The Copula-GARCH model of conditional dependencies

TheCopula-GARCHmodelofconditionaldependencies:AninternationalstockmarketapplicationEricJondeau,MichaelRockinger*SwissFinanceInstituteandUniversityofLausanne,Lausanne,SwitzerlandAbstractModelingthedependencybetweenstockmarketreturnsisadifficulttaskwhenreturnsfollowacom-plicateddynamics.Whenreturnsarenon-normal,itisoftensimplyimpossibletospecifythemultivariatedistributionrelatingtwoormorereturnseries.Inthiscontext,weproposeanewmethodologybasedoncopulafunctions,whichconsistsinestimatingfirsttheunivariatedistributionsandthenthejoiningdistri-bution.Insuchacontext,thedependencyparametercaneasilyberenderedconditionalandtimevarying.Weapplythismethodologytothedailyreturnsoffourmajorstockmarkets.Ourresultssuggestthatcon-ditionaldependencydependsonpastrealizationsforEuropeanmarketpairsonly.Forthesemarkets,de-pendencyisfoundtobemorewidelyaffectedwhenreturnsmoveinthesamedirectionthanwhentheymoveinoppositedirections.Modelingthedynamicsofthedependencyparameteralsosuggeststhatde-pendencyishigherandmorepersistentbetweenEuropeanstockmarkets.
2006PublishedbyElsevierLtd.JELclassification:C51;F37;G11Keywords:Stockindices;Internationalcorrelation;Dependency;GARCHmodel;SkewedStudent-tdistribution;Copulafunction*Correspondingauthor.UniversityofLausanne,EcoledesHEC,DepartmentofFinanceandInsurance,1015Lausanne,Switzerland.Tel.:þ41216923348;fax:þ41216923435.E-mailaddresses:[e-mail](E.Jondeau),[e-mail](M.Rockinger).0261-5606/$-seefrontmatter
2006PublishedbyElsevierLtd.doi:10.1016/j.jimonfin.2006.04.007JournalofInternationalMoneyandFinance25(2006)827e853(1990),SusmelandEngle(1994),andBekaertandHarvey(1995)havemeasuredtheinterdependenceofreturnsandvolatilitiesacrossstockmarkets.Morespecifically,LonginandSolnik(1995)havetestedthehypothesisofaconstantconditionalcorrelationbetweenalargenumberofstockmarkets.Theyfoundthatcorrelationgenerallyincreasesinperiodsofhigh-volatilityoftheU.S.market.Inaddition,inasimilarcontext,testsofaconstantcor-relationhavebeenproposedbyBeraandKim(2002)andTse(2000).RecentcontributionsbyKronerandNg(1998),EngleandSheppard(2001),Engle(2002),andTseandTsui(2002)havedevelopedGARCHmodelswithtime-varyingcovariancesorcorrelations.Asanalterna-tiveapproach,RamchandandSusmel(1998)andAngandBekaert(2002)haveestimatedamul-tivariateMarkov-switchingmodelandtestedthehypothesisofaconstantinternationalconditionalcorrelationbetweenstockmarkets.Theyobtainedthatcorrelationisgenerallyhigherinthehigh-volatilityregimethaninthelow-volatilityregime.Inthiscontext,animportantissueishowdependencybetweenstockmarketscanbemea-suredwhenreturnsarenon-normal.IntheGARCHframework,somerecentpapershavefocusedonmultivariatedistributionswhichallowforasymmetryaswellasfattails.Forinstance,multivariateskeweddistributions,andinparticulartheskewedStudent-tdistribution,havebeenstudiedbySahuetal.(2001)andBauwensandLaurent(2002).Inaddition,intheMarkov-switchingcontext,ChesnayandJondeau(2001)havetestedforaconstantcorrelationbetweenstockreturns,whileallowingforStudent-tinnovations.1Formosttypesofunivariatedistributions,however,itissimplyimpossibletospecifyamultivariateextensionthatwouldallowthedependencystructuretobecaptured.Inthispaper,wepresentanewmethodologytomeasureconditionaldependencyinaGARCHcontext.Ourmethodologybuildsonso-called‘‘copula’’functions.Thesefunctionsprovideaninterestingtooltomodelamultivariatedistri-butionwhenonlymarginaldistributionsareknown.Suchanapproachis,thus,particularlyuse-fulinsituationswheremultivariatenormalitydoesnothold.Anadditionalinterestingfeatureofcopulasistheeasewithwhichtheassociateddependencyparametercanbeconditionedandrenderedtimevarying,evenwhencomplicatedmarginaldynamicsareestimated.Weusethismethodologytoinvestigatetheimpactofcertainjointstockreturnrealizationsonthesubsequentdependencyofinternationalmarkets.Manyunivariatemodelshavebeenpro-posedtospecifythedynamicsofreturns.However,giventhefocusofthiswork,wedrawonrecentadvancesinthemodelingofconditionalreturnsthatallowsecond,third,andfourthmomentstovaryovertime.OurunivariatemodelbuildsonHansen’s(1994)seminalpaper.Inthatpaper,aso-calledskewedStudent-tdistributionisderived.Thisdistributionallowsforacontrolofasymmetryandfat-tailedness.Byrenderingthesecharacteristicsconditional,itispossibletoobtaintime-varyinghighermoments.2Thismodel,therefore,extendsEngle’s(1982)ARCHandBollerslev’s(1986)GARCHmodels.InanextensiontoHansen(1994),1Somepapersalsoconsideredhowcorrelationvarieswhenstockmarketindicesaresimultaneouslyaffectedbyverylarge(positiveornegative)fluctuations.LonginandSolnik(2001),usingextremevaluetheory,foundthatdependencyincreasesmoreduringdownsidemovementsthanduringupsidemovements.Poonetal.(2004)adoptedanalternativestatisticalframeworktotestconditionaldependencybetweenextremereturnsandshowedthatsuchataildependencymayhavebeenoverstatedoncethetime-variabil