The Analysis of Foreign Exchange Data Using Wavefo

THEANALYSISOFFOREIGNEXCHANGEDATAUSINGWAVEFORMDICTIONARIESJAMESB.RAMSEYANDZHIFENGZHANGyAbstract.Thispaperuseswaveformdictionariestodecomposethesignalscontainedwithinthreeforeignexchangeratesusingtick-by-tickobservationsobtainedworldwide.ThethreeexchangeratesexaminedaretheJapaneseYenandtheGermanDeutscheMarkagainsttheU.S.dollarandtheDeutscheMarkagainsttheYen.ThedatawereprovidedbyOlsenAssociates.AwaveformdictionaryisaclassoftransformsthatgeneralizesbothwindowedFouriertransformsandwavelets.Eachwaveformisparameterizedbylocation,frequency,andscale.Suchtransformscananalyzesignalsthathavehighlylocalizedstructuresineithertimeorfrequencyspaceaswellasbroadbandstructures;thatis,waveformscan,inprinciple,detecteverythingfromshocksrepresentedbyDiracDeltafunctions,to\chirps,shortburstsofenergywithinanarrowbandoffrequencies,tothepresenceoffrequenciesthatoccursporadically,andnallytothepresenceoffrequenciesthatholdovertheentireobservedperiod.Waveformdictionariesaremostusefulinanalyzingdatathatarenotstationaryandnon-stationarityuptosecondorderiswellrecognizedinthecontextofforeignexchangerates.Acknowledgment.ThenancialandmaterialsupportoftheC.V.StarrCenterforAppliedEconomicsisgratefullyacknowledged.1.Introduction.Dailyforeignexchangerateshavebeenthesubjectofintenseacademicanalysisforquitesometime.Recently,inaseriesofinnovativepapersthedailyanalysishasbeenextendedtointra-dailydataonpostingsofbidandaskquotesonforeignexchangerates.Allofthisliteraturehasinvolvedthreerelatedlinesofresearch.Therstisthequestionastowhatdistribution,ordistributions,arerelevantintheanalysisofthesedata.Thesecondconcernsthedegreeoftemporalcorrelation,bothformeansandforsecondmomentsintheformof\ARCHanalyses.Thethirdbranchdealswiththequestionofthedegreeofstationarityinthedata.Thereisanalternativeapproachtothesedata.Thesequenceofpricereactionstovariationsininforma-tionandthenancialneedsoftheparticipatingagentsmightberegardedasasequenceofsignalsofvarioustypesandofvaryingduration.Wemightalsopresumethatthesignalsarepartiallyobscuredbynoisethatisgeneratedbyliquiditytrades,agenterrors,andisolatedrandomshockstotheforeignexchangesystem.Aphysicalexamplewillclarifythesituation.Consider,asequenceofsignalsgeneratedwithbackgroundnoisebysomeonespeaking.Eachsyllableinvolvesadistinctsetoffrequenciesthatlastsforashortperiodoftime.Somesyllablesareofveryshortduration,somearemuchlonger.Forsomesignals,thetransitionisveryabrupt,forothersthetransitionisveryslow.Wemaypresumethattheamplitudeofthesignalvariesunpredictably.Thebackgroundmayalsoincludeafewlowfrequencyharmonics.Statistically,thesedataarehighlynon-stationaryandtherewouldbelittleevidenceforanyfrequencytoholdovertheentiretimesequenceofpoints.Ifsuchaconceptionofthemechanicsoftheforeignexchangemarketisapplicable,thentheuseofwaveletsandtheirgeneralizationtowaveformdictionarieswillprovideamostusefulapproachtotheanalysisofthesedata.Therstsectionofthispaperbrieyreviewsonlytheliteraturemostrelevanttotheobjectivesofthispaper.Thisisbecausetheliteraturehasbeenwellreviewedquiterecentlybyothers;foroneofthelatestsee[10](Guillaumeetal1994).ThenextsectiondiscussestheideaofdictionariesofwaveformsandthelinkbetweenregularFourieranalysisandthescalingpropertiesofwavelets.Thefollowingsectiondescribesthedataandindicatestheextenttowhichthesub-samplingschemechosenfacilitatestheanalysis.Theresultsofthewaveformanalysisarediscussedinthenextsectionandthepaperiscompletedwithabriefsummary.2.ReviewoftheLiterature.Themostrecentliteraturecanbeusefullyreviewedintermsofthethreegroupingsindicatedabove.MorecomprehensivereviewscanbefoundinGuillaumeetalcitedabove,[3](Bollerslevetal1990),andin[14](Mulleretal1990).Atthemoment,thereisverylittleagreementabouttheformoftheforeignexchangedataandthemechanismsthatmightgeneratethem.Thisinconclusiveresultisdueneithertoalackofinterest,nortoalackofimaginationinattemptingtoanalyzethesedata.DepartmentofEconomics,NewYorkUniversity,NewYork,NY.Internet:[e-mail]ics,StanfordUniversity,StanfordCA94305.Internet:[e-mail]1Thetruthisthatthegeneratingmechanismforthesedataislikelytobeverycomplexandtheforeignexchangemarketisknowntobeevolving.Consequently,attemptstoprovidesuccinctparsimoniousmodelsofthedataarehamperedbytheextenttowhichthesimplifyingassumptionsneededbysuchanapproachareviolated.Therstquestioniswhatistherelevantdistribution,orratherwhataretherelevantclassesofdistri-butions.Thereisstillconsiderabledisputeaboutthisfundamentalmatterasisrecordedin[14](Mulleretal1990).Forexample,[2](BootheandGlassman1987)usingdailyspotforeignexchangerates,examinedsomealternativestotheGaussiandistributionandconcludedthattheStudenttdistributionoramixtureofnormalsisthepreferredchoiceandthattherewasevidencethattheparametersinvolvedvaryovertime.Morerecently,andincontrast,[12](LoretanandPhillips1994)usedthepropertiesofthetailbehaviorofthePareto-LevyformtoexaminethecovariancestationarityofobservationsonthedailyStandardandPoor’s500indexanddailyforeignexchangespotrates.LoretanandPhillipspurs