NotesonTestingCausality

1 NotesonTestingCausalityJin-LungLinInstit uteofEconomics,AcademiaSinicaDepartmento fEconomics,NationalChengchiUniversityMay , Abstract isnotereviewsthede nition,distributiontheoryandmodelingstra tegyoftest-ingcausality. Startingwiththede nitionofGrangerCausality,wediscussvariou sissues on testing causality within stationary and nonstationary systems. In addi-tion, t Dooncausal-itytestingandreviewseveralemp iricalexamples. IntroductionTestingcausalityamongvariabl esisoneofthemostimportantand,yet,oneofth emostdi cultissuesineconomics. edi cultyarisesfromthenon-experimentalnature ofsocialscience. Fornaturalscience,researcherscanperforme xperimentswhere all other possiblecausesare kept xed except for the sole factor under in-vestigation. Byrepeatingtheprocessforeachpossiblecaus e, onecanidentifythecausalstructuresamongfa ctorsorvariables. erearenosuchluckforsocialsci-ence, erentvariablesa ectthesamevariablesimultaneouslyandrepea tedexperimentsundercontrolareinfeasible( experimen-taleconomicsisnosolution,atlea st,notyet).

2 Twomostdi cultchallengesare: .. ere always exist the possibility of ignored common factors. e causalrelationship among variables might disappear when the previously be one, philosophers and social scientists have attempted to use graphicalmodelstoaddressthesecondissue. Asforthe rstissue,timeseriesanalystslookfor rescue from the unique unidirectional property of time arrow:cause precedese ect. Based upon this concept, Clive Granger has proposed a working de -nition of causality, using the foreseeability as a yardstick which is called Grangercausality. ,Section discussesthede nitionofGrangercausality. Testing causality for stationary processes are reviewed in Section andSection focusesonnonstationaryprocesses. Weturntographicalmodelsinsec-tion . Ato-doandnot-to-dolistisputtogetherinSec tion . De ning Granger causality . Two assumptions . efuturecannotcausethepast. epastcausesthepresentorfuture.

3 (Howaboutexpectation?) . Acausecontainsuniqueinformationaboutane ectnotavailableelsewhere.. De nitionXtissaidnottoGranger-causeYtiffora llh> F(Yt+hS t)=F(yt+hS t Xt)whereFdenotestheconditionaldistributi onand t Xtisalltheinformationintheuniverseexcept seriesXt. Inplainwords, : e whole distributionFis generally di cult to handle empirically and weturntoconditionalexpectationandvarianc e. Itisde nedforallh> andnotonlyforh= . Causalityatdi erenthdoesnotimplyeachother. eyareneithersu cientnornecessary. tcontainsalltheinformationintheuniverseu ptotimetthatexcludesthepotential ignored common factors problem. e question is: how to mea-sure tinpractice? eunobservedcommonfactorsarealwaysapotent ialproblemforany niteinformationset. Instantaneouscausality t+h xt+handfeedbackisdi nedde nitionbecomeasbelow:XtdoesnotGrangercaus eYt+hwithrespecttoinformationJt,ifE(Yt+h SJt,Xt)=E(Yt+hSJt)Remark: Note that causality here is de ned asrelative to.

4 In other words, noe ortismadeto ndthecompletecausalpathandpossiblecommon factors.. Equivalent de nitionFor a l-dimension stationary process,Zt, there exists a canonical MA representa-tionZt= + (B)ut= + Qi= iut i, =Il Anecessaryandsu cientconditionforvariableknotGranger-cau sevariablejisthat jk,i= , fori= , , .Iftheprocessisinvertible,thenZt=C+A(B)Z t +ut=C+ Qi= AiZt i+utIfthereareonlytwovariables,ortwo-gro upofvariables,jandk,thenanecessaryandsu cientconditionforvariableknottoGranger-c ausevariablejisthatAjk,i= , fori= , , . econditionisgoodforallforecasthorizon, ( )processwithdimensionequalorgreaterthan ,Ajk,i= , fori= , , issu cientfornon-causalityath= butinsu cientforh> .Variablekmighta ectvariablejintwoormoreperiodinthefuture viathee ectthroughothervariables. Forexample, y ty ty t = .. y t y t y t + u tu tu t en,y = u u u = ;y =A y =.

5 ;y =A y = .. Tosummarize, . Forbivariateortwogroupsofvariables,IRana lysisisequivalenttoapplyingGranger-causa litytesttoVARmodel; . Fortestingtheimpactofonevariableontheoth erwithinahighdimensional( )system, ,foranVAR( )processwithdimensiongreaterthan ,itdoesnotsu cetochecktheupperright-handcornerelement ofthecoe cientmatrixin order to determine if the last variable is noncausal for the rst ( )andDuforandRenault( )fordetaileddiscussion. Testing causality for stationary series . Impulse response and causal orderingItiswellknownthatresidualsfromaV ARmodelaregenerallycorrelatedandap-plyin gtheCholeskydecompositionisequivalenttoa ssumingrecursivecausalor-dering from the top variable to the bottom variable. Changing the order of thevariablescouldgreatlychangetheresults oftheimpulseresponseanalysis.. Causal analysis for bivariate VARF orabivariatesystem,yt,xtde nedby ytxt = A (B)A (B)A (B)A (B) yt xt + uytuxt = (B) (B) (B) (B) uyt uxt + uytuxt xtdoes not Granger-causeytif (B)= or ,i= , fori=.

6 Iscondition is equivalent toA ,i= , fori= , , ,p. In other words, this corre-spondstotherestrictionsthatallcros s-lagscoe of illustration, we shall focus upon bivariate AR( ) process so thatAi j(B)=Ai j,i,j= , asde nedabove. eresultscanbeeasilygeneralizedtoAR(p) : . Feedback,H ,x yH = A A A A . Independent,H x yH = A A .xcausesybutydoesnotcausex,H ,y~ xH = A A A .ycausesxbutxdoesnotcausey,H ,x~ yH = A A A Caines,KengandSethi( )proposedatwo-stagetestingprocedureforde ter-miningcausaldirections. In rststage,testH (null)againstH ,H (null)againstH ,andH (null)againstH . Ifnecessary,testH (null)againstH ,andH (null)againstH . SeeLiang,ChouandLin( )foranapplication.. Causal analysis for multivariate ,KengandSethi( )providedareasonableprocedure.. For a pair(X,Y), construct bivariate VAR with order chosen to minimizemultivariate nalpredictionerror(MFPE).

7 Applythestagewiseproceduretodeterminethe causalstructureofX,Y; . IfaprocessX,hasnmultiplecausalvariables, y ,..,yn,rankthesevariablesaccordingtothed ecreasingorderoftheirspeci cgravitywhichistheinverseofMFPE(X,yi); . Foreachcausedvariableprocess,X, rstconstructtheoptimalunivariateARmodel using FPE to determine the lag order. e, add the causal variable,one at a time according to their causal rank and use FPE to determine theoptimal orders at each step. Finally, we get the optimal ordered univariatemultivariateARmodelofXagainsti tscausalvariables; . Pool all the optimal univariate AR models above and apply the Full Infor-mation Maximum Likelihood (FIML) method to estimate the system. Fi-nallyperformthediagnosticcheckingwith thewholesystemasmaintainedmodel.. Causal analysis for Vector ARMA model (h= )LetXben vectorgeneratedby (B)Xt= (B)at XidoesnotcauseXjifandonlyifdet( i(z), (j)(z))= where i(B)istheithcolumnofthematrix (z)and (j)(z)isthematrix (z) (two-group)case, (B) (B) (B) (B) XitX t = (B) (B) (B) (B) a ta t en,XidoesnotcauseXjifandonlyif (z) (z) (z) (z)= Ifn =n = , en,XidoesnotcauseXjifandonlyif (z) (z) (z) (z)= Generaltestingproceduresis.

8 BuildamultivariateARMA modelforXt, . Derive the noncausality conditions in term of AR and MA parameters, sayRj( l)= ,j= ,..,K . Chooseatestcriterion,Wald, ( l)=( Rj(B) lS l l)k kLetV( l)be the asymptotic covariance matrix of N( l= l). en the WaldandLRteststatisticsare: W=NR( l) [T( l) V( l)T( l)] R( l), LR= (L( ,X) L( ,X))where istheMLEof undertheconstraintofnoncausality. Toillustrate,letXtbeainvertible -dimensionalARMA( , )model. B B B B X tX t = B B B B a ta t X doesnotcauseX ifandonlyif (z) (z) (z) (z)= ( )z+( )z = = , = Forthevector, l=( , , , ) ,thematrixT( l)= mightnotbenonsingularunderthenullofH X doesnotcauseX .Remarks: econditionsareweakerthan = = = isanecessaryconditionforH , = = issu cientcondi-tionand = ,& = aresu cientforH.

9 LetH X doesnotcauseX . Considerthefollowinghypotheses:H = ;H = = H , = , and = H = en,H = H H ,H H H ,H H H .Testingprocedures: . TestH atlevel . IfH isrejected,thenH isrejected. Stop.. IfH is not rejected, testH at level . IfH is not rejected,H cannot berejected. Stop . IfH isrejected,test H = atlevel . If H isrejected,thenH isalsorejected. If H isnotrejected,thenH isalsonotrejected. Causal analysis for nonstationary processes e asymptotic normal or distribution in previous section is build upon theassumption that the underlying processesXtis stationary. e existence of unitroot and cointegration might make the traditional asymptotic inference , I shall brie y review unit root and cointegration and their relevance withtesting causality. In essence, cointegration, causality test, VAR model and IR arecloselyrelatedandshouldbeconsideredjo intly.

10 Unit root:What is unit root? e time seriesytas de ned inAp(B)yt=C(B) thas an unit root ifAp( )= ,C( ) .Why do we care about unit root? Foryt, the existence of unit roots implies that a shock in thas permanentimpactsonyt. Ifythasaunitroot,thenthetraditionalasymp toticnormalityresultsusuallynolongerappl y. Weneeddi erentasymptotictheorems.. Cointegration:What is cointegration?When linear combination of two I( ) process become an I( ) process, then do we care about cointegration? Cointegrationimpliesexistenceoflong-rune quilibrium; Cointegrationimpliescommonstochastictren d; Withcointegration,wecanseparateshort-and long-runrelationshipamongvariables; Cointegrationcanbeusedtoimprovelong-runf orecastaccuracy; Cointegrationimpliesrestrictionsonthepar ametersandproperaccountingoftheserestric tionscouldimproveestimatione (p)serieswithrcointegrationvector (p r).Ap(B)Yt=Ut Yt= Yt +p Qi= i Yt i+ Dt+UtYt=CtQi= (Ut+ Di)+C (B)(Ut+ Dt)+P Y Ap( )= = C= ( ) Cointegration introduces one additional causal channel (error correctionterm)foronevariabletoa ecttheothervariables.