On some probabilistic properties of periodic GARCH

arXiv:0709.2983v1[math.PR]19Sep2007OnsomeprobabilisticpropertiesofperiodicGARCHprocessesAbdelouahabBibi*AbdelhakimAknouche***D´epartementdeMath´ematiques,Universit´eMentourideConstantine,AlgeriaE-mail:[e-mail]**Facult´edeMath´ematiques,Universit´eU.S.T.H.B.,Algiers,AlgeriaE-mail:[e-mail]inessomeprobabilisticpropertiesoftheclassofperiodicGARCHprocesses(PGARCH)whichfeatureperiodicityinconditionalheteroskedasticity.Inthesemodels,theparametersareal-lowedtoswitchbetweendifferentregimes,sothattheirstructuresharesmanypropertieswithperiodicARMAprocess(PARMA).Weexaminethestrictandsecondorderperiodicstationarities,theex-istenceofhigher-ordermoments,thecovariancestructure,thegeometricergodicityandβ−mixingofthePGARCH(p,q)processundergeneralandtractableassumptions.Someexamplesareproposedtoillustratethevariousconcepts.Keywords.PeriodicGARCHProcesses;PeriodicStationarity;GeometricErgodicity;Higher-OrderMoments.AMS(2000)SubjectClassification.Primary62M10;Secondary62M05.1IntroductionConsideraperiodicGARCH(p1,...,ps,q1,...,qs)process(xt)t∈Zwithperiods0andordersp=(p1,...,ps)andq=(q1,...,qs),definedonsomeprobabilityspace(Ω,A,P)withthenon-linearperiodicdifferenceequation:∀t∈Z:xst+v=εst+vphst+vhst+v=α0(v)+pvPi=1αi(v)x2st+v−i+qvPj=1βj(v)hst+v−j(1.1)where(εt)t∈Zisasequenceofindependentandidenticallydistributed(i.i.d.)randomvariablesdefinedonthesameprobabilityspace(Ω,A,P)suchthatE{εt}=Eε3t =0,andEε2t =1(theseconditionsareobviouslysatisfiedif(εt)t∈ZisGaussian).Inthedifferenceequation(1.1)xst+vreferstoxtduringthev−th‘season’,1≤v≤sofperiodt,α0(v),α1(v),...,αpv(v)andβ1(v),...,βqv(v)arethemodelcoefficientsatseasonvsuchthatforall1v=1,...,s,α0(v)0,αi(v)≥0,i=1,...,pv,andβj(v)≥0,j=1,...,qv.Moreover,weassumethatεkisindependentofxtforkt.Weusetheperiodicnotations(xst+v),(εst+v),(hst+v),(αi(v),0≤i≤pv),and(βi(v),1≤i≤qv)toemphasizetheperiodicityinthemodel.Thereisnolossofgeneralityintakingpvandqvtobeconstantinv.Ifpvorqvchangewithv,onecansetp=max1≤v≤spv,q=max1≤v≤sqvandtakeαk(v)=0forpvk≤pandβk(v)=0forqvk≤q,sointhesequel,weshallconsidertheperiodicGARCHwithconstantorderspandq.SinceBollerslevandGhysels(1996),thistypeofnon-linearmodelshasbecomeanappealingtoolforinvestigatingbothvolatilityanddistinctseasonalpatterns,andhasbeenappliedinvariousdisciplinessuchasfinanceandmonetaryeconomics(seee.g.BollerslevandGhysels,1996andFransesandPaap,2000).Whenweconsideraperiodicmodelasadatageneratingprocess,itisimportanttofindconditionsensuringthe(periodic)stationarity,ergodicityandtheexistenceofhighermomentsforfurtherstatisticalanalysis.VariousprobabilisticpropertiesofstandardGARCHmodelshavebeenstudiedextensivelybymanyauthors(seee.g.,ChenandAn,1998,BougerolandPicard,1992a,1992bandCarrascoandChen,2002andthereferencestherein).Inthepresentpaper,wefocusonstudyingthefundamentalprobabilisticpropertiesofthePGARCHprocess(xt)t∈Zgeneratedby(1.1)so,inSection2,wepresentavectorialrepresentationfromwhichwederivesomesufficientconditionsforthestrictstationarity.InSections3and4,necessaryandsufficientconditionsforthesecondorderstationarityandtheexistenceofhigherordermomentsaregiven.Section5isdevotedtocovariancestructure.InSection6weprovideconditionsunderwhichstrictlystationarysolutionsareexponentialβ-mixingwithfinitehigherordermoments.WeconcludeinSection7.Somenotationsareusedthroughoutthepaper:I(k)denotestheidentitymatrixoforderkandO(k×l)denotesthematrixoforderk×lwhoseelementsarezeroes,forsimplicitywesetO(k):=O(k×k),ρ(A)referstothespectralradiusofasquarematrixA,i.e.,themaximumeigenvalueofamatrixAinabsolutevalue,Vec(A)istheusualcolumn-stackingvectorofthematrixA,k.kreferstothestandard(Euclidean)norminRnortheuniforminducednorminthespaceM(n)ofn×nmatrices,⊗denotestheKroneckerproductofmatrices,andA⊗m=A⊗A⊗...⊗A(m−times),foranyintegerm≥1.Foranyp≥1,Lp=Lp(Ω,A,P)denotestheHilbertspaceofrandomvariablesXdefinedontheprobabilityspace(Ω,A,P)suchthatkXkp={E|X|p}1/p+∞.Wealsousethefollowingpropertyofmatrixoperation,Vec(ABC′)=(C⊗A)Vec(B),where′isthematrixtranspose.22TheMarkovianrepresentationandstrictstationarityLet(xt)t∈Zbeaprocessconformingtothemodel(1.1).Settingyst+v=x2st+vandηst+v=ε2st+v,weobtainfrom(1.1)thefollowingrepresentationyst+v=pXi=1αi(v)ηst+vyst+v−i+qXj=1βj(v)ηst+vhst+v−j+α0(v)ηst+v.(2.1)Equation(2.1)isintractablewhenwewanttoexaminetheprobabilisticstructureofthisrepresenta-tion.Instead,wewillworkwiththecorrespondingstate-spacerepresentation.Letd=p+qanddefineηt=(ηst+1,...,ηst+s)′,yst+v=(yst+v,...,yst+v−p+1,hst+v,...,hst+v−q+1)′andBvηt=(α0(v)ηst+v,0,...,0,α0(v),0,...,0)′asvectorsinRs,RdandRdrespectively,andsetφv(ηt)=Av(ηt)Bv(ηt)AvBv!d×dwhereAv(ηt)=α1(v)ηst+vα2(v)ηst+v...αp(v)ηst+v10...00.........0010p×p,Bv(ηt)=β1(v)ηst+v...βq(v)ηst+v0...0.........0···0p×qarep×pandp×qmatrixvaluedpolynomialfunctionsofηtandwhereAv=α1(v)...αp(v)0...0.........0···0q×p,Bv=β1(v)β2(v)...βq(v)10...00.........0010q×qthismeansthatAv(x),Bv(x)andBv(x)haveentriesandcoordinates,respectively,whicharepolynomialfunctionsofthecoordinat