金融计量经济学(双语版)(全套课件)

Chapter5Univariatetimeseriesmodellingandforecasting1课件1introduction单变量时间序列模型只利用变量的过去信息和可能的误差项的当前和过去值来建模和预测的一类模型(设定)。与结构模型不同；通常不依赖于经济和金融理论用于描述被观测数据的经验性相关特征ARIMA（AutoRegressiveIntegratedMovingAverage）是一类重要的时间序列模型Box-Jenkins1976当结构模型不适用时，时间序列模型却很有用如引起因变量变化的因素中包含不可观测因素，解释变量等观测频率较低。结构模型常常不适用于进行预测本章主要解决两个问题一个给定参数的时间序列模型，其变动特征是什么？给定一组具有确定性特征的数据，描述它们的合适模型是什么？2课件AStrictlyStationaryProcessAstrictlystationaryprocessisonewhere

Foranyt1,t2,…,tn∈Z,anym∈Z,n=1,2,…AWeaklyStationaryProcessIfaseriessatisfiesthenextthreeequations,itissaidtobeweaklyorcovariancestationary1.E(yt)=

, t=1,2,...,

2.3.

t1

,t22SomeNotationandConcepts3课件Soiftheprocessiscovariancestationary,allthevariancesarethesameandallthecovariancesdependonthedifferencebetweent1

andt2.Themoments ,s=0,1,2,... areknownasthecovariancefunction.Thecovariances,

s,areknownasautocovariances.

However,thevalueoftheautocovariancesdependontheunitsofmeasurementofyt.Itisthusmoreconvenienttousetheautocorrelationswhicharetheautocovariancesnormalisedbydividingbythevariance: ,s=0,1,2,... Ifweplot

sagainsts=0,1,2,...thenweobtaintheautocorrelationfunction(acf)orcorrelogram.

SomeNotationandConcepts4课件

AWhiteNoiseProcess

5课件6课件AnACFExample(p234)7课件Letut(t=1,2,3,...)beasequenceofindependentlyandidenticallydistributed(iid)randomvariableswithE(ut)=0andVar(ut)=

2,then yt=

+ut+

1ut-1

+

2ut-2+...+

qut-q

isaqthordermovingaveragemodelMA(q).Orusingthelagoperatornotation: Lyt=yt-1

Liyt=yt-i通常，可以将常数项从方程中去掉，而并不失一般性。3MovingAverageProcesses

8课件移动平均过程的性质Itspropertiesare E(yt

)=

Var(yt)=

0=(1+)

2 Covariances 自相关函数9课件

ConsiderthefollowingMA(2)process:

whereutisazeromeanwhitenoiseprocesswithvariance. (i)CalculatethemeanandvarianceofXt (ii)Derivetheautocorrelationfunctionforthisprocess(i.e.expresstheautocorrelations,

1,

2,...asfunctionsoftheparameters

1and

2). (iii)If

1=-0.5and

2=0.25,sketchtheacfofXt.ExampleofanMAProcess

10课件(i)IfE(ut)=0,thenE(ut-i)=0

i.So E(Xt)=E(ut+

1ut-1+

2ut-2)=E(ut)+

1E(ut-1)+

2E(ut-2)=0

Var(Xt) =E[Xt-E(Xt)][Xt-E(Xt)] Var(Xt) =E[(Xt)(Xt)] =E[(ut+

1ut-1+

2ut-2)(ut+

1ut-1+

2ut-2)] =E[+cross-products] ButE[cross-products]=0,sinceCov(ut,ut-s)=0fors

0.So

Var(Xt) =

0=E[] = =Solution11课件Solution(cont’d)12课件

3 =E[Xt][Xt-3] =E[(ut+

1ut-1+

2ut-2)(ut-3+

1ut-4+

2ut-5)] =0

So

s=0fors>2.

nowcalculatetheautocorrelations:Solution(cont’d)13课件(iii)For

1=-0.5and

2=0.25,substitutingtheseintotheformulaeabovegives

1

=-0.476,

2=0.190. Thustheacfplotwillappearasfollows:ACFPlot14课件Anautoregressivemodeloforderp,AR(p)canbeexpressedas Orusingthelagoperatornotation: Lyt=yt-1

Liyt=yt-i

or or where

4AutoregressiveProcesses

15课件平稳性使AR模型具有一些很好的性质。如前期误差项对当前值的影响随时间递减。TheconditionforstationarityofageneralAR(p)modelisthattherootsof特征方程

alllieoutsidetheunitcircle.Example1:Isyt=yt-1+utstationary? Thecharacteristicrootis1,soitisaunitrootprocess(sonon-stationary)Example2:p241AstationaryAR(p)modelisrequiredforittohaveanMA(

)representation.

TheStationaryCondition

foranARModel

16课件Statesthatanystationaryseriescanbedecomposedintothesumoftwounrelatedprocesses,apurelydeterministicpartandapurelystochasticpart,whichwillbeanMA(

).

FortheAR(p)model,,ignoringtheintercept,theWolddecompositionis

where,

可以证明，算子多项式R(L)的集合与代数多项式R(z)的集合是同结构的，因此可以对算子L做加、减、乘和比率运算。Wold’sDecompositionTheorem

17课件Themomentsofanautoregressiveprocessareasfollows.Themeanisgivenby*TheautocovariancesandautocorrelationfunctionscanbeobtainedbysolvingwhatareknownastheYule-Walkerequations: *IftheARmodelisstationary,theautocorrelationfunctionwilldecayexponentiallytozero.TheMomentsofanAutoregressiveProcess18课件ConsiderthefollowingsimpleAR(1)model

(i)Calculatethe(unconditional)meanofyt.Fortheremainderofthequestion,set

=0forsimplicity.(ii)Calculatethe(unconditional)varianceofyt.(iii)Derivetheautocorrelationfunctionforyt.

19课件(i) E(yt)=E(

+

1yt-1) =

+

1E(yt-1) Butalso

E(yt)=

+

1

(

+

1E(yt-2)) =

+

1

+

12

E(yt-2))

=

+

1

+

12

(

+

1E(yt-3)) =

+

1

+

12

+

13

E(yt-3)

AninfinitenumberofsuchsubstitutionswouldgiveE(yt)=

(1+

1+

12

+...)+

1

y0 Solongasthemodelisstationary,i.e.,then

1

=0. SoE(yt)=

(1+

1+

12

+...)=Solution

20课件(ii)Calculatingthevarianceofyt:*

FromWold’sdecompositiontheorem:

Solongas,thiswillconverge.

Solution(cont’d)

21课件Var(yt) =E[yt-E(yt)][yt-E(yt)]butE(yt)=0,sincewearesetting

=0.Var(yt)=E[(yt)(yt)]*有简便方法

=E[ ] =E[ =E[ = = =Solution(cont’d)

22课件(iii)Turningnowtocalculatingtheacf,firstcalculatetheautocovariances:（*用简便方法）

1=Cov(yt,yt-1)=E[yt-E(yt)][yt-1-E(yt-1)]

1=E[ytyt-1]

1=E[ ] =E[ = =Solution(cont’d)

23课件Solution(cont’d)Forthesecondautocorrelationcoefficient,

2=Cov(yt,yt-2)=E[yt-E(yt)][yt-2-E(yt-2)]Usingthesamerulesasappliedaboveforthelag1covariance

2=E[ytyt-2]

=E[ ] =E[ == =24课件Solution(cont’d)Ifthesestepswererepeatedfor

3,thefollowingexpressionwouldbeobtained

3=andforanylags,theautocovariancewouldbegivenby

s=Theacfcannowbeobtainedbydividingthecovariancesbythevariance:25课件Solution(cont’d)

0=

1=

2=

3=…

s=26课件Measuresthecorrelationbetweenanobservationkperiodsagoandthecurrentobservation,aftercontrollingforobservationsatintermediatelags(i.e.alllags<k).yt-k与

yt之间的偏自相关函数

kk

是在给定yt-k+1,yt-k+2,…,yt-1

的条件下，yt-k与

yt之间的部分相关。So

kkmeasuresthecorrelationbetweenytandyt-kafterremovingtheeffectsofyt-k+1,yt-k+2,…,yt-1.

或者说，偏自相关函数

kk

是对yt-k与

yt之间未被yt-k+1,yt-k+2,…,yt-1所解释的相关的度量。Atlag1,theacf=pacfalwaysAtlag2,

22=(

2-

12)/(1-

12)Forlags3+,theformulaearemorecomplex.

5ThePartialAutocorrelationFunction(denoted

kk)

27课件ThepacfisusefulfortellingthedifferencebetweenanARprocessandanMAprocess.InthecaseofanAR(p),therearedirectconnectionsbetweenytandyt-sonlyfors

p.SoforanAR(p),thetheoreticalpacfwillbezeroafterlagp.InthecaseofanMA(q),thiscanbewrittenasanAR(

),sotherearedirectconnectionsbetweenytandallitspreviousvalues.ForanMA(q),thetheoreticalpacfwillbegeometricallydeclining.ThePartialAutocorrelationFunction28课件TheinvertibilityconditionIfMA(q)processcanbeexpressedasanAR(∞),thenMA(q)isinvertible.MA(q)的可逆性条件：特征方程根的绝对值大于1。 从而有29课件BycombiningtheAR(p)andMA(q)models,wecanobtainanARMA(p,q)model:

where and or with

6ARMAProcesses

30课件ARMA过程的特征是AR和MA的组合。可逆性条件：Similartothestationaritycondition,wetypicallyrequiretheMA(q)partofthemodeltohaverootsof

(z)=0greaterthanoneinabsolutevalue.

ThemeanofanARMAseriesisgivenby

TheautocorrelationfunctionforanARMAprocesswilldisplaycombinationsofbehaviourderivedfromtheARandMAparts,butforlagsbeyondq,theacfwillsimplybeidenticaltotheindividualAR(p)model.

ARMA过程的特征31课件Anautoregressiveprocesshasageometricallydecayingacf:拖尾numberofspikes尖峰信号ofpacf=ARorder：截尾

AmovingaverageprocesshasNumberofspikesofacf=MAorder：截尾ageometricallydecayingpacf：拖尾AARMAprocesshasageometricallydecayingacf:拖尾ageometricallydecayingpacf：拖尾SummaryoftheBehaviouroftheacfandpacfforARandMAProcesses32课件

Theacfandpacfareestimatedusing100,000simulatedobservationswithdisturbancesdrawnfromanormaldistribution.

ACFandPACFforanMA(1)Model:yt=–0.5ut-1+ut33课件ACFandPACFforanMA(2)Model:

yt=0.5ut-1-

0.25ut-2+ut

34课件ACFandPACFforaslowlydecayingAR(1)Model:

yt=0.9

yt-1+ut

35课件ACFandPACFforamorerapidlydecayingAR(1)Model:yt=0.5

yt-1+ut

36课件ACFandPACFforaAR(1)ModelwithNegativeCoefficient:yt=-0.5

yt-1+ut

37课件ACFandPACFforaNon-stationaryModel

(aunitcoefficient):yt=yt-1+ut

38课件ACFandPACFforanARMA(1,1):

yt=0.5yt-1+0.5ut-1+ut

39课件Chapter6Multivariatemodels401MotivationsAllthemodelswehavelookedatthusfarhavebeensingleequationsmodelsoftheform y=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/supplied Pt=price，St=priceofasubstitutegoodTt=somevariableembodyingthestateoftechnology41Assumingthatthemarketalwaysclears,anddroppingthetimesubscriptsforsimplicity (4) (5) ThisisasimultaneousSTRUCTURALFORMofthemodel.

Thepointisthatpriceandquantityaredeterminedsimultaneously(priceaffectsquantityandquantityaffectsprice).

PandQareendogenousvariables,whileSandTareexogenous.

WecanobtainREDUCEDFORMequationscorrespondingto(4)and(5)bysolvingequations(4)and(5)forPandforQ.SimultaneousEquationsModels:

TheStructuralForm

42SolvingforQ, (6)

SolvingforP, (7)

Rearranging(6),

(8)

ObtainingtheReducedForm43Multiplying(7)throughby

,

(9)

(8)and(9)arethereducedformequationsforPandQ.ObtainingtheReducedForm44Butwhatwouldhappenifwehadestimatedequations(4)and(5),i.e.thestructuralformequations,separatelyusingOLS?

BothequationsdependonP.OneoftheCLRMassumptionswasthatE(X

u)=0,whereXisamatrixcontainingallthevariablesontheRHSoftheequation.

Itisclearfrom(8)thatPisrelatedtotheerrorsin(4)and(5)-i.e.itisstochastic.

WhatwouldbetheconsequencesfortheOLSestimator,,ifweignorethesimultaneity?

2SimultaneousEquationsBias

45Recallthat andSothat

Takingexpectations,

IftheX’sarenon-stochastic,E(X

u)=0,whichwouldbethecaseinasingleequationsystem,sothat ,whichistheconditionforunbiasedness.SimultaneousEquationsBias46But....iftheequationispartofasystem,thenE(X

u)

0,ingeneral.Conclusion:ApplicationofOLStostructuralequationswhicharepartofasimultaneoussystemwillleadtobiasedcoefficientestimates.

IstheOLSestimatorstillconsistent,eventhoughitisbiased?No-Infacttheestimatorisinconsistentaswell.Henceitwouldnotbepossibletoestimateequations(4)and(5)validlyusingOLS.SimultaneousEquationsBias47SoWhatCanWeDo?Takingequations(8)and(9),wecanrewritethemas (10)

(11)

WeCANestimateequations(10)&(11)usingOLSsincealltheRHSvariablesareexogenous.

But...weprobablydon’tcarewhatthevaluesofthe

coefficientsare;whatwewantedweretheoriginalparametersinthestructuralequations-

,

,

,

,

,

.3AvoidingSimultaneousEquationsBias48

CanWeRetrievetheOriginalCoefficientsfromthe

’s? Shortanswer:sometimes.

wesometimesencounteranotherproblem:identification.*Considerthefollowingdemandandsupplyequations

Supplyequation (12) Demandequation (13)

Wecannottellwhichiswhich!BothequationsareUNIDENTIFIEDorUNDERIDENTIFIED.Theproblemisthatwedonothaveenoughinformationfromtheequationstoestimate4parameters.Noticethatwewouldnothavehadthisproblemwithequations(4)and(5)sincetheyhavedifferentexogenousvariables.

4IdentificationofSimultaneousEquations49Wecouldhavethreepossiblesituations:

1.Anequationisunidentified·

like(12)or(13)·

wecannotgetthestructuralcoefficientsfromthereducedformestimates

2.Anequationisexactlyidentified·

e.g.(4)or(5)·

cangetuniquestructuralformcoefficientestimates

3.Anequationisover-identified·

Examplegivenlater·

Morethanonesetofstructuralcoefficientscouldbeobtainedfromthereducedform.

WhatDetermineswhetheranEquationisIdentifiedornot?50Howdowetellifanequationisidentifiedornot?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?51StatementoftheOrderConditionLetGdenotethenumberofstructuralequations.AnequationisjustidentifiedifthenumberofvariablesexcludedfromanequationisG-1.

IfmorethanG-1areabsent,itisover-identified.IflessthanG-1areabsent,itisnotidentified.

ExampletheY’sareendogenous,whiletheX’sareexogenous.Determinewhethereachequationisover-,under-,orjust-identified.

(14)-(16)

Statementoftheordercondition52

SolutionG=3;If#excludedvariables=2,theeqnisjustidentifiedIf#excludedvariables>2,theeqnisover-identifiedIf#excludedvariables<2,theeqnisnotidentified

Equation14: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。Exampleoftheordercondition535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545556Considerthefollowingsystemofequations:

(21-23)

Assumethattheerrortermsarenotcorrelatedwitheachother.CanweestimatetheequationsindividuallyusingOLS?

Equation21:Containsnoendogenousvariables,soX1andX2arenotcorrelatedwithu1.SowecanuseOLSon(21).Equation22:ContainsendogenousY1togetherwithexogenousX1andX2.WecanuseOLSon(22)ifalltheRHSvariablesin(22)areuncorrelatedwiththatequation’serrorterm.Infact,Y1isnotcorrelatedwithu2becausethereisnoY2terminequation(21).SowecanuseOLSon(22).6RecursiveSystems

57Equation23:ContainsbothY1andY2;werequirethesetobeuncorrelatedwithu3.Bysimilarargumentstotheabove,equations(21)and(22)donotcontainY3,sowecanuseOLSon(23).

ThisisknownasaRECURSIVEorTRIANGULARsystem.Wedonothaveasimultaneityproblemhere.

Butinpracticenotmanysystemsofequationswillberecursive...RecursiveSystems58IndirectLeastSquares(ILS)CannotuseOLSonstructuralequations,butwecanvalidlyapplyittothereducedformequations.

Ifthesystemisjustidentified,ILSinvolvesestimatingthereducedformequationsusingOLS,andthenusingthemtosubstitutebacktoobtainthestructuralparameters.

However,ILSisnotusedmuchbecause 1.Solvingbacktogetthestructuralparameterscanbetedious. 2.Mostsimultaneousequationssystemsareover-identified.

7EstimationproceduresforSystems59Infact,wecanusethistechniqueforjust-identifiedandover-identifiedsystems.

Twostageleastsquares(2SLSorTSLS)isdoneintwostages:

Stage1:ObtainandestimatethereducedformequationsusingOLS.Savethefittedvaluesforthedependentvariables.

Stage2:Estimatethestructuralequations,butreplaceanyRHSendogenousvariableswiththeirstage1fittedvalues.

EstimationofSystems

UsingTwo-StageLeastSquares

60Example:Sayequations(14)-(16)arerequired.

Stage1:Estimatethereducedformequations(17)-(19)individuallybyOLSandobtainthefittedvalues,.

Stage2:ReplacetheRHSendogenousvariableswiththeirstage1estimatedvalues:

(24)-(26)

Nowandwillnotbecorrelatedwithu1,willnotbecorrelatedwithu2,andwillnotbecorrelatedwithu3.EstimationofSystems

UsingTwo-StageLeastSquares61EstimationofSystems

UsingTwo-StageLeastSquares

62RecallthatthereasonwecannotuseOLSdirectlyonthestructuralequationsisthattheendogenousvariablesarecorrelatedwiththeerrors.

OnesolutiontothiswouldbenottouseY2orY3,butrathertousesomeothervariablesinstead.

Wewanttheseothervariablestobe(highly)correlatedwithY2andY3,butnotcorrelatedwiththeerrors-theyarecalledINSTRUMENTS.

SaywefoundsuitableinstrumentsforY2andY3,z2andz3respectively.Wedonotusetheinstrumentsdirectly,butrunregressionsoftheform

(27)&(28)

InstrumentalVariables

63Obtainthefittedvaluesfrom(27)&(28),and,andreplaceY2andY3withtheseinthestructuralequation.

Itistypicaltousemorethanoneinstrumentperendogenousvariable.

Iftheinstrumentsarethevariablesinthereducedformequations,thenIVisequivalentto2SLS.InstrumentalVariables64WhatHappensifWeUseIV/2SLSUnnecessarily?Thecoefficientestimateswillstillbeconsistent,butwillbeinefficientcomparedtothosethatjustusedOLSdirectly.

TheProblemWithIVWhataretheinstruments? Solution:2SLSiseasier.

OtherEstimationTechniques

1.3SLS-allowsfornon-zerocovariancesbetweentheerrorterms.2.LIML-estimatingreducedformequationsbymaximumlikelihood3.FIML-estimatingalltheequationssimultaneouslyusingmaximumlikelihoodInstrumentalVariables65AnaturalgeneralisationofautoregressivemodelspopularisedbySims.AVARisinasenseasystemsregressionmodeli.e.thereismorethanonedependentvariable.

SimplestcaseisabivariateVAR

whereuitisaniiddisturbancetermwithE(uit)=0,i=1,2;E(u1tu2t)=0.

TheanalysiscouldbeextendedtoaVAR(g)model,orsothattherearegvariablesandgequations.8VectorAutoregressiveModels

66OneimportantfeatureofVARsisthecompactnesswithwhichwecanwritethenotation.Forexample,considerthecasefromabovewherek=1.

Wecanwritethisas

or

yt =

0 +

1yt-1+ut g

1 g

1 g

g

g

1g

1VectorAutoregressiveModels:

NotationandConcepts67VAR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+...+

k

yt-k+ut g

1g

1g

gg

1g

g

g

1g

gg

1g

1Wecanalsoextendthistothecasewherethemodelincludesfirstdifferencetermsandcointegratingrelationships(aVECM).VectorAutoregressiveModels:

NotationandConcepts

68AdvantagesofVARModelling -Donotneedtospecifywhichvariablesareendogenousorexogenous-allareendogenous -Allowsthevalueofavariabletodependonmorethanjustitsownlagsorcombinationsofwhitenoiseterms,somoregeneralthanARMAmodelling -Providedthattherearenocontemporaneoustermsontherighthandsideoftheequations,cansimplyuseOLSseparatelyoneachequation，因为方程右边的变量都是前定变量。

-Forecastsareoftenbetterthan“traditionalstructural”models.VARModelsComparedwithStructuralEquationsModels69ProblemswithVAR’s -VAR’sarea-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?

VARModelsComparedwithStructuralEquationsModels70ChoosingtheOptimalLagLength71ChoosingtheOptimalLagLength72InformationCriteriaforVARLagLengthSelection

Multivariateversionsoftheinformationcriteriaarerequired.Thesecanbedefinedas:

whereallnotationisasaboveandk

isthetotalnumberofregressorsinallequations,whichwillbeequaltog2k+gforgequations,eachwithklagsofthegvariables,plusaconstanttermineachequation.Thevaluesoftheinformationcriteriaareconstructedfor0,1,…lags(uptosomepre-specifiedmaximum).73DoestheVARIncludeContemporaneousTerms?

Sofar,wehaveassumedtheVARisoftheform

Butwhatiftheequationshadacontemporaneousfeedbackterm?

WecanwritethisasThisVARisinprimitive/structuralform.74PrimitiveversusStandardFormVARsWecantakethecontemporaneoustermsovertotheLHSandwriteor Byt=

0+

1yt-1+utWecanthenpre-multiplybothsidesbyB-1togive

yt=B-1

0+B-1

1yt-1+B-1utor yt=A0+A1yt-1+etThisisknownasastandardformVAR,whichwecanestimateusingOLS.75VAR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α12或α22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767778ImpulseResponsesVARmodelsareoftendifficulttointerpret:onesolutionistoconstructtheimpulseresponsesandvariancedecompositions.ImpulseresponsestraceouttheresponsivenessofthedependentvariablesintheVARtoshockstotheerrorterm.

Aunitshockisappliedtoeachvariableanditseffectsarenoted.ConsiderforexampleasimplebivariateVAR(1):Achangeinu1twillimmediatelychangey1.Itwillchangey2andalsoy1duringthenextperiod.Wecanexaminehowlongandtowhatdegreeashocktoagivenequationhasonallofthevariablesinthesystem.eg.p34179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:

80

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81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：

–2–1012345………t82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。83VarianceDecompositionsVariancedecompositionsofferaslightlydifferentmethodofexaminingVARdynamics.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.84TheOrderingoftheVariablesButforcalculatingimpulseresponsesandvariancedecompositions,theorderingofthevariablesisimportant.Themainreasonforthisisthatabove,weassumedthattheVARerrortermswerestatisticallyindependentofoneanother.Thisisgenerallynottrue,however.Theerrortermswilltypicallybecorrelatedtosomedegree.Therefore,thenotionofexaminingtheeffectoftheinnovationsseparatelyhaslittlemeaning,sincetheyhaveacommoncomponent.Whatisdoneisto“orthogonalise”theinnovations.InthebivariateVAR,thisproblemwouldbeapproachedbyattributingalloftheeffectofthecommoncomponenttothefirstofth