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1、Tsay-Driver-12013/10/2812:29page105#1CHAPTER3VectorAutoregressiveMoving-AverageTimeSeriesForparsimoniousparameterizationandfurthersimpli?cationinmodelingmulti-variatetimeseries,weconsiderinthischapterthevectormoving-average(VMA)modelsandthevectorautoregressivemoving-av
2、erage(VARMA)models.SeeEqua-tion(1.21).Westudypropertiesofthemodelsanddiscusstheirimplications.WealsoaddresssomenewchallengesfacingtheVARMAmodelsthatdonotoccurinstudyingVARmodelsofChapter2.SimilartothecaseofVARmodels,westartwithsimplemodelsandprovidejusti?cationsforusingthe
3、models.Wethenpro-videresultsforthegeneralstationaryandinvertibleVARMA(p,q)models.Someofthegeneralizationsareobtainedbyexpandingthedimensionoftheunderlyingtimeseriesinordertoreducetheorderofamodel.SpecialattentionispaidtotheexactlikelihoodfunctionofaVARMAmodelandthecalculat
4、ionofthelikelihoodfunction.WealsostudylineartransformationandtemporalaggregationofaVARMAmodel.Realexamplesandsomesimulationsareusedtodemonstratetheapplica-tionsandtoemphasizethemainpointsoftheVARMAmodel.Finally,weanalyzesomerealtimeseriesviatheVARMAmodels.Heavymatrixnotati
5、onisusedinSections3.9and3.10todescribethelikelihoodfunctionofaVARMAtimeseries.ForreaderswhofocusonapplicationsofVARMAmodels,thesetwosectionscanbeskippedonthe?rstread.MultivariateTimeSeriesAnalysis:WithRandFinancialApplications,FirstEdition.RueyS.Tsay.c2014JohnWiley&Sons,I
6、nc.Published2014byJohnWiley&Sons,Inc.105Tsay-Driver-12013/10/2812:29page106#2106vectorautoregressivemoving-averagetimeseries3.1VECTORMAMODELSAk-dimensionaltimeseriesztfollowsaVMAmodeloforderqifqzt=μ+at?θiat?i,(3.1)i=1whereμisaconstantvectordenotingthemeanofzt,θiar
7、ek×kmatriceswithθq=0,and{at}isawhitenoiseseriesde?nedinEquation(2.1)withVar(at)=Σa=[σa,ij],whichispositivede?nite.Usingtheback-shiftoperatorthemodelbecomesz=μ+θ(B)a,whereθ(B)=I?qθBiisamatrixpolynomialttki=1iofdegreeq.VMAmodelsexistformanyreasons.In?nance,itiswellknowntha
8、tthebidandaskbouncecanintroducenegativelag-1serialcorrelationinhigh-frequencyreturns,fore
