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JournalofEconomicDynamicsandControl2(1980)329-352.QNorth-HollandTESTINGFORCAUSALITYAPersonalViewpointC.W.J.GRANGERReceivedFebruary1980,linalversionreceivedJune1980Ageneraldefinitionofcausahtyisintroducedandthenspecializedtobecomeoperational.Byconsideringsimpleexamplesanumhcr~4,I~\;I~I;I+x.tnd;tlsoditliculties,withthedefinitionarcdiscussed.TestsbasedontheJC~‘IIIIIIO~~arcthouc~m\~dcrcdandtheuseofpost-sampledataemphasized,ratherthanrelyingonthesamedatatohtamodelanduseittotestcausality.ItissuggestedthataBayesianviewpointshouldbetakenininterpretingtheresultsofthesetests.Finally,theresultsofastudyrelatingadvertisingandconsumptionarebrieflypresented.1.TheproblemandadefinitionMoststatisticiansmeettheconceptofcausalityearlyintheircareersas,whendiscussingtheinterpretationofacorrelationcoefficientoraregression,mosttextbookswarnthatanobservedrelationshipdoesnotallowonetosayanythingaboutcausationbetweenthevariables.Ofcoursethiswarninghasmuchtorecommendit,butconsiderthefollowingspecialsituation:supposethatXandYaretheonlytworandomvariablesintheuniverseandthatastrongcorrelationisobservedbetweenthem.FurthersupposethatGod,oranacceptablesubstitute,tellsonethatXdoesnotcauseYleavingopenthepossibilityofYcausingX.Inthecircumstances,thestrongobservedcorrelationmightleadtoacceptanceofthepropositionthatYdoescauseX.ThispossibilityoccursbecauseoftheextrastructureimposedonthesituationbytheknowledgethatXdoesnotcauseY.Aswillbeseen,thewaystructureisimposedwillbeimportantindefinitionsofcausality.Thetextbooks,havinggivenacautionarywarningaboutcausality,virtuallyneverthengoonwithapositivestatementoftheform‘theproceduretotestforcausalityis...‘,althoughafewdosaythatcausalitycanbedetectedfromaproperlyconductedexperiment.Theobviousreasonforthelackofsuchpositivestatementsisthatthereisnogenerallyacceptedprocedurefortestingforcausality,partiallybecauseofalackofadefinitionofthisconceptthatisuniversallyliked.Attitudestowardscausalitydifferwidely,fromthedefeatistonethatitisimpossibletodefinecausality,letalonetestforit,tothepopulistviewpointthateveryonehastheirownpersonaldefinitionandsoitisunlikelythatagenerallyacceptabledefinitionexists.Itisclearlyatopicinwhichindividualtastespredominate,anditwouldbeimpropertotrytoforceresearchworkerstoacceptadefinitionwithwhichtheyfeeluneasy.Myownexperienceisthat,unlikeart,causalityisaconceptwhosedefinitionpeopleknowwhattheydonotlikebutfewknowwhattheydolike.Itmightthereforebehelpfultopresentadefinitionthatsomeofusappeartothinkhassomeacceptablefeaturessothatitcanbepubliclydebatedandcomparedwithalternativedefinitions.Foreaseofexposition,auniverseisconsideredinwhichallvariablesaremeasuredjustatprespecifiedtimepointsatconstantintervalsr=1,2,.WhenattimeII,letalltheknowledgeintheuniverseavailableatthattimebedenotedR,,anddenotebyQ,,-Y,thisinformationexceptthevaluestakenbyavariablexuptotimeII,whereYI~Q,,.Q,,includesnovariatesmeasuredattimepointst~,althoughitmaywellcontainexpectationsorforecastsofsuchvalues.However,theseexpectationswillsimplybefunctionsofR,,.R,,willcertainlybemultivariateandY,couldbe,andbothwillbestochasticvariables.Toprovidestructuretothesituation,thefollowingaxiomswillbeassumedtohold:AxiomA.Thepastandpresentmaycausethefuture,butthefuturecannotcausethepast.AsionB.Q,,containsnoredundantinformation,sothatifsomevariableZ,,isfunctionallyrelatedtooneormoreothervariables,inadeterministicfashion,thenZ,,shouldbeexcludedfromQ,,.Thus,forexample,iftemperatureismeasuredhourlyatsomelocationbothindegreesFahrenheitanddegreesCentigrade,thereisnopointinincludingbothofthesevariablesintheuniversalinformationset.SupposethatweareinterestedinthepropositionthatthevariableYcausesthevariableX.Attime17,thevalueXn+,willbe,ingeneral,arandomvariableandsocanbecharacterizedbyprobabilitystatementsoftheformProb(X,,,EA)forasetA.Thissuggeststhefollowing:GenerdDefinitiort.Y,issaidtocauseX,,,ifProb(X,,+1~AIQ,)#Prob(X,+leAl.Q,,-Y,)forsomeA.Forcausationtooccur,thevariableY,needstohavesomeuniqueinformationaboutwhatvalueX,,,,willtakeintheimmediatefuture.Theultimateobjectiveistoproduceanoperationaldefinition,whichthisiscertainlynot,byaddingsufficientlimitations.Thisprocesswillbediscussedinsection3,andthedefinitionwillalsobedefendedthere.Inthefollowingsectionsomemoregeneralbackgroundmaterialwillbeintroducedwhichwill,hopefully,makethedefencealittleeasier.2.AvarietyofviewpointsoncausalityTheobviousplacetolookfordefinitionsofcausalityanddiscussionsoftheconceptisthewritingsofphilosophersonthetopic,ofwhichtherehavebeenplentyfromAristotleonwards.AusefuldiscussionofpartsofthisliteraturecanbefoundinBunch(1963).Ithinkthatitisfairtosaythatthephilosophershavenotreachedaconsensusofopiniononthetopic,havenotfoundadefinitionthatamajoritycanacceptand,inparticular,havenotproducedmuchthatisusefultopractisingscientists.Mostoftheexamplestraditionallyusedbyphilosopherscomefromclassicalphysicsorchemistry,suchasaskingwhatcausestheflamewhenamatchisstruck,ornotingthatapplyingheattoametalrodcausesittobecomelonger.Muchoftheliteratureattemptstodiscussuniquecausesindeterministi