Support Vector Regression Based GARCH Model with A

1SupportVectorRegressionBasedGARCHModelwithApplicationtoForecastingVolatilityofFinancialReturnsSHIYICHEN*ANDKIHOJEONG**April,2007ABSTRACTInrecentyears,supportvectorregression(SVR),anovelneuralnetwork(NN)technique,hasbeensuccessfullyusedforfinancialforecasting.ThispaperdealswiththeapplicationofSVRinvolatilityforecasting.BasedonarecurrentSVR,aGARCHmethodisproposedandiscomparedwithamovingaverage(MA),arecurrentNNandaparametricGACHintermsoftheirabilitytoforecastfinancialmarketsvolatility.TherealdatainthisstudyusesBritishPound-USDollar(GBP)dailyexchangeratesfromJuly2,2003toJune30,2005andNewYorkStockExchange(NYSE)dailycompositeindexfromJuly3,2003toJune30,2005.Theexperimentshowsthat,underbothvaryingandfixedforecastingschemes,theSVR-basedGARCHoutperformstheMA,therecurrentNNandtheparametricGARCHbasedonthecriteriaofmeanabsoluteerror(MAE)anddirectionalaccuracy(DA).NostructuredwaybeingavailabletochoosethefreeparametersofSVR,thesensitivityofperformanceisalsoexaminedtothefreeparameters.KEYWORDS:recurrentsupportvectorregression;GARCHmodel;volatilityforecasting*(Correspondingauthor,Potentialspeaker)ChinaCenterforEconomicStudies(CCES),FudanUniversity,Shanghai200433,China;Email:[e-mail];TEL:[phone]**SchoolofEconomicsandTrade,KyungpookNationalUniversity,Daegu702-701,KoreaEmail:[e-mail];TEL:[phone]I.INTRODUCTIONVolatilityisimportantinfinancialmarketssinceitisakeyvariableinportfoliooptimization,securitiesvaluation,andriskmanagement.Muchattentionofacademicsandpractitionershasbeenfocusedonmodelingandforecastingvolatilityinthelastfewdecades.Sofarintheliterature,thepredominantmodelofthepastwasGARCHmodelbyBollerslev(1986),whogeneralizestheseminalideaonARCHbyEngle(1982),anditsvariousparametricextensions.ThepopularityofGARCHmodelisduetoitsabilitytocapturemanyoftheempiricallystylizedfactsoffinancialtimeseries,suchastime-varyingvolatility,persistenceandvolatilityclustering(Marcucci,2005);seeBollerslev,ChouandKroner(1992)forliteraturesurveys.EvidenceontheforecastingabilityofGARCHmodelissomewhatmixed.AndersonandBollerslev(1998)showthatGARCHmodelprovidesgoodvolatilityforecast.Conversely,someempiricalstudiesshowthatGARCHmodeltendstogivepoorforecastingperformances;forexample,Figlewski(1997),Cumbyetal.(1993),Jorion(1995,1996),BrailsfordandFaff(1996),andMcMillanetal.(2000).ToimprovetheforecastingabilityofGARCHmodel,somealternativeapproacheshavebeenadvocatedfromtheperspectiveofestimationandforecasting.Neuralnetwork(NN)isonesuchmethod.Inrecentyears,NNshavebeensuccessfullyusedforforecastingfinancialtimeseries;forrecentwork,seeFernandez-Rodriguezetal.(2000)andRefenesandWhite(1998).ThemainappealofNNsistheirflexibilityinapproximatinganynon-linearfunctionarbitrarilywellwithoutaprioriassumptionsaboutthepropertiesofthedata;seeHorniketal.(1989)foradiscussionoftheNNuniversalapproximationproperty.Motivatedbytheirgoodpropertyandpromisingresultsinabroadrangeoffinancialapplications,variousNN-basedGARCHmodelshavebeensuggestedandappliedtoforecastingvolatility.ThebasicfindingsupportsthatNN-basedGARCHoutperformstraditionalGARCHmodelsinforecastingconditionalvolatility;seeDonaldsonandKamstra(1997),Schittenkopfetal.(2000),Taylor(2000),DunisandHuang(2002).However,NNsuffersfromanumberofweaknessesincludingtheneedforalargenumberofcontrollingparameters,difficultyinobtainingaglobalsolutionandthedangerofover-fitting(TayandCao,2001).Theover-fittingproblemisaconsequenceoftheoptimizationalgorithmsusedforparameterselectionandthestatisticalmeasuresusedtoselectthebestmodel.Recently,anovelNNalgorithm,calledsupportvectormachine(SVM),wasdevelopedbyVapnikandhisco-workers(1995,1997)andisgainingpopularityduetomanyattractivefeatures.WhilethetraditionalNNimplementstheempiricalriskminimization(ERM)principle,SVMimplementsthestructuralriskminimization(SRM)principlewhichseekstominimizeanupperboundontheVapnik-Chervonenkis(VC)dimension(generalizationerror),asopposedtoERMthatminimizestheerroronthein-sampleestimatingdata;,refertoGunn(1998)foragoodintroductiontoSVMandrelatedconcepts.BasedonSRMprinciple,SVMachievesabalancebetweenthetrainingerrorandgeneralizationerror,leadingtobetterforecastingperformancethantraditionalNN.SelectingthebestmodelinSVMisequivalenttosolvingaquadraticprogramming,whichgivesSVManothermeritofauniqueglobalsolution.SVMwasoriginallydevelopedforclassificationproblems(SVC)3andthenextendedtoregressionproblems(SVR).ThemainpurposeofthispaperistoformulatesometypesofSVR-basedGARCHmodelsandtocomparetheforecastingperformancetotheresultsobtainedfromamovingaverage(MA),arecurrentNNandaparametricGACH(MLE).Recently,Pérez-Cruzetal.(2003)alsoproposedaSVR-basedGARCH(1,1)modelandshowedthattheproposedmethodprovidedbettervolatilityforecaststhanparametricGARCHmodel.However,theyusedfeedforwardSVRprocedurewhichhasthesamestructureasautoressive(AR)processandhaspoorabilitytomodelthelong-timememory(Haykin,1999).Inthispaper,weapplytherecurrentSVRprocedure,firstlyproposedbyChenandJeong(2005),whichcanintroduceARMAstructureintoeithermeanfunctionorconditionalvariance.T