Multivariate-models金融计量经济学导论

ChrisBrooks2002,陈磊2004,6-1,Chapter6,Multivariatemodels,ChrisBrooks2002,陈磊2004,6-2,1Motivations,Allthemodelswehavelookedatthusfarhavebeensingleequationsmodelsoftheformy=X+uAllofthevariablescontainedintheXmatrixareassumedtobeEXOGENOUS.由系统外因素决定的变量yisanENDOGENOUSvariable.既影响系统同时又被该系统及其外部因素所影响的变量.Anexample-thedemandandsupplyofagood:(1)(2)(3)、=quantityofthegooddemanded/suppliedPt=price，St=priceofasubstitutegoodTt=somevariableembodyingthestateoftechnology,ChrisBrooks2002,陈磊2004,6-3,Assumingthatthemarketalwaysclears,anddroppingthetimesubscriptsforsimplicity(4)(5)ThisisasimultaneousSTRUCTURALFORMofthemodel.Thepointisthatpriceandquantityaredeterminedsimultaneously(priceaffectsquantityandquantityaffectsprice).PandQareendogenousvariables,whileSandTareexogenous.WecanobtainREDUCEDFORMequationscorrespondingto(4)and(5)bysolvingequations(4)and(5)forPandforQ.,SimultaneousEquationsModels:TheStructuralForm,ChrisBrooks2002,陈磊2004,6-4,SolvingforQ,(6)SolvingforP,(7)Rearranging(6),(8),ObtainingtheReducedForm,ChrisBrooks2002,陈磊2004,6-5,Multiplying(7)throughby,(9)(8)and(9)arethereducedformequationsforPandQ.,ObtainingtheReducedForm,ChrisBrooks2002,陈磊2004,6-6,Butwhatwouldhappenifwehadestimatedequations(4)and(5),i.e.thestructuralformequations,separatelyusingOLS?BothequationsdependonP.OneoftheCLRMassumptionswasthatE(Xu)=0,whereXisamatrixcontainingallthevariablesontheRHSoftheequation.Itisclearfrom(8)thatPisrelatedtotheerrorsin(4)and(5)-i.e.itisstochastic.WhatwouldbetheconsequencesfortheOLSestimator,ifweignorethesimultaneity?,2SimultaneousEquationsBias,ChrisBrooks2002,陈磊2004,6-7,RecallthatandSothatTakingexpectations,IftheXsarenon-stochastic,E(Xu)=0,whichwouldbethecaseinasingleequationsystem,sothat,whichistheconditionforunbiasedness.,SimultaneousEquationsBias,ChrisBrooks2002,陈磊2004,6-8,But.iftheequationispartofasystem,thenE(Xu)0,ingeneral.Conclusion:ApplicationofOLStostructuralequationswhicharepartofasimultaneoussystemwillleadtobiasedcoefficientestimates.IstheOLSestimatorstillconsistent,eventhoughitisbiased?No-Infacttheestimatorisinconsistentaswell.Henceitwouldnotbepossibletoestimateequations(4)and(5)validlyusingOLS.,SimultaneousEquationsBias,ChrisBrooks2002,陈磊2004,6-9,SoWhatCanWeDo?Takingequations(8)and(9),wecanrewritethemas(10)(11)WeCANestimateequations(10)whatwewantedweretheoriginalparametersinthestructuralequations-,.,3AvoidingSimultaneousEquationsBias,ChrisBrooks2002,陈磊2004,6-10,CanWeRetrievetheOriginalCoefficientsfromthes?Shortanswer:sometimes.wesometimesencounteranotherproblem:identification.*ConsiderthefollowingdemandandsupplyequationsSupplyequation(12)Demandequation(13)Wecannottellwhichiswhich!BothequationsareUNIDENTIFIEDorUNDERIDENTIFIED.Theproblemisthatwedonothaveenoughinformationfromtheequationstoestimate4parameters.Noticethatwewouldnothavehadthisproblemwithequations(4)and(5)sincetheyhavedifferentexogenousvariables.,4IdentificationofSimultaneousEquations,ChrisBrooks2002,陈磊2004,6-11,Wecouldhavethreepossiblesituations:1.Anequationisunidentifiedlike(12)or(13)wecannotgetthestructuralcoefficientsfromthereducedformestimates2.Anequationisexactlyidentifiede.g.(4)or(5)cangetuniquestructuralformcoefficientestimates3.Anequationisover-identifiedExamplegivenlaterMorethanonesetofstructuralcoefficientscouldbeobtainedfromthereducedform.,WhatDetermineswhetheranEquationisIdentifiedornot?,ChrisBrooks2002,陈磊2004,6-12,Howdowetellifanequationisidentifiedornot?Therearetwoconditionswecouldlookat:-Theorder阶condition-isanecessarybutnotsufficientconditionforanequationtobeidentified.-Therank秩condition-isanecessaryandsufficientconditionforidentification.在G个内生变量、G个方程的联立方程组模型中，某一方程是可识别的，当且仅当该方程没有包含的变量在其他方程中对应系数组成的矩阵的秩为G-1。对于相对简单的方程系统，这两个规则将得到同样的结论。事实上，大多数经济和金融方程系统都是过度识别的。,WhatDetermineswhetheranEquationisIdentifiedornot?,ChrisBrooks2002,陈磊2004,6-13,StatementoftheOrderConditionLetGdenotethenumberofstructuralequations.AnequationisjustidentifiedifthenumberofvariablesexcludedfromanequationisG-1.IfmorethanG-1areabsent,itisover-identified.IflessthanG-1areabsent,itisnotidentified.ExampletheYsareendogenous,whiletheXsareexogenous.Determinewhethereachequationisover-,under-,orjust-identified.(14)-(16),Statementoftheordercondition,ChrisBrooks2002,陈磊2004,6-14,SolutionG=3;If#excludedvariables=2,theeqnisjustidentifiedIf#excludedvariables2,theeqnisover-identifiedIf#excludedvariables2,theeqnisnotidentifiedEquation14:NotidentifiedEquation15:JustidentifiedEquation16:Over-identified如果模型中每个结构方程都是可识别的，则称结构型联立方程组模型是可识别的。,Exampleoftheordercondition,ChrisBrooks2002,陈磊2004,6-15,5外生性的定义,Leamer（1985）：p310变量X对变量Y是外生的，如果变量Y关于X的条件分布不随产生X的过程的变化而改变。外生性的两种形式：前定变量：与方程中的当前和未来误差项独立。严格外生变量：与方程中任何时期的误差项独立。前定变量的通常定义：包括外生变量和滞后的内生变量,ChrisBrooks2002,陈磊2004,6-16,Howdowetellwhethervariablesreallyneedtobetreatedasendogenousornot?Consideragainequations(14)-(16).Equation(14)containsY2andY3-butdowereallyneedequationsforthem?WecanformallytestthisusingaHausmantestasfollows:1.Obtainthereducedformequationscorrespondingto(14)-(16).Thereducedformsturnouttobe:(17)-(19)Estimatethereducedformequations(17)-(19)usingOLS,andobtainthefittedvalues,5TestsforExogeneity,ChrisBrooks2002,陈磊2004,6-17,2.Runtheregressioncorrespondingtoequation(14).3.Runtheregression(14)again,butnowalsoincludingthefittedvaluesasadditionalregressors:(20)4.UseanF-testtotestthejointrestrictionthat2=0,and3=0.Ifthenullhypothesisisrejected,Y2andY3shouldbetreatedasendogenous.,TestsforExogeneity,ChrisBrooks2002,陈磊2004,6-18,Considerthefollowingsystemofequations:(21-23)Assumethattheerrortermsarenotcorrelatedwitheachother.CanweestimatetheequationsindividuallyusingOLS?Equation21:Containsnoendogenousvariables,soX1andX2arenotcorrelatedwithu1.SowecanuseOLSon(21).Equation22:ContainsendogenousY1togetherwithexogenousX1andX2.WecanuseOLSon(22)ifalltheRHSvariablesin(22)areuncorrelatedwiththatequationserrorterm.Infact,Y1isnotcorrelatedwithu2becausethereisnoY2terminequation(21).SowecanuseOLSon(22).,6RecursiveSystems,ChrisBrooks2002,陈磊2004,6-19,Equation23:ContainsbothY1andY2;werequirethesetobeuncorrelatedwithu3.Bysimilarargumentstotheabove,equations(21)and(22)donotcontainY3,sowecanuseOLSon(23).ThisisknownasaRECURSIVEorTRIANGULARsystem.Wedonothaveasimultaneityproblemhere.Butinpracticenotmanysystemsofequationswillberecursive.,RecursiveSystems,ChrisBrooks2002,陈磊2004,6-20,IndirectLeastSquares(ILS)CannotuseOLSonstructuralequations,butwecanvalidlyapplyittothereducedformequations.Ifthesystemisjustidentified,ILSinvolvesestimatingthereducedformequationsusingOLS,andthenusingthemtosubstitutebacktoobtainthestructuralparameters.However,ILSisnotusedmuchbecause1.Solvingbacktogetthestructuralparameterscanbetedious.2.Mostsimultaneousequationssystemsareover-identified.,7EstimationproceduresforSystems,ChrisBrooks2002,陈磊2004,6-21,Infact,wecanusethistechniqueforjust-identifiedandover-identifiedsystems.Twostageleastsquares(2SLSorTSLS)isdoneintwostages:Stage1:ObtainandestimatethereducedformequationsusingOLS.Savethefittedvaluesforthedependentvariables.Stage2:Estimatethestructuralequations,butreplaceanyRHSendogenousvariableswiththeirstage1fittedvalues.,EstimationofSystemsUsingTwo-StageLeastSquares,ChrisBrooks2002,陈磊2004,6-22,Example:Sayequations(14)-(16)arerequired.Stage1:Estimatethereducedformequations(17)-(19)individuallybyOLSandobtainthefittedvalues,.Stage2:ReplacetheRHSendogenousvariableswiththeirstage1estimatedvalues:(24)-(26)Nowandwillnotbecorrelatedwithu1,willnotbecorrelatedwithu2,andwillnotbecorrelatedwithu3.,EstimationofSystemsUsingTwo-StageLeastSquares,ChrisBrooks2002,陈磊2004,6-23,TSLS是比较经济、易用的方法。如果在第一阶段估计时所得到的R2非常高，那么古典OLS估计量与TSLS估计量将非常接近；如果在第一阶段估计时所得到的R2非常低，TSLS估计量将没有太大的实际意义。TSLS估计量是有偏估计量，但却是一致估计量。ItisstillofconcerninthecontextofsimultaneoussystemswhethertheCLRMassumptionsaresupportedbythedata.Ifthedisturbancesinthestructuralequationsareautocorrelated,the2SLSestimatorisnotevenconsistent.ThestandarderrorestimatesalsoneedtobemodifiedcomparedwiththeirOLScounterparts,butoncethishasbeendone,wecanusetheusualt-andF-teststotesthypothesesaboutthestructuralformcoefficients.,EstimationofSystemsUsingTwo-StageLeastSquares,ChrisBrooks2002,陈磊2004,6-24,RecallthatthereasonwecannotuseOLSdirectlyonthestructuralequationsisthattheendogenousvariablesarecorrelatedwiththeerrors.OnesolutiontothiswouldbenottouseY2orY3,butrathertousesomeothervariablesinstead.Wewanttheseothervariablestobe(highly)correlatedwithY2andY3,butnotcorrelatedwiththeerrors-theyarecalledINSTRUMENTS.SaywefoundsuitableinstrumentsforY2andY3,z2andz3respectively.Wedonotusetheinstrumentsdirectly,butrunregressionsoftheform(27)E(u1tu2t)=0.TheanalysiscouldbeextendedtoaVAR(g)model,orsothattherearegvariablesandgequations.,8VectorAutoregressiveModels,ChrisBrooks2002,陈磊2004,6-28,OneimportantfeatureofVARsisthecompactnesswithwhichwecanwritethenotation.Forexample,considerthecasefromabovewherek=1.Wecanwritethisasoryt=0+1yt-1+utg1g1ggg1g1,VectorAutoregressiveModels:NotationandConcepts,ChrisBrooks2002,陈磊2004,6-29,VAR模型还具有灵活性和易于一般化的重要特点.例如，模型可以扩展到包含移动平均误差项，即VARMA。Thismodelcanbeextendedtothecasewherethereareklagsofeachvariableineachequation:yt=0+1yt-1+2yt-2+.+kyt-k+utg1g1ggg1ggg1ggg1g1Wecanalsoextendthistothecasewherethemodelincludesfirstdifferencetermsandcointegratingrelationships(aVECM).,VectorAutoregressiveModels:NotationandConcepts,ChrisBrooks2002,陈磊2004,6-30,AdvantagesofVARModelling-Donotneedtospecifywhichvariablesareendogenousorexogenous-allareendogenous-Allowsthevalueofavariabletodependonmorethanjustitsownlagsorcombinationsofwhitenoiseterms,somoregeneralthanARMAmodelling-Providedthattherearenocontemporaneoustermsontherighthandsideoftheequations,cansimplyuseOLSseparatelyoneachequation，因为方程右边的变量都是前定变量。-Forecastsareoftenbetterthan“traditionalstructural”models.,VARModelsComparedwithStructuralEquationsModels,ChrisBrooks2002,陈磊2004,6-31,ProblemswithVARs-VARsarea-theoretical(asareARMAmodels)。VAR模型较少用于理论分析和政策建议。-Howdoyoudecidetheappropriatelaglength?-Somanyparameters!Ifwehavegequationsforgvariablesandwehaveklagsofeachofthevariablesineachequation,wehavetoestimate(g+kg2)parameters.e.g.g=3,k=3,parameters=30-DoweneedtoensureallcomponentsoftheVARarestationary?-Howdoweinterpretthecoefficients?,VARModelsComparedwithStructuralEquationsModels,ChrisBrooks2002,陈磊2004,6-32,ChoosingtheOptimalLagLength,Cross-EquationRestrictionsInthespiritof(unrestricted)VARmodelling,eachequationshouldhavethesamelaglengthSupposethatabivariateVAR(8)estimatedusingquarterlydatahas8lagsofthetwovariablesineachequation,andwewanttoexaminearestrictionthatthecoefficientsonlags5through8arejointlyzero.ThiscanbedoneusingalikelihoodratiotestDenotethevariance-covariancematrixofresiduals(givenbyE),as.Thelikelihoodratiotestforthisjointhypothesisisgivenby,ChrisBrooks2002,陈磊2004,6-33,ChoosingtheOptimalLagLength,whereisthevariance-covariancematrixoftheresidualsfortherestrictedmodel(with4lags),isthevariance-covariancematrixofresidualsfortheunrestrictedVAR(with8lags),andTisthesamplesize.Theteststatisticisasymptoticallydistributedasa2withdegreesoffreedomequaltothetotalnumberofrestrictions.IntheVARcaseabove,wearerestricting4lagsoftwovariablesineachofthetwoequations=atotalof4*2*2=16restrictions.InthegeneralcasewherewehaveaVARwithgequations,andwewanttoimposetherestrictionthatthelastqlagshavezerocoefficients,therewouldbeg2qrestrictionsaltogetherDisadvantages:ConductingtheLRtestiscumbersomeandrequiresanormalityassumptionforthedisturbances.,ChrisBrooks2002,陈磊2004,6-34,InformationCriteriaforVARLagLengthSelection,Multivariateversionsoftheinformationcriteriaarerequired.Thesecanbedefinedas:whereallnotationisasaboveandkisthetotalnumberofregressorsinallequations,whichwillbeequaltog2k+gforgequations,eachwithklagsofthegvariables,plusaconstanttermineachequation.Thevaluesoftheinformationcriteriaareconstructedfor0,1,lags(uptosomepre-specifiedmaximum).,ChrisBrooks2002,陈磊2004,6-35,DoestheVARIncludeContemporaneousTerms?,Sofar,wehaveassumedtheVARisoftheformButwhatiftheequationshadacontemporaneousfeedbackterm?WecanwritethisasThisVARisinprimitive/structuralform.,ChrisBrooks2002,陈磊2004,6-36,PrimitiveversusStandardFormVARs,WecantakethecontemporaneoustermsovertotheLHSandwriteorByt=0+1yt-1+utWecanthenpre-multiplybothsidesbyB-1togiveyt=B-10+B-11yt-1+B-1utoryt=A0+A1yt-1+etThisisknownasastandardformVAR,whichwecanestimateusingOLS.,ChrisBrooks2002,陈磊2004,6-37,VAR模型的识别,粗略地讲，结构型VAR模型的识别问题是指，能否从一个简约模型的估计值反导出原来的结构模型的系数。结构型VAR模型是不可识别的，因为两个方程的等号右边具有相同的前定变量为了解决这个问题，需要加入一定的约束条件。即同期项的一个系数12或22须设为0，使距阵B为三角形。最好是依据经济理论加入约束条件,ChrisBrooks2002,陈磊2004,6-38,BlockSignificanceandCausalityTests,Itislikelythat,whenaVARincludesmanylagsofvariables,itwillbedifficulttoseewhichsetsofvariableshavesignificanteffectsoneachdependentvariableandwhichdonot.Forillustration,considerthefollowingbivariateVAR(3):Wemightbeinterestedintestingthefollowinghypotheses,andtheirimpliedrestrictionsontheparameters:,ChrisBrooks2002,陈磊2004,6-39,BlockSignificanceandCausalityTests,EachofthesefourjointhypothesescanbetestedwithintheF-testframework.ThesetestscouldalsobereferredtoasGrangercausalitytests.Grangercausalitytestsseektoanswerquestionssuchas“Dochangesiny1causechangesiny2?”Ify1causesy2,lagsofy1shouldbesignificantintheequationfory2.Ifthisisthecase,wesaythaty1“Granger-causes”y2.Ify2causesy1,lagsofy2shouldbesignificantintheequationfory1.Ifbothsetsoflagsaresignificant,thereis“bi-directionalcausality”.Ify2causesy1,buty1doesnotcausesy2,theny2isstrongexogenous(intheequationfory1).Ifneithery2causesy1,nory1causesy2,theny1y2areindependent此处因果性并不意味着一个变量的变动引起另一个变量的变动,ChrisBrooks2002,陈磊2004,6-40,ImpulseResponses,VARmodelsareoftendifficulttointerpret:onesolutionistoconstructtheimpulseresponsesandvariancedecompositions.ImpulseresponsestraceouttheresponsivenessofthedependentvariablesintheVARtoshockstotheerrorterm.Aunitshockisappliedtoeachvariableanditseffectsarenoted.ConsiderforexampleasimplebivariateVAR(1):Achangeinu1twillimmediatelychangey1.Itwillchangey2andalsoy1duringthenextperiod.Wecanexaminehowlongandtowhatdegreeashocktoagivenequationhasonallofthevariablesinthesystem.eg.p341,ChrisBrooks2002,陈磊2004,6-41,多变量VAR模型也可改写为这里yt是一个k维内生变量向量，t是协方差矩阵为的扰动向量。,一般VAR模型的脉冲响应函数,假如VAR(p)可逆，我们可以得到VMA()的表达式:,ChrisBrooks2002,陈磊2004,6-42,VMA表达式的系数可按下面的方式给出：VAR的系数A和VMA的系数必须满足下面关系：其中，。关于的条件递归定义了VMA系数：,从而可知VMA的系数可以由VAR的系数递归得到。,一般VAR模型的脉冲响应函数,ChrisBrooks2002,陈磊2004,6-43,考虑VMA()的表达式设，y的第i个变量可以写成：其中k是变量个数。仅考虑2个变量(k=2)的情形：,现在假定在基期给一个单位的脉冲，即：21012345t,ChrisBrooks2002,陈磊2004,6-44,由的脉冲引起的的响应函数：,由上述推导可知由的脉冲引起的的响应函数序列是由VMA()中系数矩阵第2行，第1列的元素组成,q=1,2,。因此，一般地，由的脉冲引起的的响应函数可以求出如下：其中，代表着对第j个变量的单位冲击引起第i个变量的第q期滞后反映。,ChrisBrooks2002,陈磊2004,6-45,VarianceDecompositions,VariancedecompositionsofferaslightlydifferentmethodofexaminingVARdynamics.Theygivetheproportionofthemovementsinthedependentvariablesthatareduetotheir“own”shocks,versusshockstotheothervariables.Thisisdonebydetermininghowmuchofthes-stepaheadforecasterrorvarianceforeachvariableisexplainedbyinnovationstoeachexplanatoryvariable(s=1,2,).ThevariancedecompositiongivesinformationabouttherelativeimportanceofeachshocktothevariablesintheVAR.脉冲响应函数和方差分解常常提供一定程度上的相似信息.,ChrisBrooks2002,陈磊2004,6-46,TheOrderingoftheVariables,Butforcalculatingimpulseresponsesandvariancedecompositions,theorderingofthevariablesisimportant.Themainreasonforthisisthatabove,weassumedthattheVARerrortermswerestatisticallyindependentofoneanother.Thisisgenerallynottrue,however.Theerrortermswilltypicallybecorrelatedtosomedegree.Therefore,thenotionofexaminingtheeffectoftheinnovationsseparatelyhaslittlemeaning,sincetheyhaveacommoncomponent.Whatisdoneisto“orthogonalise”theinnovations.InthebivariateVAR,thisproblemwouldbeapproachedbyattributingalloftheeffectofthecommoncomponenttothefirstofthetwovariablesintheVAR.Inthegeneralcasewheretherearemorevariables,thesituationismorecomplexbuttheinterpretationisthesame.,ChrisBrooks2002,陈磊2004,6-47,由以上讨论可知，对第i个变量的冲击不仅直接影响第i个