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1、Chapter5Presentingtimeseriesanalysis5.1BasicprinciplesoflineartimeseriesWeconsidertheassetreturnstobeacollectionofrandomvariablesovertime,obtainingthetimeseriesfrtginthecaseoflogreturns.Lineartimeseriesanalysisisafirststeptounderstandingthedynamicstructureofsuchaseries(seeBoxetal
2、.[1994]).Thatis,foranassetreturnrt,simplemodelsattemptatcapturingthelinearrelationshipbetweenrtandsomeinformationavailablepriortotimet.Forinstance,theinformationmaycontainthehistoricalvaluesofrtandtherandomvectorYthatdescribestheeconomicenvironmentunderwhichtheassetpriceisdeterm
3、ined.Asaresult,correlationsbetweenthevariableofinterestanditspastvaluesbecomethefocusoflineartimeseriesanalysis,andarereferredtoasserialcorrelationsorautocorrelations.Hence,Linearmodelscanbeusedtoanalysethedynamicstructureofsuchaserieswiththehelpofautocorrelationfunction,andfore
4、castingcanthenbeperformed(seeBrockwelletal.[1996]).5.1.1StationarityWhilethefoundationoftimeseriesanalysisisstationarity,autocorrelationsarebasictoolsforstudyingthisstationarity.Atimeseriesfxt;Zgissaidtobestronglystationary,orstrictlystationary,ifthejointdistributionof(xt1;::;xt
5、k)isidenticaltothatof(xt1+h;::;ytk+h)forallh(xt1;::;xtk)=(xt1+h;::;ytk+h)wherekisanarbitrarypositiveintegerand(t1;::;tk)isacollectionofkpositiveintegers.Thus,strictstationarityrequiresthatthejointdistributionof(xt1;::;xtk)isinvariantundertimeshift.Sincethisconditionisdifficulttov
6、erifyempirically,aweakerversionofstationarityisoftenassumed.Thetimeseriesfxt;Zgisweaklystationaryifboththemeanofxtandthecovariancebetweenxtandxt karetime-invariant,wherekisanarbitraryinteger.Thatis,fxtgisweaklystationaryifE[xt]=andCov(xt;xt k)=kwhereisconstantandkisindependent
7、oft.Thatis,weassumethatthefirsttwomomentsofxtarefinite.Inthespecialcasewherextisnormallydistributed,thentheweakstationarityisequivalenttostrictstationarity.Thecovariancekiscalledthelag-kautocovarianceofxtandhasthefollowingproperties:•0=Var(xt)• k=k202QuantitativeAnalyticsThelatter
8、holdsbecauseCov(xt;xt ( k))=Cov(xt ( k);xt)=Cov