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1、230IntroductoryEconometricsforFinancebecomesinsignificantafteraroundlag5.Recallingthattheautocorre-lationcoefficientforthisAR(1)atlagsisequalto(−0.5)s,thiswillbepositiveforevens,andnegativeforodds.Onlythefirstpacfcoefficientissignificant(andnegative).Figure
2、5.7plotstheacfandpacfforanon-stationaryseries(seechapter7foranextensivediscussion)thathasaunitcoefficientonthelaggeddependentvariable.Theresultisthatshockstoyneverdieaway,andpersistindefinitelyinthesystem.Consequently,theacffunctionre-mainsrelativelyflata
3、tunity,evenuptolag10.Infact,evenbylag10,theautocorrelationcoefficienthasfallenonlyto0.9989.Notealsothatonsomeoccasions,theacfdoesdieaway,ratherthanlookinglikefigure5.7,evenforsuchanon-stationaryprocess,owingtoitsinherentinstabilitycombinedwithfinitecomput
4、erprecision.Thepacf,however,issignificantonlyforlag1,correctlysuggestingthatanautoregressivemodelwithnomovingaveragetermismostappropriate.Finally,figure5.8plotstheacfandpacfforamixedARMAprocess.Asonewouldexpectofsuchaprocess,boththeacfandthepacfdeclinege
5、ometrically--theacfasaresultoftheARpartandthepacfasaresultoftheMApart.ThecoefficientsontheARandMAare,however,sufficientlysmallthatbothacfandpacfcoefficientshavebecomeinsignificantbylag6.5.7BuildingARMAmodels:theBox–JenkinsapproachAlthoughtheexistenceofARMA
6、modelspredatesthem,BoxandJenkins(1976)werethefirsttoapproachthetaskofestimatinganARMAmodelinasystematicmanner.Theirapproachwasapracticalandpragmaticone,involvingthreesteps:(1)Identification(2)Estimation(3)Diagnosticchecking.Thesestepsarenowexplainedingre
7、aterdetail.Step1Thisinvolvesdeterminingtheorderofthemodelrequiredtocapturethedy-namicfeaturesofthedata.Graphicalproceduresareused(plottingthedataovertimeandplottingtheacfandpacf)todeterminethemostap-propriatespecification.Univariatetimeseriesmodellingan
8、dforecasting231Step2Thisinvolvesestimationoftheparametersofthemodelspecifiedinstep1.Thiscanbedoneusingleastsquaresoranothertechnique,knownasmaximumlikelihood,dependingonthemodel.Step3Thisinvolvesmodelchecking--i.e.determiningwhetherthemo
