1、The Econometric Modelling of Financial Time SeriesThisisthesubstantiallyrevisedandupdatedsecondeditionofTerenceMillsbest-sellinggraduatetextbookTheEconometricModellingofFinancialTimeSeries.Thebookprovidesdetailedcoverageofthevarietyofmodelsthatarecurrentlybeingusedintheempiricalanalysisof?nancialmar
2、kets.Coveringbond,equityandforeignexchangemarkets,itisaimedatscholarsandpracti-tionerswishingtoacquireanunderstandingofthelatestresearchtechniquesand?ndings,andalsograduatestudentswishingtoresearchinto?nancialmarkets.Thissecondeditionincludesagreatdealofnewmaterialthathasbeendevel-opedinthelastsixye
3、ars,andalsoprovidesamorein-depthtreatmentoftwocrucial,andrelated,areas:thetheoryofintegratedprocessesandcointegration.Completelynewmaterialdiscussesthedistributionalpropertiesofassetreturnsandrecentandnoveltechniquesofanalysingandinterpretingvectorautoregres-sionsthatcontainintegratedandpossiblycoin
4、tegratedvariables.TerenceMillsisProfessorofEconomicsatLoughboroughUniversity.HeobtainedhisPh.8D.attheUniversityofWarwick,lecturedattheUniversityofLeeds,andhasheldprofessorialappointmentsatCityUniversityBusinessSchoolandtheUniversityofHull.Hehasover100publications,includingarticlesintheAmericanEconom
5、icReview,JournalofEconometricsandJournalofAppliedEconometrics.Dataappendixavailableon/.lboro.ac.uk/departments/ec/cup/.TheEconometricModellingofFinancialTimeSeriesSecondeditionTerenceC.MillsProfessorofEconomics,LoughboroughUniversityPUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AN
6、D ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia /.cambridge.org ? Cambridge University Press 1999 This edition ? Cam
7、bridge University Press (Virtual Publishing) 2003 First published in printed format 1993 Second edition 1999 A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 62413 4 hardback Original ISBN 0 521 62492 4 paperb
8、ack (Second edition) ISBN 0 511 01200 4 virtual (eBooks4 Edition) ContentsPrefacetosecondeditionpagevii1Introduction12Univariatelinearstochasticmodels:basicconcepts82.1Stochasticprocesses,ergodicityandstationarity82.2Stochasticdifferenceequations112.3ARMAprocesses132.4Linearstochasticprocesses282.5A
9、RMAmodelbuilding282.6Non-stationaryprocessesandARIMAmodels372.7ARIMAmodelling482.8ForecastingusingARIMAmodels533Univariatelinearstochasticmodels:furthertopics613.1Determiningtheorderofintegrationofatimeseries623.2Decomposingtimeseries:unobservedcomponentmodelsandsignalextraction993.3Measuresofpersis
10、tenceandtrendreversion1073.4Fractionalintegrationandlongmemoryprocesses1144Univariatenon-linearstochasticmodels1224.1Martingales,randomwalksandnon-linearity1224.2Testingtherandomwalkhypothesis1244.3Stochasticvolatility1264.4ARCHprocesses1314.5Othernon-linearunivariatemodels1534.6Testingfornon-linear
11、ity171vviContents5Modellingreturndistributions1775.1Descriptiveanalysisofthreereturnseries1775.2Twomodelsforreturndistributions1785.3Determiningthetailshapeofareturnsdistribution1845.4Empiricalevidenceontailindices1885.5Testingforcovariancestationarity1935.6Modellingthecentralpartofreturnsdistributi
12、ons1965.7Dataanalyticmodellingofskewnessandkurtosis1985.8Distributionalpropertiesofabsolutereturns2005.9Summaryandfurtherextensions2036Regressiontechniquesfornon-integrated?nancialtimeseries2056.1Regressionmodels2056.2ARCH-in-meanregressionmodels2186.3Misspeci?cationtesting2216.4Robustestimation2336
13、.5Themultivariatelinearregressionmodel2356.6Vectorautoregressions2386.7Variancedecompositions,innovationaccountingandstructuralVARs2456.8VectorARMAmodels2487Regressiontechniquesforintegrated?nancialtimeseries2537.1Spuriousregression2537.2Cointegratedprocesses2607.3Testingforcointegrationinregression
14、2687.4Estimatingcointegratingregressions2737.5VARswithintegratedvariables2797.6CausalitytestinginVECMs2977.7Fullymodi?edVARestimation2997.8Impulseresponseasymptoticsinnon-stationaryVARs3028Furthertopicsintheanalysisofintegrated?nancialtimeseries3068.1Testingforasinglelong-runrelationship3068.2Common
15、trendsandcycles3108.3EstimatingpermanentandtransitorycomponentsofaVECM3158.4Presentvaluemodels,excessvolatilityandcointegration3188.5Generalisationsandextensionsofcointegrationanderrorcorrectionmodels334Dataappendix340References342PrefacetothesecondeditionInthesixyearssinceIcompletedthemanuscriptfor
16、the?rsteditionofTheEconometricModellingofFinancialTimeSeries,therehavebeenmanyadvancesintimeserieseconometrics,someofwhichhavebeenindirectresponsetofeaturesfoundinthedatacomingfrom?nancialmarkets,whileothershavefoundreadyapplicationin?nancial?elds.Incorporatingthesedevelopments,andomittingsometechni
17、questhathavesincefallenoutoffavour,hasledtoasecondeditionratherdifferenttothe?rst.Althoughthebasicstructureofthebookremainsthesame,itisworthpointingoutthesechangesinsomedetail.Chapters1and2remainessentiallythesame,althoughexampleshavebeenupdated.Itisinchapter3thatthe?rstmajorchangesappear.Likeitorno
18、t,theanalysisofunitrootshascometodominatemuchoftimeserieseconometricsduringthe1990s.Althoughmuchoftherecentpub-lishedresearchonunitrootshasbeenhighlytechnicalwithoutbeingparticularlyilluminating,Ifeltthatamoreformalanalysisofunitroottestingandinferencethanappearedinthe?rsteditionwaswarranted.Tothise
19、nd,thereisnowamoredetailedtreatmentoftheasymptoticdis-tributionsofthevariousteststatisticsandanemphasisonMonteCarlosimulationtoobtainestimatesofthesedistributionsandhencecriticalvalues.Newerdevelopmentsincorporatingbreakingandevolvingtrendsandstochasticunitrootsarealsodiscussed.Thischapteralsoinclud
20、esarathermoredetailedtreatmentoffractionallyintegratedprocesses.Evidenceofnon-linearityin?nancialtimeserieshasaccumulatedovertheyears,althoughtheinitialenthusiasmatthestartofthedecadeforchaoticdynamicshasratherdissipated.StochasticvariancemodelsandthemanyextensionsoftheARCHprocesshavebecomeverypopul
21、ar,thelatterleadingtoanimpressivearrayofacronyms!Bothoftheseclassesofmodelsareanalysedingreaterdetailthanbeforeinchapter4.Arti?cialneuralnetworksalsohavetheirdevotees,andthesearealsonowbrie?ydiscussedinthischapter.Non-linearitygoeshand-in-handwithviiviiiPrefacetothesecondeditionnon-normality,andthec
22、ompletelynewchapter5looksatvariousmeth-odsofmodellingreturndistributionsandtransformationsofreturns,introducingawealthoftechniquesthathaveonlyrecentlymadetheirappearanceintheliterature.Muchofthematerialofchapter6(previouslychapter5)remainsasbefore,althoughnewsectionsonGMMandrobustestimationandtest-i
23、ngparameterstabilityareincludedandalternativemethodsofcomput-ingvariancedecompositionsareintroduced.Asdevelopmentsintheanalysisofunitrootsinunivariatetimeseriesledtomanychangesinchapter3,soanalogousdevelopmentsintheanalysisofunitrootsandcointegrationinamultivariateframeworkhasledtoacompleterewriting
24、ofthechapter,nowchapter7,onregressiontechniquesforintegrated?nancialtimeseries.Amoreformalapproachistakenhereaswell,withMonteCarlosimulationagainbeingusedtobringoutthedifferencesinthevariousasymptoticdistributionsofestimatorsthatresultfromalter-nativesetups.TheVECMframeworkisnowusedtoanalysecointegr
25、a-tioninnon-stationaryVARs,andadetailedtreatmentofcausalitytesting,alternativeestimationmethods,andimpulseresponseasympto-ticsisalsoprovided.The?nalchapter,chapter8,introducesfurthertopicsintheanalysisofintegrated?nancialtimeseries,suchascommontrendsandcyclesandestimatingpermanentandtransitorycompon
26、ents,andlooksatsomenon-lineargeneralisationsofcointegrationanderrorcorrectionmechanisms.Italsoincludesmaterialonpresentvaluemodelsandexcessvolatility,butonlywithintheVARframework:thediscussionofthe?rstgenerationofexcessvolatilitytestshasbeenomittedfromthiseditionastheynolongerappeartobeused.Manynewe
27、xampleshavebeendevelopedusingseveralnewdatasets.IwouldliketothankChrisBrooksandScottSpearsforgenerouslysup-plyingtheirexchangerateandstockmarketdata,respectively,andRaphaelMarkellosforprovidingtheneuralnetworkexample.Mythanksalsogotothekindreviewersofthe?rsteditionandtothemanypeoplewhohavecontactedm
28、esincethebookappeared:theirencouragingcom-mentsconvincedmethatasecondeditionwasworthwhileembarkingupon.Sincetheappearanceofthe?rsteditionIhaveoncemorechangeduniversities.Loughboroughhasprovidedmewithaverycongenialatmo-sphereinwhichtoresearchandeventobeabletoteachsomeofthematerialcontainedinthebook!T
29、heproductionteamatCambridgeUniversityPresscontinuetodoa?nejob,andPatrickMcCartan,andnowAshwinRattan,canalwaysberelieduponforencouragementandsomeentertaininggossip!Finally,butofcoursetheyshouldbeattheheadofanylistofacknowledgements,mythanksandlovegotomyfamily,TheaandAlexa,andtomymother,Rose.1Introduc
30、tionTheaimofthisbookistoprovidetheresearcherin?nancialmarketswiththetechniquesnecessarytoundertaketheempiricalanalysisof?nancialtimeseries.Toaccomplishthisaimweintroduceanddevelopbothuni-variatemodellingtechniquesandmultivariatemethods,includingthoseregressiontechniquesfortimeseriesthatseemtobeparti
31、cularlyrelevanttothe?nancearea.Whydoweconcentrateexclusivelyontimeseriestechniqueswhen,forexample,cross-sectionalmodellingplaysanimportantroleinempiricalinvestigationsoftheCapitalAssetPricingModel(CAPM):see,asanearlyandin?uentialexample,FamaandMacBeth(1973)?Ouransweristhat,apartfromtheusualconsidera
32、tionsofpersonalexpertiseandinter-est,plusmanuscriptlengthconsiderations,itisbecausetimeseriesana-lysis,inbothitstheoreticalandempiricalaspects,hasbeenformanyyearsanintegralpartofthestudyof?nancialmarkets,withempiricalresearchbeginningwiththepapersbyWorking(1934),Cowles(1933,1944)andCowlesandJones(19
33、37).Workingfocusedattentiononapreviouslynotedcharacteristicofcommodityandstockprices:namely,thattheyresemblecumulationsofpurelyrandomchanges.Cowlesinvestigatedtheabilityofmarketanalystsand?nancialservicestopredictfuturepricechanges,?ndingthattherewaslittleevidencethattheycould.CowlesandJonesreported
34、evidenceofpositivecorrelationbetweensuccessivepricechangesbut,asCowles(1960)waslatertoremark,thiswasprobablyduetotheirtakingmonthlyaveragesofdailyorweeklypricesbeforecomputingchanges:aspuriouscorrelationphenomenonanalysedbyWorking(1960).Thepredictabilityofpricechangeshassincebecomeamajorthemeof?nanc
35、ialresearchbut,surprisingly,littlemorewaspublisheduntilKendalls(1953)study,inwhichhefoundthattheweeklychangesinawidevarietyof?nancialpricescouldnotbepredictedfromeitherpastchangesintheseriesorfrompastchangesinotherpriceseries.Thisseems12TheEconometricModellingofFinancialTimeSeriestohavebeenthe?rstex
36、plicitreportingofthisoft-quotedpropertyof?nancialprices,althoughfurtherimpetustoresearchonpricepredict-abilitywasonlyprovidedbythepublicationofthepapersbyRoberts(1959)andOsborne(1959).Theformerpresentsalargelyheuristicargu-mentforwhysuccessivepricechangesshouldbeindependent,whilethelatterdevelopsthe
37、propositionthatitisnotabsolutepricechangesbutthelogarithmicpricechangeswhichareindependentofeachother:withtheauxiliaryassumptionthatthechangesthemselvesarenormallydis-tributed,thisimpliesthatpricesaregeneratedasBrownianmotion.Thestimulationprovidedbythesepaperswassuchthatnumerousarticlesappearedover
38、thenextfewyearsinvestigatingthehypothesisthatpricechanges(orlogarithmicpricechanges)areindependent,ahypothesisthatcametobetermedtherandomwalkmodel,inrecogni-tionofthesimilarityoftheevolutionofapriceseriestotherandomstaggerofadrunk.Indeed,thetermrandomwalkisbelievedtohave?rstbeenusedinanexchangeofcor
39、respondenceappearinginNaturein1905(seePearsonandRayleigh,1905),whichwasconcernedabouttheoptimalsearchstrategyfor?ndingadrunkwhohadbeenleftinthemiddleofa?eld.Thesolutionistostartexactlywherethedrunkhadbeenplaced,asthatpointisanunbiasedestimateofthedrunksfuturepositionsincehewillpresumablystaggeralong
40、inanunpredictableandrandomfashion.ThemostnaturalwaytostateformallytherandomwalkmodelisasP?P?a?1:1?tt?1twherePisthepriceobservedatthebeginningoftimetandaisanerrortttermwhichhaszeromeanandwhosevaluesareindependentofeachother.Thepricechange, P?P?P,isthussimplyaandhencettt?1tisindependentofpastpricechan
41、ges.Notethat,bysuccessivebackwardsubstitutionin(1.1),wecanwritethecurrentpriceasthecumulationofallpasterrors,i.e.XtP?atii?1sothattherandomwalkmodelimpliesthatpricesareindeedgeneratedbyWorkingscumulationofpurelyrandomchanges.OsbornesmodelofBrownianmotionimpliesthatequation(1.1)holdsforthelogarithmsof
42、Pand,further,thataisdrawnfromazeromeannormaldistributiontthavingconstantvariance.Introduction3MostoftheearlypapersinthisareaarecontainedinthecollectionofCootner(1964),whileGrangerandMorgenstern(1970)provideadetaileddevelopmentandempiricalexaminationoftherandomwalkmodelandvariousofitsre?nements.Amazi
43、ngly,muchofthisworkhadbeenanticipatedbytheFrenchmathematicianLouisBachelier(1900,EnglishtranslationinCootner,1964)inaremarkablePh.D.thesisinwhichhedevelopedanelaboratemathematicaltheoryofspeculativeprices,whichhethentestedonthepricingofFrenchgovernmentbonds,?ndingthatsuchpriceswereconsistentwithther
44、andomwalkmodel.WhatmadethethesisevenmoreremarkablewasthatitalsodevelopedmanyofthemathematicalpropertiesofBrownianmotionwhichhadbeenthoughttohave?rstbeenderivedsomeyearslaterinthephysicalsciences,particularlybyEinstein!Yet,asMandelbrot(1989)remarks,Bachelierhadgreatdif?cultyinevengettinghimselfaunive
45、rsityappoint-ment,letalonegettinghistheoriesdisseminatedthroughouttheacademiccommunity!Itshouldbeemphasisedthattherandomwalkmodelisonlyahypoth-esisabouthow?nancialpricesmove.Onewayinwhichitcanbetestedisbyexaminingtheautocorrelationpropertiesofpricechanges:see,forexample,Fama(1965).Amoregeneralperspe
46、ctiveistoview(1.1)asaparticularmodelwithintheclassofautoregressive-integrated-movingaverage(ARIMA)modelspopularisedbyBoxandJenkins(1976).Chapter2thusdevelopsthetheoryofsuchmodelswithinthegeneralcontextof(univariate)linearstochasticprocesses.Weshouldavoidgivingtheimpressionthattheonly?nancialtimeseri
47、esofinterestarestockprices.Thereare?nancialmarketsotherthanthoseforstocks,mostnotablyforbondsandforeigncurrency,buttherealsoexistthevariousfutures,commodityandderivativemar-kets,allofwhichprovideinterestingandimportantseriestoanalyse.Forcertainofthese,itisbynomeansimplausiblethatmodelsotherthanthera
48、ndomwalkmaybeappropriateor,indeed,modelsfromaclassotherthantheARIMA.Chapter3discussesvarioustopicsinthegeneralana-lysisoflinearstochasticmodels,mostnotablythatofdeterminingtheorderofintegrationofaseriesand,associatedwiththis,theappropriatewayofmodellingtrendsandstructuralbreaks.Italsoconsidersmethod
49、sofdecomposinganobservedseriesintotwoormoreunobservedcompo-nentsandofdeterminingtheextentofthememoryofaseries,bywhichismeantthebehaviouroftheseriesatlowfrequenciesor,equivalently,intheverylongrun.Avarietyofexamplestakenfromthe?nanciallitera-tureareprovidedthroughoutthechapter.Duringthe1960smuchresearchwasalsocarriedoutonthetheoreticalfoundationsof?nancialmarkets,leadingtothedevelopmentofthe4TheEconometricModellingofFinancialTimeSeriestheoryofef?cientcapitalmarkets.AsLeRoy(1989)discusses,thisledtosomeseriousquestionsbeingrais
