Monte Carlo posterior integration in GARCH models

MonteCarloPosteriorIntegrationinGARCHModelsPeterMullerandAndyPolePeterMullerisAssistantProfessor,InstituteofStatisticsandDeci-sionSciences,DukeUniversity,Durham,NC27708-0251.AndyPoleisManagingDirector,InvictusPartners,LLC,590MadisonAvenue,37thFloor,NewYorkNY10022.ThisresearchwaspartiallysupportedbyNSFgrantDMS-9305699atDukeUniversity.SummaryRecentdevelopmentsinestimatingnon-linear,non-normaldynamicmod-elshaveusedGibbssamplingschemesoverthespaceofallparametersinvolvedinthemodel(Carlin,PolsonandStoer,1992)aswellasMonteCarlointegra-tionbasedonpropagatingaMonteCarlosampleontheparametervectortthroughthestagesofthedynamicmodel.Bothapproacheshaveadvantages.TherstbecauseitenablesaconvenientGibbssamplerimplementation.Thelatterbecauseitsplitstheproblemintoaseriesofsimulationproblems,oneforeachtimestep,thushavingcomputationaleortonlyincreaselinearlywiththelengthofthetimeseries.Also,propagatingaMonteCarlosamplethroughthedynamicmodelallowsforunrestrictedgeneralityinthedistributionalformofevolutionnoiseandlikelihood.InthispaperwedevelopschemesalongbothlinestoapplytotheanalysisofGARCH(generalizedautoregressiveconditionalheteroskedasticity)modelsfordailyexchangeratedata.KEYWORDS:Exchangeratedata;Generaldynamicmodel;MarkovchainMonteCarlo;Metropolis;Statespacemodel.1INTRODUCTIONARCH(autoregressiveconditionalheteroskedasticity)modelshaveachievedaconsiderablefollowingintheeconometricsandnanceliteraturesincetheirintroductionbyEngle(1982).AnARCHmodelisadiscretestochasticprocesswiththecharacteristicfeaturethatthevarianceattimetissometime-varyingfunctionofthetimet1informationset.Below,in(1)wegiveamorespecicformaldenition.Applicationsandstudieswerestimulatedbythegeneralisa-tiontoGARCHmodelsinBollerslev(1986).Therearenowover300papersinthemainstreamstatisticsandeconometricsjournalsdiscussingtheoreticalpropertiesofmanyformulationsofGARCHmodelaswellasnumerousappli-cations.AnexcellentsurveyoftheliteratureisBollerslevetal.(1992);someofthereviewmateriallaterinthissectionisbasedonthatarticle.Thecontributionofthispaperistwofold.Firstly,novelcombinationsofMarkovchainMonteCarlotechniquesaredeveloped.Thoughpromptedby,anddevelopedherespecicallyfor,dynamicGARCHmodels,thealgorithmsareusefulforawiderangeofnon-standardsequentialanalyses,dynamicornot.ThemethodsbuildonandarerelatedtoMarkovchainMonteCarlotechniquesdiscussedinCarlin,PolsonandStoer(1992),Jacquier,PolsonandRossi(1994and1995),andTierney(1994).Secondly,theclassof(mul-tivariate)GARCHmodelsisextendedtoadynamicsettingtherebyallowingmuchgreatermodellingexibility.Wenotethatdynamicmodellinghasprovedimmenselyusefulwithtraditionallinearregressiontypemodels,particularlyinforecastingapplications.FromthatwesuggestthatdynamicformulationsoftheGARCHmodelfamilyhavesimilarlygoodpotentialforadvancingGARCHbasedstudies.Theapplicationtoexchangeratesdiscussedinthispaperlends1supporttothisclaim,althoughmoreworkisrequiredtoexplorecovariancestructures.TheimplementationofthedynamicmodelisbasedonmethodsdenedinMuller(1992).Intheremainderofthisintroductionwereviewthesalientappliedliteraturetogiveanoverviewofthesettingforourworkandtomotivateourobjectivesforthepaper.InSection2weintroducethedatasetthatwillbeusedforillus-tration.InSection3wediscusstheestimationofasimpleGARCH(1,1)modelbyanindependencechaintypeMarkovchainMonteCarloalgorithm.AndinSection4weextendthebasicGARCH(1,1)modeltoamultivariatedynamicmodelwhichweestimatebyaMonteCarloalgorithmbasedonsimulatingthedynamicmodel.1.1ReviewThebasicARCHmodelintroducedinEngle(1982)allowsfortheerrortermsinatimeseriesmodeltohaveatimevaryingvariance.Specicallythevarianceisdirectlyrelatedtopreviousvaluesoftheobservationseriessuchthathigherseriesvaluesleadtohigherstochasticvarianceinsucceedingperiods.Letftgbeastochasticprocesswithmean0andvarianceht,tind[0;ht]:(1)Thevarianceisrelatedtoprevioussquaredvaluesoftheprocessinanautore-gressivescheme,ht=0+12t1+:::+p2tp:ThisisthebasicARCH(p)model.Bollerslev(1986)extendedtheARCHfor-mulationtoincludelaggedvaluesofthevarianceitselfinthevarianceequationtherebyprovidingamoreexiblemodelform.TheGARCH(p,q)modelfor2varianceisht=0+pXi=1i2ti+qXj=1jhtjOneofthemotivationsfortheGARCHformulationwasthelargenumbersoflagsrequiredinmanyapplicationsofARCH.GARCHmodelssharewithclassicaltimeseriesARMAmodelsthepropertythatsomehighorderautore-gressivemodelsarewellapproximatedbylowordermovingaveragemodels.InmostapplicationsofGARCHarstordermodel,GARCH(1,1),isfoundtobeappropriate.WeintroduceaMarkovchainMonteCarloschemetoestimateGARCH(1,1)modelsinSection3.Mostoftheliteratureadoptsthegaussiandistributionfortheconditionalsamplingdistribution,tN[0;ht]:However,severalalternativeshavealsobeeninvestigated.Bollerslev(1987)suggeststhestudenttdistribution,Jorion(1988)usesanormal-Poissonmix-ture,BaillieandBollerslev(1989)usethepowerexponentialmodel,Hsieh(1989)usesanormal-lognormalmixturedistribution,andNelson(1990)usesthegeneralisedexponentialdistribution.ThebasicmultivariateGARCH(p,q)modelwasrststudiedbyBollerslev,Engle,andWooldridge(1988).Let