Empirical Modeling of Extreme Events from Return{V

EmpiricalModelingofExtremeEventsfromReturn{VolumeTimeSeriesinStockMarketPeterBuhlmannDepartmentofStatisticsUniversityofCaliforniaBerkeleyBerkeley,CA94720-3860June1996AbstractWeproposethediscretizationofreal-valuednancialtimeseriesintofewordinalvaluesandusenon-linearlikelihoodmodelingforsparseMarkovchainswithintheframeworkofgeneralizedlinearmodelsforcategoricaltimeseries.Weanalyzedailyreturnandvolumedataandestimatetheprobabilitystructureoftheprocessofextremelower,extremeupperandthecomplementaryusualevents.Knowingthewholeprobabilitylawofsuchordinal-valuedvectorprocessesofextremeeventsofreturnandvolumeallowsustoquantifynon-linearassociations.Inparticu-lar,wenda(newkindof)asymmetryinthereturn{volumerelationshipwhichisapartialanswertoaresearchissuegivenbyKarpo(1987).Wealsoproposeasimplepredictionalgorithmwhichisbasedonanempiricallyselectedmodel.Keywords:Asymmetricvolume{returnrelation;Finance;Generalizedlinearmodel;Markovchain;Non-linearassociation;PredictionResearchsupportedinpartbytheSwissNationalScienceFoundation.11IntroductionManyofthetimeseriesoccurringinnancesuerfromthefactthattherearenosimplemechanisticmodelswhichsubstantiallyreducethecomplexityofthedataandstillexplainitwell.Eitherthemodelsaretoosimpletoexplainthenatureofnancialphenomenaorthemodelistoocomplexsothatover-ttingoccurs.Over-ttingresultsinhighvariabilityofestimatesandreproducibilityinasimilarsituationcannotbesatisfactorilyachieved.Ifwebelieveinanintrinsichighcomplexityofthenancialnature,weseemtobeforcedtoworkwithmodelshavingmanyparametersandhencewithestimatesbeinghighlyvariable.Weproposeonewayoutofthisimpassebythefollowingquestion:Whymodelingthewholestructureofanobservednancialtimeseries,ratherthanmodelingonlysomestructure,suchasextremeevents?Weproposehereadiscretizationofareal-valuednancialtimeseriesintoonlythreeordinalcategories,correspondingtoextremelower,extremeupperandthecomplementary(usual)events.Thisresultsinahugereductionofthenumberofparametersforamodelandstillallowstoexplorequestionsaboutextremeevents,suchasunusualvaluesorlargepositiveornegativeincrements.WeexemplifythemethodbyanalyzingdailydataofreturnfromtheDowJonesindexandvolumefromtheNewYorkStockExchange(NYSE).Foranalyzingthethreeevents(lowerextreme,upperextremeandusual)weusealikelihoodmodelingapproachbasedonahigherorderMarkovianassumption,andmodelcumulativeprobabilitieswithintheframeworkofgeneralizedlinearmodelswithlaggedvariablestreatedasfactors.Theword‘linear’canbemisleading,ourmodelclassisverygeneralandbroad,sincewearetreatinglaggedvariablesasfactors.ThisthenincludesprocessessimilartoarbitraryniteorderMarkovchains.ByputtinganaturalhierarchyonsuchmodelswetypicallymodelsparseratherthanfullMarkovchains,whichallowstoavoidthecurseofdimensionality,cf.section3.1.Suchmodelsareveryexible,onecanalsoincludecontinuouscovariatesandexplanatoryexogenousfactorsfordescribingthedynamics.Foragoodoverviewintheindependentset-up,cf.McCullaghandNelder(1989).Withinthismodelclass,theselectionofamodelcanbesupportedbyconsideringmeasuresofpredictivepowersuchasAkaike’sinformationcriterion(AIC),cf.Akaike(1973).Havingselectedamodel,wealsoproposeasimplepredictionalgorithm.Bydiscretizingwepayapricebyrestrictingthefocustoextremeevents(andthecomplementaryusualevents).Ontheotherhandwegainalotintheprocessofempiricalmodelsearchingandtting.Moreover,ourconclusionsareprobabilisticstatementsratherthantheoftenusedcorrelationwhichmeasuresonlylinearassociation.WeobtainbyourmethodfullyprobabilisticinterpretationsfortherelationshipofextremeeventsofreturnfromtheDowJonesandvolumefromtheNYSE.Thisgivesnewinsightintothestructureofnancialmarkets.Inparticular,ourempiricalresultsareananswertoaresearchquestionposedbyKarpo(1987):forourdata,thevolume{returnrelationship,atleastforextremeevents,isinanewsenseasymmetric.Thispaperisorganizedasfollows.Insection2wedescribethedataset,insection3weexplainthemodelingandpredictiontechniques,insection4wereportourempiricalndingsfortheanalyzeddataset,insection5wedrawsomeconclusionsandinsection6webrieyoutlinesomemoremathematicalandcomputationaldetails.22ThedataThedataisaboutdailyreturnoftheDowJonesindexanddailyvolumeoftheNewYorkStockExchange(NYSE).ThedailymeasurementsarefromtheperiodJuly1962throughJune1988,correspondingto6430days2.Whatweterm‘volume’isthestandardizedaggregateturnoverontheNYSE,Volt=log(Ft)log(t1Xs=t100Fs=100);whereFsequalsthefractionofsharestradedondays.ThereturnisdenedasRett=log(Dt)log(Dt1);whereDtistheindexoftheDowJonesondayt.ThisisthesamedatawiththesamestandardizationasinWeigendandLeBaron(1994).Figure1showsthesestandardizednancialtimeseries;thetimet=6253,wherereturnRett0:2correspondstothe1987crash.Besidesthisspecialstructurearoundthecrash,bothserieslookquitestationary.3ModelingSincemechanistictheoriesforvolumeorreturnarenotbroadlyaccepted,wewanttouseamodelingapproachwhichisinanonparametricspirit.Ourmodelsinsections3.3and3.4caneasilybeextendedtoincludesomeusualparametricparts,yieldingthenakindofsemiparametricmodel.3.1ThecurseofdimensionalityforfullMarkov