一种多元GARCH模型及其应用研究

华中科技大学硕士学位论文一种多元GARCH模型及其应用研究姓名：杜福林申请学位级别：硕士专业：数量经济学指导教师：周少甫20041031IIIGARCHEngleABBGARCHGARCHBollerslevDCCMVGARCHIVAbstractInthearticleatfirstwereviewedrelevantdocumentsaboutmultivariableGARCHmodel,andintroducedanewkindofestimatorputforwardbyEnglerecently.ThenweanalysisedthecorrelationofthemainstockindexinChinesemarket,itwasfoundthatconditionalcorrelationcandescribeaccuratelyalongtimemovingtrendoftwovariables.BecauseofthebasichomogeneityofShenzhenandShanghaistockmarket,thecorrelationbetweenthemisverystrongatsometime,butnotstillverysteady,thisdemonstratedbecauseeachmarkethaditsowntrusteeshipsandexchangesystemandhadnotformedtheunifiedstockmarketyet,sothecompanycannotissuetheirstocksatthesametimeontwomarket,andthetradercan'ttradetrans-market(hardlyhavingconditionandchanceofsteppingmarketing),thisdidn’tcontrolfavorablelythemarketrisk.AndthecorrelationoftheA-shareB-sharemarketofourcountryissmall,butseldomnegativerelatively.thoughcorrelationgotweakeninsometimes,intotaltrend,thecorrelationincreasedsteadylyallthetime,thisisconsistentwiththedevelopmentprocessofB-sharemarket.Thecorrelationoftwomarketsisstillunstable,provedthattheindentitybetweenthemhasnotbeenproducedyet,sotheamalgamationoftwomarketisstillearly.Infinancialresearch,byuseofamarketmodeltoestimatethataconstandbetacoefficienthaslonghistory,butsomeincreasednewlyevidencesshownotonlythepersonalsharebutalsoportfolio’sbetacoefficientchangeswithtime,thenwecansetupconditionalbetaseriesofbytheconditionalvarianceinformationwhichGARCHmodelgenerated.Inthepastdocuments,becauseofunmaturemultivariableGARCHtheory,whenstudyingthecorrelationbetweenthepersonalshareandmarketindex,peoplegenerallyadoptedBollerslev’sconstantcorrelationmodel,butsomeresearchersfoundthattheconstandcorrelationassumptioncannotbesupportedbytherealfinancialdata,sointhearticle,inordertoestimatetimevaryingbetacoefficientofseveralstocksintheVShanghaistockmarket,weassumedthatthesporttrendbetweeneachstockandmarketindexobeyedaDCC-MVGARCHprocess,thusgainedthecomparativelyaccuratetimevaryingbetacoefficient.Finally,wesuppliedaCAPMformulabasedontimevaryingbetacoefficient.Keywords:HeterogeneityModelDynamicconditionalcorrelationTimevaryingbetacoefficientCAPM111.1CAPMAPTGARCHGARCH,,,GARCHEngleDCCMVGARCH.2GARCHDCCMVGARCHARCHCAPMAPTARCH──ARCHARCHCAPMGARCH1.220GARCHBollerslevGARCHGARCHEngleKroner1995BEKKBEKK3GARCHBollerslevConstantConditionalCorrelationmodelEngle2002DynamicaConditionalCorrelationBollerslev1.3GARCHEngleGARCHCAPMGARCHDCCMatlab6.542GARCH2.1GARCHARCHEngle1982[1]ARCHt-1,...},,,{22111-----=ΨtttttxyxytrARCH1r-Ψtt),0(thN1.1∑=-+=qiititrh120aa1.2nii,...,1,0,00=≥aa0thARCHtrqq1986Bollerslev1.2ARCH——GARCHGeneralizedAutoRegressiveConditionalHeteroskedasticity[2]GARCH∑∑=-=-++=pjjtjqiitithh1120rbaa1.30th5pjqiji,...,1,0,...,1,000=≥=≥baa1.4GARCHp,qpqGARCHARCHGARCHGARCHARCH2.2GARCHnXtItXXttXt|I1-tXi.i.dNutHt(1.5)NutHtutHtGARCHHtwCLHtDL[rt'rt](1.6)C(L)=∑=piiiLC1DL∑=qiiiLD1rt=Xt-utGARCHGARCH11——BollerslevGARCHGARCH[3]VechHtVech(w)CLVechHt-1+D(L)Vech(rtrt)(1.7)6Vech•n×n|→2)1(+×nn×1wn×nC(L)=∑=piiiLC1DL∑=qiiiLD1CiDir×rrnn1/2r×[1+r(p+q)]21EngleKroner(1995)C1D11Vech-GARCHVech-GARCHDVech-GARCHGARCHpq3r9BollerslevEngleWooldridge1998GARCH1,1HtEngleKroner1995BEKK[4]HtΓ'Γ∑=piiC1Ht-i'iC∑=qjjD1rt-jrt-jDj(1.8)rn×n×p+q,11ΓCiHtVechGARCHBEKKVechBEKKBEKKEngleNgRothschild1990GARCHtij,sji,wliljtpp,sVech⎟⎟⎠⎞⎜⎜⎝⎛abba=(a,b,a)BollerslevEngleKraftKroner7tpp,sppwb1,-tpps2,tpra(1.9)GARCH1,1BEKKGARCHBollerslev1990[5]BollerslevGARCHHt=2112,1,212,2,2,21,2,,ttntttntntntntsssssssss⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦LLMMMML(1.10)2itsiw2,1,jtipjji-=∑sb∑=-qjjtiji12,,eai=1…nsij,t=tjtiji,,,ssrij=1…ni≠j(p+q)n+n(n1)/27GARCHConstantConditionalCorrelationmodelBeraKim1996Tse1998[6]82.3DCCMVGARCH2002EngleSheppard—DynamicConditionalCorrelation[7]k0Htrt|1-ℑtΝ0Ht1-ℑtrttHt=DtRtDtRt=(Q*)-1Qt(Q*)-1Qt=[1∑=Mmm1a∑=Nnn1b]Q∑=Mmm1a'mtmt--ee∑=Nnn1bQt-nQ=T-1∑=Titt1'ee(1.11)RtDt=diag(ith)hit=∑∑=-=-++iiQqqitiqpppitipih112rbaw'ter'D1-tQRttji,,rtijq,/tjjtiiqq,,RtQ*tiitqQttijq,tiiq,tjjq,manbDCCmnqp+1×nqp+8qp=1DCCGARCHqDCCGARCHpqGARCH9jfff,,,21kΛjf,ifiqiipiiibbaaw,,,,,,11ΛΛiGARCHRtk×kIkRt[8]QL1|frt∑=Ttk1)2log((21plog(|Ik|)2log(|Dt|)r'D1-tIkD1-trt)∑=Ttk1)2log((21p2log(|Dt|)r'D2-trt∑=Ttk1)2log((21p∑=+knititithrh12))(log()∑=knk1)2log((21p∑=+Ttititithrh12))(log()(1.12)QL2tr,|∧fj∑=Ttk1)2log((21p2log(|Dt|)log(|Rt|)r'D1-tR1-tD1-trt∑=Ttk1)2log((21p2log(|Dt|)log(|R1-t|)'teR1-tte)(1.13)∧flog(|R1-t|)'teR1-tteDCCQ*2Ltr,|∧fj∑=Tt1(21log(|R1-t|)'teR1-tte)(1.14)NeweyMcFadden1994GMMDCCDCC[9]A10q=(00,jf)Θ=Φ×ΨΘA20fE[ln1f(tr,f)]0jE[ln2f(tr,f)]10A3ln1f(tr,f)ln2f(tr,f)A4E[Φ∈fsup),(ln1ftrf]E[Θ∈qsup),(ln2qtrf]1A1—A4Pn0ff→∧(∧∧yf,n)nPn→∧q0qA5(i)E[),(ln01fftrf∇]0E[∞∇]),(ln201fftrf(ii)E[),(ln02qqtrf∇]0E[∞∇]),(ln202qqtrfA6(i)11A)],(ln[01ffftrfE∇)1(O(ii)22A)],(ln[02qyytrfE∇)1(ODCC2A1—A6{tf1}{tf2})(0∧∧-qqnnA~),0(1*'*1*--ABAN⎥⎦⎤⎢⎣⎡∇∇∇=)(ln)(ln0)(ln020201*qqfyyfyfffffA⎥⎦⎤⎢⎣⎡2212110AAA⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡∇∇=∑=-222112111002'2/101'2/1*)},,(ln),,(ln{varBBBrfnrfnBntttyffyfn∧q1**1*--ABANeweyMcFadden1**1*--ABAiGARCHi1133.1[10]t100AtsAr1r22s2