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1、c06-ModelingUnivariatePage201Thursday,October26,20062:05PMCHAPTER6ModelingUnivariateTimeSeriesnthischapterwediscusstechniquesformodelingunivariatetimeIseries.Thesetechniquesare,forexample,employedforshort-termpredictionofassetpricesorreturnsortotestthemarket
2、-efficiencyhypothesis.Werestrictthediscussiontolineartimesseriesmodelsandfocusontheclassofautoregressivemovingaverage(ARMA)modelsAlthoughfinancialtimeseriestypicallyexhibitstructuresthataremorecomplexthanthoseprovidedbyARMAprocesses,ARMAmodelsareafirststartingp
3、ointandoftenserveasabenchmarkagainstmorecom-plexapproaches.Westartbyintroducingsometechnicalbackground,definitions,propertiesofARMAprocesses,andvariousmodelsbelongingtothisclass.ThepracticalstepsforderivingamodelfromdatausingtheBox-Jenkinsapproacharepresented
4、inthenextchapter.DIFFERENCEEQUATIONSInlineartimeseriesanalysisitiscommonlyassumedthatatimeseriestobemodeledcanberepresentedorapproximatedbyalineardifferenceequation.Inthissection,weintroducethenotationforlineardifferenceequationsandapproachestotheirsolutions
5、.NotationConsiderasituationwherethevalueofatimeseriesattimet,yt,isalinearfunctionofthelastpvaluesofyandofexogenousterms,denotedbyεt.Wewriteyt=a1yt–1+a2yt–2+···+apyt–p+εt(6.1)201c06-ModelingUnivariatePage202Thursday,October26,20062:05PM202FINANCIALECONOMETRIC
6、SExpressionsoftype(6.1)arecalleddifferenceequations.Iftheexogenoustermsarezero,(6.1)iscalledanhomogenousdifferenceequation.Iftheexogenoustermisawhitenoise,expression(6.1)repre-sentsanautoregressiveprocessoforderp,whichwillbedetailedlater.Let’snowintroducethe
7、lagoperatornotation.Thelagoperator,denotedbyL,isanoperatorthatshiftsthetimeindexbackwardbyoneunit.1Applyingthelagoperatortoavariableattimet,weobtainthevalueofthevariableattimet–1:Lyt=yt–1ApplyingL2amountstolaggingthevariabletwice.i.e.,L2y=L(Ly)=ttLyt–1=yt–2.
8、Moreformally,thelagoperatortransformsonetimeseries,say∞{}ytt=–∞intoanotherseries,say∞{}xtt=–∞wherext=yt–1.Aconstantccanbeviewedasaspecialseries,namelyseries∞{}ytt=–∞withyt=cforallt,andwecanapply
