arma and arch models with infinite variance innovations

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1、c15-ARMA-ARCHInfPage495Thursday,October26,20062:12PMCHAPTER15ARMAandARCHModelswithInfinite-VarianceInnovationsnChapter6wedescribedautoregressivemovingaverage(ARMA)Iprocessesandtheirpropertieswithregardstostationarityandestima-tion.Inthepreviouschapterweintroducedastablenon-Gaussiandis-tribu

2、tedrandomvariable(exhibitingheavytails)alongwithmajorpropertiesofstabilityandpowerlawdecayoftailswhichimplythatitssecondmomentisinfinite.1ARMAmodelscanthenbeextendedbycon-sideringerrortermsthatfollowastablenon-Gaussiandistribution,giv-ingriseto“infinitevarianceautoregressivemovingaveragemode

3、ls.”Suchheavy-tailedprocessesareencounteredineconomicsandfinanceanditisofpracticalinteresttoanalyzetheirproperties.However,thestatisticaltheoryoftheinfinitevariancemodelsisfundamentallydiffer-entfromthatofmodelswithfinitevariances.InthischapterwedescribeARMAmodelsandautoregressiveconditionalh

4、eteroskedastic(ARCH)modelswithinfinitevarianceinnovationsandoutlinerelevantproper-tiesalongwithestimationapproaches.INFINITEVARIANCEAUTOREGRESSIVEPROCESSESRecallfromequation(6.8)inChapter6thatastationaryautoregressive(AR(p))timeseries{yt}isrepresentedbydifferenceequationyt=a0+a1yt–1+a2yt–2+

5、…+apyt–p+εt(15.1)1Thereareversionsofthestablerandomvariablescalled“modifiedtemperedsta-ble”withfinitesecondmoment.JanRosinski,“TemperingStableProcesses,”inO.E.Barndorff-Nielsen(ed.),SecondMaPhyStoConferenceonLévyProcesses:TheoryandApplications,2002,pp.215–220.495c15-ARMA-ARCHInfPage496Thur

6、sday,October26,20062:12PM496FINANCIALECONOMETRICSwhere{εt}isasequenceofindependentandidenticallydistributed(IID)errorsandφ=(a0,a1,…,ap)′isanunknownparametervectorwithtrue2valueφ0.Whenthesecondmomentoftheerrorterm,E()εt,isfinite,wehaveshownthatvariousestimators(e.g.,least-squaresestimatorsan

7、dmaximumlikelihoodestimators)ofφ0areasymptoticallynormal2andseveralmethodsareavailableforstatisticalinference.WhenE()εtisinfinite,wecallthemodelgivenbyequation(15.1)theinfinitevari-anceautoregressivemodel(IVARmodel).Suchheavy-tailedmodelshavebeenencounteredineco