempiricalresearchsuggeststhatthevolatilityof¯nancialmarketsdisplayssometypeofpersistencethatcannotbeappropriatelycapturedbyclassicalGARCHmodels.Inparticular,thesemodelsusuallyindicatehighpersistenceinthecondi-tionalvolatility.Thispersistence,aswasnotedbyHamiltonandSusmel(1994),Gray(1996),andKlaassen(2002),isnotcompatiblewiththepoorforecastingresultsofthesemodels.Fur-thermore,Diebold(1986)andLamoureuxandLastrapes(1990),amongothers,arguethatthenearintegratedbehavioroftheconditionalvariancemayoriginatefromstructuralchangesinthevarianceprocess,whicharenotaccountedforbystandardGARCHmodels.MikoschandStarica(2004)showthatestimatingaGARCH(1,1)modelonasampledisplayingstructuralchangesintheunconditionalvolatilitydoesindeedcreateanintegratedGARCH(IGARCH)e®ect.These¯ndingsclearlyindicateapotentialsourceofmisspeci¯cation,totheextentthatthestructuralformoftheconditionalmeanandvarianceisrelativelyin°exibleandheld¯xedthroughouttheentiresampleperiod.Forexample,theexistenceofshiftsinthevarianceprocessovertimecaninducevolatilitypersistence(seeWongandLi(2001)andLanneandSaikkonen(2003)).HencetheestimatesofaGARCHmodelsu®erfromasubstantialupwardbiasinthepersistenceparameter.Therefore,modelsinwhichtheparametersareallowedtochangeovertimemaybemoreappropriateformodellingvolatility.Inthisperspective,severalmodelsthatarebasedonamixtureofdistributionshavebeenproposed.Schwert(1989)considersamodelinwhichreturnsmayhaveeitherahighoralowvariance,andswitchesbetweenthesestatesaredeterminedbyatwo-stateMarkovprocess.HamiltonandSusmel(1994)andCai(1994)introduceanARCHmodelwithregime-switching1parametersinordertotakeintoaccountsuddenchangesinvolatility.TheyuseanARCHspeci¯cationinsteadofaGARCHtoavoidtheproblemofpathdependenceoftheconditionalvolatilityontherulingregime.Later,atractableMarkov-switchingGARCHmodelwaspresentedbyGray(1996)andamodi¯cationofhismodelwassuggestedbyKlaassen(2002),seealsoBollen,Gray,andWhaley(2000),Dueker(1997)andHaas,Mittnik,andPaolella(2004b).SeveralauthorshavealsoexaminedtheclassofmixturesofnormalGARCHmodels,i.e.modelswhereerrorshaveaconditionaldistributionthatisamixtureofnormaloneswithGARCHvariancecomponentsandtheprobabilitythateachobservationbelongstoagivenvolatilityregimeisconstant.VlaarandPalm(1993)werethe¯rsttosuggestamixtureoftwonormaldistributionswherethedi®erencebetweentheconditionalvariancesineachstateisconstant.AnotherversionwasproposedbyBauwens,Bos,andvanDijk(1999)whoconsideramixtureGARCHinwhichthetwoconditionalvariancesareproportionaltoeachother.Recently,Haas,Mittnik,andPaolella(2004a)speci¯edageneralframeworkforthesemodels,allowingforinterdependencebetweenthevariancecomponentsineachregime.Theobjectiveofthispaperistodevelopmodelsthatbetterdescribethevolatilitybehaviorandtoextendtherecentliteratureonswitchingvolatilitymodels.Weproposearegime-switchingGARCH(RS-GARCH)model,inwhichtheparametersa