时间序列分析（英文）

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1、LectureNotesonTIMESERIESANALYSISbyProfessorHowellTong°c2004CHAPTERONEGENERAL(0)ExamplesofTimeSeries.Trytofindsomemoreexamplesbyyourself!(1)t2f0;§1;§2;:::g=T,calledtheparameterspace,T.XttakesrealvaluessothatstatespaceisR.fXt:t=0;§1;§2;:::gdenotesasequenceofrandomvariables(r.v.s.)indexe

2、dbyt,i.e.foreacht,Xtisarealvaluedr.v.fXtgisastochasticprocess.Iftdenotestimes,fXt:t=0;§1;§2;:::giscalledatimeseries(indiscretetime),tobeabbreviatedbyT.S.(2)fXtgisstrictlystationary(´stationaryinthestrictsense)ifforanyt1;t2;:::;tn2T,(anypositiveintegern),andanyk2T,thejointdistribution

3、offXt1;Xt2;:::;XtngisthesameasthejointdistributionoffXt1+k;Xt2+k;:::Xtn+kg.Wehavestatisticalequilibrium.(3)fXtgissecondorderstationary(´stationaryinwidesense´covariancestationary´widesensestationary)if(i)E(Xt)=¹<1;independentoft(ii)VarX=¾2<1;independentoftandcov(X;X)isafunctionof¿onl

4、y:ttt+¿ThiscourseismainlyconcernedwithwidesensestationaryT.S.Forsimplicity,wecallthemstationaryT.S.)N.B.Strictstationarity+existenceof¹;¾26(widesensestationarity.14(4)cov(Xt;Xt+¿)=°(¿),autocovariancefunctionatlag¿.f°(¿)¿=0;§1;§2;:::giscalledtheautocovariancefunction(a.r.v.f.).°(¿)=°(

5、¡¿)becausecov(Xt;Xt+¿)=cov(Xt+¿;Xt).4½(¿)=°(¿)=°(0),normalisation.f½(¿):¿=0;§1;§2;:::giscalledtheautocorrelationfunction(a.c.r.f.).Sometimeswrite°¿for°(¿)and½¿for½(¿).(i)°(0)=¾2=varX(obvious)t(ii)j°(¿)j·°(0)all¿(byCauchy-Schwartzinequality)(iii)°(¿)=°(¡¿)8¿(done)(i0)½(0)=1(ii0)j½(¿)j

6、·18¿(iii0)½(¡¿)=½(¿)8¿f°(¿)gisapositivesemi¡definitefunctioninthesensethatforanyt1;t2;:::;tn2Tand8z1;z2;:::;zn2RPnPnr=1s=1°(tr¡ts)zrzs¸0For:0·Var(z1Xt1+z2Xt2+:::+znXtn)XnXn=cov(Xtr;Xts)zrzsr=1s=1PnPn=r=1s=1°(tr¡ts)zrzskf°(¿)gisthea:c:v:f:ofastationaryT:S:,f°(¿)gispositivesemi¡definite:

7、CharacterisationTheorem.SeeBrockwell&Davis,p.27.(5)OftenweassumethatX;X;:::;XarejointlyGaussianforanyn2Z+andanyt1t2tnt1;t2;:::;tn2T.ThenfXt:t=0;§1;:::giscalledaGaussianT.S..2ForGaussianT.S.strictstationarity´widesensestationarity.3CHAPTERTWOSTANDARDMODELS(0)StrictWhiteNoiseProcess/Wh

8、iteNoiseProc