Stochastic volatility models Conditional normality

StochasticVolatilityModels:ConditionalNormalityversusHeavy-TailedDistributionsRomanLiesenfeldandRobertC.JungUniversitatTubingenySeptember1997AbstractMostoftheempiricalapplicationsofthestochaticvolatility(SV)modelarebasedontheassumptionthattheconditionaldistributionofreturnsgiventhelatentvolatilityprocessisnormal.InthispapertheSVmodelbasedonaconditionalnormaldistributioniscomparedwithSVspecicationsusingconditionalheavy-taileddistributions,especiallyStudent’st-distributionandthegeneralizederrordistribution.ToestimatetheSVspecicationsasimulatedmaximumlikelihoodapproachisapplied.TheresultsbasedonGermanstockmarketdatarevealthattheSVmodelwithaconditionalnormaldistributiondoesnotadequatelyaccountforthetwofollowingempiricalfactssimultaneously:theleptokurticdistributionofthereturnsandlowbutslowlydecayingautocorrelationfunctionsofthesquaredreturns.ItisshownthattheseempiricalfactsaremoreadequatelycapturedbyaSVmodelwithaconditionalheavy-taileddistribution.Finally,itturnsoutthatthechoiceoftheconditionaldistributionhassystematiceectsontheparameterestimatesofthevolatilityprocess.KEYWORDS:Volatilitypersistenceofreturns;Leptokurticreturndistribution;Simulatedmaximumlikelihood;MonteCarlointegration.JEL-CLASSIFICATION:C22,C52,C15.ResultsinthispaperarerelatedtoaprojectwhichisnanciallysupportedbytheDeutscheForschungs-gemeinschaft.TheauthorswouldliketothankWalterKramer,MartinKukukandGerdRonningforhelpfulcomments.ThispaperhasbeenpresentedattheEconometricSocietymeetinginToulouse,August1997.yAddressforcorrespondence:WirtschaftswissenschaftlicheFakultat,UniversitatTubingen,Mohlstr.36,72074Tubingen,GERMANY.E-mail:[e-mail]@uni-tuebingen.de.1.INTRODUCTIONThestochasticvolatility(SV)modelintroducedbyTaylor(1986)isusedtoaccountforthewelldocumentedautoregressivebehaviorinthevolatilityofnancialreturnseries.Itrepresentsanalternativetotheautoregressiveconditionallyheteroskedastic(ARCH)modelofEngle(1982)orBollerslev’s(1986)generalizedARCH(GARCH)model.ThestandardversionofthisSVmodelisgivenbyrt=+exp(t=2)ut;uti:i:d:(0;1)(1a)t=+t1+vt;vti:i:d:N(0;1);(1b)wherertisthereturnondaytandtisthelogvolatility.Theparameterrepresentsthepre-dictablepartofthereturns.Theerrorprocessesutandvtaremutuallyandseriallyindependentwithmeanzeroandunitvariance.Both,utandvtareunobservable.Hencet,whichisassumedtofollowaGaussianAR(1)-processwithapersistenceparameter,isalsounobservable.Forjj1theSVmodeliscovariancestationary.Theparametermeasuresthestandarddeviationofvolatilityshocksandisassumedtobegreaterthanzero.MostoftheempiricalapplicationsoftheSVmodelarebasedontheadditionalassumptionthatutisnormallydistributedleadingtoanormaldistributionforthedailyreturnsconditionalont;seeforexampleTaylor(1986,1994),MahieuandSchotman(1994),Jaquier,PolsonandRossi(1994)andKim,ShephardandChib(1996).InthestudiesofRuiz(1994)andHarvey,RuizandShephard(1994)theSVmodelisextendedtoallowtheconditionaldistributionofthereturnstobemoreheavy-tailedthanthenormaldistributionbyusingthead-hocassumptionofascaledStudentt-distributionforut.ThepurposeofthispaperistoanalyzetheabilityoftheSVmodeltocaptureadequatelythefollowingempiricalregularitiesofnancialreturnseries.First,theleptokurticdistributionofdailyreturnsmeaningthatitisexcessivelypeakedaroundzeroandthatitexhibitsfattertailsthanthecorrespondingnormaldistribution.Second,theautoregressivebehaviorofthevolatilityindicatedbyatypicallylowbutveryslowlydecayingautocorrelationfunctionofthesquaredreturns.WewilldemonstratethattheSVmodelwithaconditionalnormaldistributionforthereturns(SV-normal)istoorestrictivetoaccountadequatelyforbothabovementionedempiricalregularitiessimultaneously.Furthermorewewillshowthatthesubstitutionoftheconditionalnormaldistributionofthereturnsbyaconditionalheavy-taileddistributionas,forexample,theStudentt-distributionandthegeneralizederrordistribution(GED)canhelptocaptureadequatelybothempiricalregularities.Finally,itturnsoutthattheassumptionconcerningthe1conditionaldistributionofthereturnsaectstheestimatesoftheparameterswhichgovernsthevolatilityprocess.SincethelatentvolatilityprocesstisassumedtobeseriallycorrelatedthemarginallikelihoodoftheSVmodelisgivenbyahighdimensionalintegralwhichmakestheestimationbystandardmaximumlikelihood(ML)infeasible.Hence,toestimatetheSVmodelwithdierentconditio-nalreturndistributionsweusethesimulatedmaximumlikelihood(SML)approachdevelopedbyDanielssonandRichard(1993).Thisestimationstrategyallowstoadoptthestandardin-strumentsofinferencedevelopedfortheMLmethod.Thepaperisorganizedasfollows.Section2containsabriefdescriptionofthedatausedthroug-houtthepaper.Section3analyzesthetheoreticalpredictionsoftheSVmodelwithaconditionalnormalandconditionalheavy-taileddistributionsconcerningthekurtosisofthereturnsandtheautocorrelationofthesquaredreturns.Thepredictedmomentsarecomparedwiththecor-respondingempiricalmoments.Section4describesthesimulatedmaximumlikelihood(SML)estimationtechniqueusedtoestimatetheparametersoftheSVmodel.Section5presentstheresultsoftheSMLe