金融计量ChrisBrooks

LearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingFinancialEconometricsLecture2:UnivariateTimeSeriesDayongZhangResearchInstituteofEconomicsandManagementAutumn,2012SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingLearningOutcomes◮Explainthedefiningcharacteristicsofvarioustypesofstochasticprocesses;◮IdentifyAutoregressiveandMovingAverageProcesses◮BuildtheARMAmodelusingBox-JenkinsApproach◮ProduceforecastforARMAmodelsandevaluatetheaccuracyofpredictionsSouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesaStrictlyStationaryProcessStationarityisaveryimportantissueintimeseriesdata.LetFY(yt1,...,ytT)representstheCDFofthejointdistributionofYtattimet1,t2,...,tT∈Z,astrictlystationaryprocessisonewhere,foranyk∈ZandT=1,2,...FY(yt1,...,ytT)=FY(yt1+k,...,ytT+k)(1)Theprobabilitydistributionforthesequenceytisthesameasthatforyt+k∀k;inotherwords:astrictlystationaryprocesshasaprobabilitydistributionthatistime-invariant.SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesAWeaklyStationaryProcessIfaseriessatisfythefollowingconditions,itissaidtobeweaklyorcovariancestationary1.E(yt)=μ2.E(yt−μ)(yt−μ)=σ2∞3.E(yt1−μ)(yt2−μ)=γt2−t1∀t1,t2thatis:aconstantmean,constantvarianceandaconstantautocovariancestructure.SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesAutocovarianceandAutocorrelationTheautocovarianceisdefinedasE(yt−E(yt))(yt−s−E(yt−s))=γs,s=0,1,...(2)anditisknownastheautocovariancefunction.Normalizeusingthevariance,wehavetheautocorrelationsτs=γsγ0,s=0,1,...(3)Thegraphofτsplottedagainsts=0,1,...isknownasACForcorrelogram.SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesAWhiteNoiseProcessAwhitenoiseprocesshasconstantmeanandvarianceandzeroautocovariance:1.E(yt)=μ2.E(yt−μ)(yt−μ)=σ2∞3.γt−r=σ2ift=rand0otherwiseIfitisassumedthatytisnormallydistributed,thenthesampleautocorrelationcoefficientsarealsoapproximatelynormallydistributedˆτs∼N(0,1/T)(4)Thiscouldbeusedtoconstructanon-rejectionregionsuchas±1.96×1√Tat95%.SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesPortmanteauTestBoxandPierce(1970)developedtheQ-statistictotestthejointhypothesisthatallmoftheautocorrelationcoefficientsaresimultaneouslyequaltozero:Q=TmXk=1ˆτ2k(5)TheQ-statisticisasymptoticallydistributedasaχ2m.TheBox-PierceQ-statistichaspoorpropertiesinsmallsample,Ljung-Box(1978)modifiedtheQ-statistictoQ∗=T(T+2)mXk=1ˆτ2kT−k∼χ2m(6)SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesMovingAverageProcessLetut(t=1,2,3,...)beawhitenoiseprocess,thenyt=μ+ut+θ1ut−1+θ2ut−2+···+θqut−q(7)isaqthordermovingaveragemodel,denotedMA(q).SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesLagOperatorItisconvenienttousethelagoperator(orbackshiftoperator)tomanipulatetimeseriesdata.DefineLasthelagoperator,sothatLyt=yt−1denotethatytislaggedonce,andL2yt=yt−2.Usingthisnotation,theMA(q)processabovecanberewrittentoyt=μ+(1+θ1L+θ2L2+···+θqLq)ut=μ+θ(L)ut(8)whereθ(L)=(1+θ1L+θ2L2+···+θqLq)SouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesthePropertiesofMA(q)Sinceutisawhitenoiseprocess,wecouldeasilyworkoutthefollowing:1.E(yt)=μ;2.var(yt)=γ0=(1+θ21+···+θ2q)σ2;3.covarianceγs=(θs+θs+1θ1+θs+2θ2+···+θqθq−s)fors=1,2,...,q0forsqSouthwesternUniversityofFinanceandEconomicsFinancialEconometricsLectureNotes2:Uni.TimeSeriesLearningOutcomesNotationandConceptsARMAModelandtheBox-JenkinsApproachForecastingStationaryMovingAverageProcessAutoregressiveProcessesMAExampleConsiderthefollowingMA(2)processyt=ut+θ1ut−1+θ2ut−2(9)Givenutisazeromeanwhitenoiseprocesswithvarianceσ2◮Calculatethemeanandvarianceofyt◮DerivetheACFforthisp