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1、c08-ARCHModelsPage279Thursday,October26,20062:07PMCHAPTER8AutoregressiveConditionalHeteroskedasticModelsnlinearregressionanalysis,astandardassumptionisthatthevarianceofIallsquarederrortermsisthesame.Thisassumptioniscalledhomoske-dasticity(constantvariance).However,manytimeseriesdataexhibithet-ero
2、skedasticity,wherethevariancesoftheerrortermsarenotequal,andinwhichtheerrortermsmaybeexpectedtobelargerforsomeobservationsorperiodsofthedatathanforothers.Theissueisthenhowtoconstructmodelsthataccommodateheteroskedasticitysothatvalidcoefficientesti-matesandmodelsareobtainedforthevarianceoftheerrort
3、erms.Autore-gressiveconditionalheteroskedasticity(ARCH)modelsarethetopicofthischapter.Theyhaveproventobeveryusefulinfinancetomodelreturnvarianceorvolatilityofmajorassetclassesincludingequity,fixedincome,andforeignexchange.Understandingthebehaviorofthevarianceofthereturnprocessisimportantforforecast
4、ingaswellaspricingoption-typederivativeinstrumentssincethevarianceisaproxyforrisk.Althoughassetreturns,suchasstockandexchangeratereturns,appeartofollowamartingaledifferencesequence,observationofthedailyreturnplotsshowsthattheamplitudeofthereturnsvariesacrosstime.Awidelyobservedphenomenoninfinancec
5、onfirmingthisfactistheso-calledvolatilityclustering.Thisreferstothetendencyoflargechangesinassetprices(eitherpositiveornegative)tobefollowedbylargechangesandsmallchangestobefollowedbysmallchanges.Hence,thereistempo-raldependenceinassetreturns.Typically,theyarenotevenclosetobeingindependentlyandide
6、nticallydistributed(IID).Thispatterninthevolatil-ityofassetreturnswasfirstreportedbyMandelbrot.1Time-varyingvola-1BenoitB.Mandelbrot,“TheVariationofCertainSpeculativePrices,”JournalofBusiness36(1963),pp.394–419.279c08-ARCHModelsPage280Thursday,October26,20062:07PM280FINANCIALECONOMETRICStilityandh
7、eavytailsfoundindailyassetreturnsdataaretwoofthetypicalstylizedfactsassociatedwithfinancialreturnseries.2TheARCHmodelanditsgeneralization,thegeneralizedautoregres-siveconditionalheteroskedasticity(GARCH)model,provideaco
