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SPSS数据统计分析与实践主讲：周涛副教授北京师范大学资源学院2007-12-11教学网站：第十七章：时间序列数据的自相关分析（AutocorrelationinTimeSeriesData）Contents:Contents:1.IllustrationofAutocorrelationProblems2.First-OrderAutoregressiveErrorModel3.Durbin-WatsonTestforAutocorrelation4.SPSSExampleforDurbin-WatsonTest5.RemedialMeasuresforAutocorrelation6.SPSSExampleforAutoregression7.SPSSAutoregression过程的几点补充AutocorrelationzThebasicregressionmodelsconsideredsofarhaveassumedthattherandomerrortermsεiareeitheruncorrelatedrandomvariablesorindependentnormalrandomvariables.zButmanyregressionapplicationinvolvetimeseriesdata.Forsuchdata,theassumptionofuncorrelatedorindependenterrortermisoftennotappropriate.zErrortermscorrelatedovertimearesaidtobeautocorrelatedautocorrelatedorseriallycorrelatedseriallycorrelated.ProblemsofAutocorrelation1.Theestimatedregressioncoefficientsarestillunbiased,buttheynolongerhavetheminimumvariancepropertyandmaybequiteinefficient.2.MSEmayseriouslyunderestimatethevarianceoftheerrorterms.3.s{bk}calculatedaccordingtoordinaryleastsquaresproceduresmayseriouslyunderestimatethetruestandarddeviationoftheestimatedregressioncoefficient.4.ConfidenceintervalsandtestsusingthetandFdistributionsarenolongerstrictlyapplicable.1.IllustrationofAutocorrelationProblemsExample—whenislargeToillustratetheseproblemsintuitively,weconsiderthesimplelinearregressionmodelwithtimeseriesdata:Here,YtandXtareobservationsforperiodt.Letusassumethattheerrortermsεiarepositivelyautocorrelatedasfollows:zThe,calleddisturbances,areindependentnormalrandomvariables.Thus,anyerrortermisthesumofthepreviouserrortermandanewdisturbanceterm.Weshallassumeherethatthehavemean0andvariance1.tttXYεββ++=10tttu+=−1εεtutε1−tεtutu0εExample—whenislargeInTable13.1,column1,weshow10randomobservationsonthenormalvariableutwithmean0andvariance1,obtainedfromastandardnormalrandomnumbersgenerator.Supposenowthatε0=3.0;weobtainthen:KK8.27.05.35.35.00.3212101=−=+==+=+=uuεεεεTab13.1ExampleofPositivelyAutocorrelatedErrorTerms.0εExamplezTheerrortermsinTable13.1,column2,areplottedinFigure13.1.Theirpositiverelationovertimeisshownbythefactthatadjacenterrortermstendtobeofthesamesignandmagnitude.Figure1.ExampleofPositivelyAutocorrelatedErrorTermsExamplezSupposethatXtintheregressionmodelrepresentstime,suchthatX1=1,X2=2,etc.zFurther,supposeweknowthatβ0=2andβ1=0.5,sothatthetrueregressionfunctioinisE{Y}=2+0.5X.zTheobservedYvaluebasedontheerrortermsincolumn2ofTable13.1areshownincolumn3.Forexample,Y0=2+0.5(0)+3.0=5.0ExamplezFigure13.2acontainsthetrueregressionlineE{Y}=2+0.5XandtheobservedYvalueshowninTable13.1,column3.zFigure13.2bcontainstheestimatedregressionline,fittedbyordinaryleastsquaresmethods,andrepeatstheobservedYvalues.XY07.085.5ˆ−=Fig.13.2(a)TrueRegressionLineandObservationwhenε0=3Fig.13.2(b)FittedRegressionLineandObservationwhenε0=3ExamplezNoticethatthefittedregressionlinedifferssharplyfromthetrueregressionlinebecausetheinitialε0valuewaslargeandthesucceedingpositivelyautocorrelatederrortermstendedtobelargeforsometime.zThispersistencypatterninthepositivelyautocorrelatederrortermsleadstoafittedregressionlinefarfromthetrueone.XY07.085.5ˆ−=Fig.13.2(a)TrueRegressionLineandObservationwhenε0=3Fig.13.2(b)FittedRegressionLineandObservationwhenε0=3Example—whenissmallWhentheinitialε0valueissmall,say,ε0=-0.2,andthedisturbancesdifferent,asharplydifferentfittedregressionlinemighthavebeenobtainedbecauseofthepersistencypattern,asshowninFigure13.2c.0εXY451.0773.1ˆ+=Fig.13.2(C)FittedRegressionLineandObservationswithε0=-0.2andDifferentDisturbancesConclusionszThevariationfromtermsmaybesosubstantialastoleadtolargevariancesoftheestimatedregressioncoefficientswhenOLSmethodsareused.zAnotherkeyproblemwithapplyingOLSmethodswhentheerrortermsarepositivelyautocorrelated,asmentionedbefore,isthattheMSEmayseriouslyunderestimatethevarianceoftheεt.2.First-OrderAutoregressiveErrorModelSimpleLinearRegressionThegeneralizedsimplelinearregressionmodelforonepredictorvariablewhentherandomerrortermsfollowafirst-orderautoregressive,orAR(1),processis:ttttttuXY+=++=−110ρεεεββ(1)Where:ρisaparametersuchthat|ρ|1Theparameterρiscalledtheautocorrelationparameter.utareindependentN(0,σ2)MultipleRegressionThegeneralizedmultipleregressionmodelwhentherandomerrortermsfollowafirst-orderautoregressiveprocessis:ttttptptttuXXXY+=+++++=−−−11,122110ρεεεββββL(2)Where:|ρ|1utareindependentN(0,σ2)Thus,weseethatgeneralizedmultipleregressionmodel(2)isidenticaltotheearliermultipleregressionmodelexceptforthestructureoftheerrorterms.PropertiesofErrorTermsRegressionmodel(1)and(2)aregeneralizedregressionmodels(广义回归模型)becausetheerrortermsintheremodelsarecorrelated.However,theerrortermsstillhavemeanzeroandconstantvariance.2221}{0}{ρσεσε−==ttE(3)(4)Notethatthevarianceoftheerrortermshereisafunctionoftheautocorrelationparameterρ.PropertiesofErrorTermsThecovariancebetweenadjacenterrortermsεtandεt-1is:⎟⎟⎠⎞⎜⎜⎝⎛−=−2211),{ρσρεεσtt(5)Thecoefficientofcorrelationbetweenadjacenterrortermsεtandεt-1isdefinedasfollows:ρρσρσρσρεσεσεεσεερ=−−⎟⎟⎠⎞⎜⎜⎝⎛−==−−−222222111111}{}{),{),{tttt