presenting time series analysis-filtering and forecasting with wavelet analysis

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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;Zgisweaklystationaryifboththemeanofxtandthecovariancebetweenxtandxtkaretime-invariant,wherekisanarbitraryinteger.Thatis,fxtgisweaklystationaryifE[xt]=andCov(xt;xtk)=kwhereisconstantandkisindependent

7、oft.Thatis,weassumethatthefirsttwomomentsofxtarefinite.Inthespecialcasewherextisnormallydistributed,thentheweakstationarityisequivalenttostrictstationarity.Thecovariancekiscalledthelag-kautocovarianceofxtandhasthefollowingproperties:•0=Var(xt)•k=k202QuantitativeAnalyticsThelatter

8、holdsbecauseCov(xt;xt(k))=Cov(xt(k);xt)=Cov