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1ARandom-WalkorColor-ChaosontheStockMarket?-Time-FrequencyAnalysisofS&PIndexesPingChenIlyaPrigogineCenterforStudiesinStatisticalMechanics&ComplexSystemsTheUniversityofTexas,Austin,Texas78712E-Mail:pchen@physics.utexas.eduStudiesinNonlinearDynamics&Econometrics,1(2),87-103(1996)JELNumericalClassification:130(fluctuations),211(econometrics).JELClassificationSystemforJournalArticles:C14(nonparametricmethods),C22(time-seriesmodels,methods),C50(generaleconometricmodeling),E32(businessfluctuations:cycles).AbstractTherandom-walk(white-noise)modelandtheharmonicmodelaretwopolarmodelsinlinearsystems.Amodelinbetweeniscolorchaoswhichgeneratesirregularoscillationswithanarrowfrequency(color)band.Time-frequencyanalysisisintroducedforevolutionarytimeseriesanalysis.Thedeterministiccomponentfromnoisydatacanberecoveredbyatime-variantfilterinGaborspace.ThecharacteristicfrequencyiscalculatedfromtheWignerdecomposeddistributionseries.ItisfoundthataboutseventypercentoffluctuationsinStandard&Poorstockpriceindexes,suchastheFSPCOMandFSDXPmonthlyseries,detrendedbytheHodrick-Prescott(HP)filter,canbeexplainedbydeterministiccolorchaos.Thecharacteristicperiodofpersistentcyclesisaroundthreetofouryears.Theircorrelationdimensionisabout2.5.Theexistenceofpersistentchaoticcyclesrevealsnewperspectiveofmarketresilienceandnewsourcesofeconomicuncertainties.Thenonlinearpatterninthestockmarketmaynotbewipedoutbymarketcompetitionundernonequilibriumsituationswithtrendevolutionandfrequencyshifts.Thecolor-chaosmodelofstock-marketmovementsmayestablishalinkbetweenbusiness-cycletheoryandasset-pricingtheory.I.Introduction2Financetheoryinequilibriumeconomicsisbasedontherandom-walkmodelofstockprices.However,thereisamoregeneralscenario:amixedprocesswithrandomnoiseanddeterministicpattern,includingapossibilityofdeterministicchaos.Chaosiswidelyfoundinthefieldsofphysics,chemistry,andbiology.Buttheexistenceofeconomicchaosisstillanopenissue[BarnettandChen1988,BrockandSayers1989,Ramsey,Sayers,andRothman1990,DeCosterandMitchell1991,1994,Barnettet.al.1995].Trends,noise,andtimeevolutioncausedbystructuralchangesarethemaindifficultiesineconomictimeseriesanalysis.Amoregeneralizedspectralanalysisisneededfortestingeconomicchaos[Chen1988,1993].Measurementcannotbeseparatedfromtheory.Therearetwopolarmodelsinlineardynamics:whitenoiseandharmoniccycle.Correlationanalysisandspectralanalysisarecomplementarytoolsinthestationarytime-seriesanalysis.Whitenoisehasazerocorrelationandaflatspectrumwhileaharmoniccyclehasaninfinitecorrelationandasharpspectrumwithzero-width.Obviously,realdatafallbetweenthesetwoextremes.Amajorchallengeineconomictimeseriesanalysisishowtodealwithtimeevolution.Econometricmodels,suchastheARCHandGARCHmodelswithachangingmeanandvarianceareparametricmodelsinthenonstationarystochasticapproach[Engle1982,Bollerslev1986].Ageneralizedspectralapproachismoreusefulinstudiesofdeterministicchaos[Chen1993].Itisknownthatastationarystochasticprocessdoesnothaveastationaryorcontinuousinstantaneousfrequencyintime-frequencyrepresentation.Therefore,wedonotusethetermsstationaryandnonstationarywhicharefamiliarinastochasticapproach.Anewrepresentationwillintroducesomeconceptualchanges.Therearemanyfundamentaldifferencesbetweenanonlineardeterministicapproachandalinearstochasticapproachincludingtimescales,observationreferences,andtestingphilosophy.Fromtheviewoftheoreticalstudies,thediscrete-timewhitechaosgeneratedbynonlineardifferenceequationsistractableinanalyticmathematicsandcompatiblewiththeoptimizationrationality[DayandBenhabib1980,Benhabib1990].Fromtheneedsofempiricalanalysis,thecontinuous-timecolorchaosgeneratedbynonlineardifferentialequationsismorecapableofdescribingbusinesscyclesthanwhitechaos,becausetheirerraticfluctuationsandrecurrentpatterncanbecharacterizedbynonlinearoscillationswithirregularamplitudeandanarrowfrequency(color)bandinspectrum[Chen1988,1993;Zarnowitz1993].Weintroducethetime-frequencyrepresentationasanon-parametricapproachofgeneralizedspectralanalysisfortheevolutionarytimeseries[QianandChen1996].TheWignerdistributioninquantummechanicsandtheGaborrepresentationincommunicationtheorywerepioneeredbytwoNobellaureatephysicists[Wigner1933,Gabor1948].Appliedscientistsinsignalprocessinghavemadefundamentalprogressindevelopingefficientalgorithmsoftime-frequencydistributionseries[QianandChen1994a,b,1996].Thesearepowerfultoolsinourstudiesofeconomicchaos[Chen1994,1995,1996].Indealingwithproblemsofgrowingtrendsandstrongnoise,weapplytheHodrick-Prescott(HP)filterfortrend-cycledecomposition[HodrickandPrescott1981]andtime-variantfiltersin3Gaborspaceforpatternrecognition[QianandChen1996,Sunet.al.1996].Wegotclearsignalsoflow-dimensionalcolorchaosfromStandard&Poorstockmarketindicators.Thechaossignalscanexplainabout70percentofstockvariancesfromdetrendedcycles.Itscharacteristicperiodisaroundthreetofouryears.Theircorrelationdimensionisabout2.5.Thetimepathsoftheircharacteristicperiodisusefulinanalyzingcauseandeffectfromhistoricalevents.Clearly,thecolor-chaosmodeldescribesmorefeaturesofmarketmoveme