DirectedAcyclicGraphs

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1、DirectedAcyclicGraphsDavidA.BesslerTexasA&MUniversityNovember20,2002UniversidadInternacionaldelEcuadorQuito,Ecuador1OutlineIntroductionCausalForksInvertedCausalForksD-separationMarkovPropertyTheAdjustmentProblemPolicyModelingPCAlgorithm2OutlineContinuedExample:TrafficFatalitiesCorrelationandPartialC

2、orrelationForecastingTrafficFatalitiesMoreExamples:USMoney,PricesandIncomeWorldStockMarketsConclusion3MotivationOftentimesweareuncertainaboutwhichvariablesarecausalinamodelingeffort.Theorymaytelluswhatourfundamentalcausalvariablesareinacontrolledsystem;however,itiscommonthatourdatamaynotbecollectedi

3、nacontrolledenvironment.Infactwearerarelyinvolvedwiththecollectionofourdata.4ObservationalDataInthecasewherenoexperimentalcontrolispresentinthegenerationofourdata,suchdataaresaidtobeobservational(non-experimental)andusuallysecondary,notcollectedexplicitlyforourpurposebutratherforsomeotherprimarypurp

4、ose.5UseofTheoryTheoryisagoodpotentialsourceofinformationaboutdirectionofcausalflow.However,theoryusuallyinvokestheceterisparibusconditiontoachieveresults.Dataareusuallyobservational(non-experimental)andthustheceterisparibusconditionmaynothold.Wemaynoteverknowifitholdsbecauseofunknownvariablesoperat

5、ingonoursystem(seeMalinvaudseconometrictext).6ExperimentalMethodsIfwedonotknowthetruesystem,buthaveanapproximateideathatoneormorevariablesoperateonthatsystem,thenexperimentalmethodscanyieldappropriateresults.Experimentalmethodsworkbecausetheyuserandomization,randomassignmentofsubjectstoalternativetr

6、eatments,toaccountforanyadditionalvariationassociatedwiththeunknownvariablesonthesystem.7DirectedGraphsCanBeUsedToRepresentCausationDirectedgraphshelpusassigncausalflowstoasetofobservationaldata.Theproblemunderstudyandtheorysuggestscertainvariablesoughttoberelated,evenifwedonotknowexactlyhow;i.e.wed

7、ontknowthetruesystem.8CausalModelsAreWellRepresentedByDirectedGraphsOnereasonforstudyingcausalmodels,representedhereasXY,istopredicttheconsequencesofchangingtheeffectvariable(Y)bychangingthecausevariable(X).ThepossibilityofmanipulatingYbywayofmanipulatingXisattheheartofcausation.Hausman(1998,page7)w

8、rites:“Causationseemsconnectedtointerventionandmanipulation:Onecanusecausestowiggletheireffects.”9WeNeedMoreThanAlgebraToRepresentCauseLinearalgebraissymmetricwithrespecttotheequalsign.Wecanre-writey=a+bxasx=-a/b+(1/b)y.Eitherformislegitimateforrepresentingtheinformationconveyedbytheequation.Aprefer

9、redrepresentationofcausationwouldbethesentencexy,orthewords:“ifyouchangexbyoneunityouwillchangeybybunits,ceterisparibus.”Thealgebraicstatementsuggestsasymmetrythatdoesnotholdforcausalstatements.10ArrowsCarrytheInformationAnarrowplacedwithitsbaseatXandheadatYindicatesXcausesY:XY.Bythewords“XcausesY”w

10、emeanthatonecanchangethevaluesofYbychangingthevaluesofX.ArrowsindicateaproductiveorgeneticrelationshipbetweenXandY.CausalStatementsareasymmetric:xyisnotconsistentwithyx.11ProblemswithPredictiveDefinitionsofCauseDefinitionoftheword“cause”thatfocusonpredictionalone,withoutdistinguishingbetweeninterven

11、tion(first)andsubsequentrealization,maymistakenlylabelascausalvariablesthatareassociatedonlythroughanomittedvariable.Predictionisoneattributeoftheword“cause.”Wemustbecarefulnottomakeittheonlyattribute(moreorlessasummaryofBunge1959).12Granger-typeCausalityForexample,Granger-typecausality(Granger1980)

12、focusessolelyonprediction,withoutconsideringintervention.IfwecanpredictYbetterbyusingpastvaluesofXthanbynotusingpastvaluesofX,thenXGranger-causesY.Theconsequencesofsuchfocusistoopenoneselfuptothefrustrationofunrealizedexpectationsbyattemptingpolicyonthewrongsetofvariables.13GraphAgraphisanorderedtri

13、ple.Visanon-emptysetofvertices(variables).Misanon-emptysetofmarks(symbolsattachedtotheendofundirectededges).Eisasetoforderedpairs.EachmemberofEiscalledanedge.14Verticesarevariables;EdgesarelinesVerticesconnectedbyanedgearesaidtobeadjacent.IfwehaveasetofverticesA,B,C,Dtheundirectedgraphcontainsonlyun

14、directededges(e.g.,AB).Adirectedgraphcontainsonlydirectededges:CD.15DirectedAcyclicGraphs(DAGs)Adirectedacyclicgraphisadirectedgraphthatcontainsnodirectedcyclicpaths.Anacyclicgraphhasnopaththatleadsawayfromavariableonlytoreturntothatsamevariable.ThepathABCAislabeled“cyclic”asherewemovefromAtoB,butth

15、enreturntoAbywayofC.16GraphsandProbabilitiesofVariablesDirectedacyclicgraphsarepictures(illustrations)forrepresentingconditionalindependenceasgivenbytherecursivedecomposition:nPr(v1,v2vn-1,vn)=Pr(vi|pai)i=1wherePristheprobabilityofvertices(variables)v1,v2,v3,.vnandpaitherealizationofsomesubsetofthev

16、ariablesthatprecede(comebeforeinacausalsense)viinorder(v1,v2,v3,.vn),andthesymbolrepresentstheproductoperation,withindexofoperationdenotedbelow(start)andabove(finish)thesymbol.Thinkofpaiastheparentofvariablei.17D-SeparationLetX,YandZbethreedisjointsubsetsofvariablesinadirectedacylicgraphG,andletpbea

17、nypathbetweenavertexvariableinXandavertexvariableinY,wherebypathwemeananysuccessionofedges,regardlessoftheirdirections.Zissaidtoblockpifthereisavertexwonpsatisfyingoneofthefollowing:(i)(i)whasconvergingarrowsalongp,andneitherwnoranyofitsdescendantsareonZor(ii)(ii)wdoesnothaveconvergingarrowsalongp,a

18、ndwisinZ.Furthermore,Zissaidtod-separateXfromYongraphG,written(XY|Z)G,ifandonlyifZblockseverypathfromavertexvariableinXtoavertexvariableinY.18GraphsandD-SeparationGeiger,VermaandPearl(1990)showthatthereisaone-to-onecorrespondencebetweenthesetofconditionalindependencies,XY|Z,impliedbytheabovefactoriz

19、ationandthesetoftriples,X,Y,Z,thatsatisfythed-separationcriterioningraphG.IfGisadirectedacyclicgraphwithvertexsetV,ifAandBareinVandifHisalsoinV,thenGlinearlyimpliesthecorrelationbetweenAandBconditionalonHiszeroifandonlyifAandBared-separatedgivenH.19Colliders(InvertedFork)Considerthreevariables(verti

20、ces):A,BandC.Avariableisacolliderifarrowsconvergeonit:ABC.ThevertexBisacollider,AandCared-separated,giventhenullset.Intuitively,thinkoftwotrainsonestartingatA,theotheratC.BothmovetowardB.Unconditionally,theywillcrashatB.However,ifweconditiononB,(ifwebuildaswitchstationatBwithsidetracks),weopen-upthe

21、flowfromAtoC.ConditioningonBmakesAandCd-connected(directionallyconnected).20ConditioningonChildren(ofcolliders)OpensUpInformationFlowsToo!AmendtheabovegraphgivenabovetoincludevariableD,asachildofB,suchthat:ABCDIfweconditiononDratherthanB,we,aswell,openuptheflowbetweenAandC(Pearl,2000p.17).Thisillust

22、ratesthe(i)componentofthedefinitiongivenabove.21CommonCauses(causalfork)SaywehavethreeverticesK,LandM,describedbythefollowinggraph:KLM.HereLisacommoncauseofKandM.Theunconditionalassociation(correlation)betweenKandMwillbenon-zero,astheyhaveacommoncauseL.However,ifweconditiononL(knowthevalueofL),theas

23、sociationbetweenKandMdisappears(Pearl,2000,p.17).Conditioningoncommoncausesblockstheflowofinformationbetweeneffects.22CausalchainsFinally,ifourcausalpathisoneofachain(causalchain),condition(ii)intheabovedefinitionagainapplies.IfDcausesEandEcausesF,wehavetherepresentationalflow:DEF.Theunconditionalas

24、sociation(correlation)betweenDandFwillbenon-zero,buttheassociation(correlation)betweenDandFconditionalonEwillbezero.(ForthoseintheaudiencefamiliarwithBoxandJenkinstimeseriesmethods,thisisapropertytheyexploitedintestingforARmodels)23ExampleofanInvertedCausalForkIntheexamplewestudybelowwetakedatafromP

25、eltzman(Jo.PoliticalEconomy1976).ThisisastudyofTrafficFatalitiesintheU.S.overtheperiod19471972.Roh,BesslerandGilbert(1997)findthefollowing(notasurprise):Speed(t)AlcoholConsumption(t)TrafficFatalities(t)24WhatShouldWeExpectBasedOnThePreviousDirectedGraph?Hereyeartoyearchangesinspeedandyeartoyearchang

26、esinalcoholconsumptionaredirectcausesofyeartoyearchangesintrafficfatalities.Thegraphisaninvertedfork.So,weshouldexpecttoseethatSpeedandAlcoholConsumptionarenotrelatedinunconditionaltestsofassociation.However,ifweconditiononTrafficFatalities,weshouldseeanon-zeromeasureofassociationbetweenSpeedandAlco

27、holConsumption.25OLSRegressionsOnAnInvertedFork(useolstomeasureassociation)Regression#1:Speed(t)=.01-.01*(AlcoholConsumption(t)(.002)(.053)Estimatedstandarderrorsofthecoefficientsarein().BasedonthisregressionwewouldsaySpeed(t)andAlcoholConsumption(t)arenotrelated(note:-.01/.0532.0).27ExampleofaCausa

28、lChainInanotherexample,considertherelationshipamongGDP,PovertyandMalnutrition.BasedonWorldBankdatafor80lessdevelopedcountries,wefind:GDPPovertyMalnutritionWeexpect,fromthedirectedgraphtheorygivenabove,MalnutritionandGDPwillberelatedinunconditionaltests.However,ifweconditiononpovertytheyshouldbeunrel

29、ated.Letssee!28RegressionswithCausalChainsRegression#1(fori=1,80countries)Malnutrition(i)=24.18-.003*GDP(i)(1.91)(.0006)Notethet-ratioof-.003/.0006=-5.38suggeststhatGDPisanimportantvariableinmovinglevelsofmalnutrition.29RegressionswithCausalChains,continued.Regression#2(fori=1,80countries)Malnutriti

30、on(i)=7.52-.0013*GDP(i)(2.09)(.0007)+.289*Poverty(i)(.055)Notethet-ratioof-.0013/.0007=-1.78suggests(ifweare5%ers)thatGDPisnotinformativewithrespecttomalnutritionifwehaveinformationaboutacountryspovertylevels.30MarkovPropertyKeytounderstandingtheseideasisthatd-separationallowsustowritetheprobability

31、ofourvariablesX,Y,andZintermsoftheproductoftheconditionalprobabilitiesoneachvariable(X,Y,orZ),wheretheconditioningfactoristheimmediateparentofeachvariable.Wedonothavetoconditionongrandparents,greatgrandparents,aunts,unclesorchildren.(Itishelpfulandvalidtorefertogenealogicalanalogieswhenthinkingabout

32、conditioninginformation.)31SomeprobabilitiesThefollowingdirectedgraphshavetheseassociatedprobabilityfactorizations:ABC;Pr(A,B,C)=Pr(A)Pr(C)Pr(B|C,A)DEF;Pr(D,E,F)=Pr(D)Pr(E|D)Pr(F|E)GHIJ;Pr(G,H,I,J)=Pr(G)Pr(J)Pr(H|G)Pr(I|J,H)PQ;Pr(P,Q)=Pr(P)Pr(Q)HerePr(.)referstotheprobabilityofthevariable(s)inparent

33、heses32AdjustmentProblem(fromPearl2000)WhatmustImeasureifIwanttoknowhowXaffectsY?Z1Z2Z3Z4Z5Z6Z7Z8Z9Z10XZ11YOriginalCausalGraphIllustratingthe“AdjustmentProblem”33D-SeparationisKeytoSolvingtheAdjustmentProblemAskthequestion:canIgetbacktoYviatheancestorsofXwithoutrunningintoconvergingarrows?Yes!Icanta

34、keseveralpathsfromXtoYthroughXsancestors:XZ3Z1Z4Z7YXZ6Z4Z7YXZ6Z4Z2Z5Z9YXZ6Z4Z2Z7YIhavetoconditiononvariablesto“block”thepathbacktoYfromX.Thereareseveralpossibilities:ItlookslikeZ7andZ9aretwo.Belowwegivesixstepsforsolvingthe“adjustmentproblem”.34Step1.Z7andZ9shouldbenon-descendantsofXZ1Z2Z3Z4Z5Z6Z7Z8

35、Z9Z10XZ11YZ11willnotworkasitisachildofX.35Step2.Deleteallnon-ancestorsofX,YandZ.Z1Z2Z3Z4Z5Z6Z7Z8Z9Z10XZ11YHereZisthesetofcandidate“blocking”variablesZ=Z7andZ9.36Step3.DeleteallarcsemanatingfromX.Z1Z2Z3Z4Z5Z6Z7Z8Z9Z10XZ11YHerewewillremovetheXZ11edge,asZ11isachildofX.37Step4.Connectanytwoparentssharin

36、gacommonchild.Z1Z2Z3Z4Z5Z6Z7Z8Z9Z10XZ11YHerewewillusedottedlinestoconnectparentswithacommonchild38Step5.Striparrow-headsfromalledgesZ1Z2Z3Z4Z5Z6Z7Z8Z9Z10XZ11Y39Step6.DeleteLinesintoandoutofZ7andZ9Z1Z2Z3Z4Z5WecannotgetZ6Z7fromXtoYZ8Z9Z10XZ11YHerewedeletealllinesintothevariablesthatwewishtoconditionon

37、,Z7andZ9.40TestTest:ifXisdisconnectedfromYintheremaininggraph,thenZ7andZ9aresufficientmeasurementstoconditionon.By“disconnected”wemeanthatwecannotgetfromXtoYviatheremaininglines.Z7andZ9passthetest.SowecanperformolsregressionofYonX,Z7andZ9tofindanunbiasedestimateoftheeffectofXonY.41Anothercandidate:L

38、etsTryZ4allbyItself.IfwetryjustZ4asasolecandidatevariabletoconditionon,ourlastfigurewillbeamendedasfollows:Z1Z2Z3Z4Z5Z6Z7Z8Z9ClearlyZ4Z10willnotworkXZ11Y42Why does Z4 fail our test in the previous slide?BecauseZ4opensupthepathbetweenZ1andZ2.Rememberourspeed,alcoholandtrafficfatalitiesexample,slides2

39、427.IfwerunanolsregressionofYonXandZ4wewouldfindbiasedestimatesofthecoefficientassociatedwithX.XwillbecorrelatedwitherrorsinY.Wesaythereisa“backdoorpath”betweenXandYthatwillgiveusbiasedparameterestimatesoftheeffectofXonY.43PolicyandDirectedGraphsConsiderthefollowingsimplegraph:XYUWeobserveXinanuncon

40、trolledsettingandweareinterestedinmanipulatingYbysettingthevalueofX.Theinferencetaskwehaveaseconomistsistomovefromasampleobtainedfromadistributionassociatedwithpassiveobservationtoconclusionsaboutthedistributionthatwouldobtainifaparticularpolicyisimposed.Policyisthusaskingquestionsaboutcounterfactua

41、ls.WhatvalueswillYtakeonifweforceXtotakeonavalueofXf=1?(hereweusethenotationthatXisforcedtohaveavalueof1asXf=1).44SimpleExampleofPolicywithExogenousVariableXConsiderthetablewhichisbasedonthegraphgivenabove,whereY=X+UPassiveObservations|ForcedorPolicyInducedXUYXfUXf=1YXf=11011011121121231232021012131

42、12224123NoticehowYbehaveswhenweforceavalueonX(X=1).45Whatwerewesupposetoseeintheprevioustable?Intheabovetable,Y,underpassiveobservationwhenX=1,hasthesamedistributionasYXf=1.Lookbackatthetable.HereXisaparentofYandthereisnobackdoorpathfromXtoY(sayviaU).ApolicyanalystmayconcludethatknowinghowXandYarere

43、latedinthisuncontrolled(passive)settingissufficientforpredictinghowtheywillbehaveinpolicysettings.46AcasewherewemustbecarefulVXYUHerewehaveavariableVthatcausesbothXandU.WillknowledgeofhowXandYbehaveinpassivesettingsbesufficientforpredictinghowtheywillbehaveinapolicysetting?47Considerthefollowing:Let

44、:Y=X+U;X=VandU=VPassiveObservations|ForcedorPolicyInducedVXUYVXf=1UXf=1YXf=1111211121112111211121112222421232224212322242123NoticeherethatwhenX=1intheunforcedsettingourdistributiononYis2,2,2,4,4,4.However,whenweforceX=1ourdistributiononYissometimes2,andsometimes3,butnever4.UnderthepolicysettingonXwe

45、cannotignoreV.Wehavetohaveitinourmodel,elsewewillhavepolicyresultswhicharenotwellpredictedthroughknowledgeofXandY(passivelyobserved).48ResultsonTrafficFatalitiesWefindthefollowingrelationshipamongtrafficfatalities,alcoholconsumptionandincome.alcoholconsumption(t)income(t)trafficfatalities(t)Thisgrap

46、halongwithourworkaboveonpolicyanalysissuggeststhatitisnotenoughtounderstandthealcoholtrafficfatalitieslink,butwemustaswellunderstandhowincomelevelscontributetotheproblem.49ConsidertheFollowingTwoRegressionsRegression#1:(IgnoretheIncomeconnection)TF(t)=.015+.608AC(t):R2=.38(.009)(.200)Regression#2:(I

47、ncludeIncomewithAlcoholConsumption)TF(t)=-.005+.338AC(t)+1.055IN(t);R2=.68(.008)(.160)(.276)50PolicyImplicationsFromthelastslideweconcludethatifwedirectnationalpolicytowardsreducingalcoholconsumptionwewillseeadecreaseintrafficfatalities.However,thatdecreaseislikelytofollowregression#2ratherthanregre

48、ssion#1.Infactbelowwewillseethatspeedisanevenmoreprominentmover(cause)oftrafficfatalities.(Wedidntconsiderspeedintheregressiongivenabovebecauseaswewillseespeedisexogenousrelativetoalcoholconsumptionandincomelevels).51PCAlgorithmHerewillpresentonealgorithmwhichcanbeusedtobuilddirectedgraphs.Thealgori

49、thmstartssystematicallyfromacompleteundirectedgraphandremovesedges(lines)betweenverticesbasedoncorrelationorpartialcorrelationbetweenvertices.Spirtes,Glymour and Scheines(1993)haveincorporated the notion of d-separation intoan algorithm(PC Algorithm)for buildingdirected acyclic graphs,using the noti

50、on ofsepset(definedbelow).52ACompleteUndirectedGraph(getsusstarted)OneformsacompleteundirectedgraphGonthe vertex set V.The complete undirectedgraph shows an undirected edge betweeneveryvariableofthesystem(everyvariableinV).Edgesbetweenvariablesareremovedsequentially based on zero correlation orparti

51、alcorrelation(conditionalcorrelation).ZXYHereX,Y,andZareconnectedwithlineshavingnoarrows53RemoveEdgesUsingCorrelationorConditionalCorrelationEachedgeissubjectedtoteststhatthecorrelationbetweenitsendpointsiszero:Ho:(X,Y)=0?Ifacorrelationisjudgedtobenotdifferentfromzero,weremovetheedgebetweenthetwoend

52、pointsofthecorrespondingedge.Edgessurvivingtheseunconditionalcorrelationtestsarethensubjectedtoconditionalcorrelationtests:Ho:(X,Y|Z)=0?IftheseconditionalcorrelationsequalzeropickuptheedgeX,Y.54FishersZFisherszstatisticcanbeappliedtotestforsignificancefromzero:z(i,j|k)n)=1/2(n-|k|-3)1/2xln(|1+(i,j|k

53、)|)x(|1-(i,j|k)|)-1.nisthenumberofobservationsusedtoestimatethecorrelations,(i,j|k)isthepopulationcorrelationbetweenseriesiandjconditionalonseriesk(removingtheinfluenceofserieskoneachiandj),and|k|isthenumberofvariablesink(thatweconditionon).55SepsetTheconditioningvariable(s)onremovededgesbetween two

54、 variables is called the sepset of thevariables whose edge has been removed(forvanishing zero order conditioning information thesepsetistheemptyset).xyzIfweremovetheedgebetweenxandythroughunconditionalcorrelationtest,(x,y)=0,thenthesepset(x,y)is.Ifweremovethisedgebyconditioningonz,(x,y|z)=0thenthese

55、pset(x,y)isz.56EdgeDirectionEdgesaredirectedbyconsideringtriples,suchthatXandYareadjacentasareYandZ,butXandZarenotadjacent:XYZ.Direct the edges between triples:if Y is not in thesepsetofXandZ.If,YandZareadjacent,XandZare not adjacent,and there is no arrowhead at Y,thenorientasXYZIf there is a direct

56、ed path from X to Y,and an edgebetweenYandZ,thendirect(YZ)as:YZ.57AssumptionsforPCAlgorithmtoWorkCausalSufficiency:Therearenoomittedvariablesthatcausetwoofmyincludedvariables.MarkovCondition:Wecanwriteprobabilitiesofvariablesbyconditioningjustoneachvariablesparents.(wediscussedthisabove).Faithfulnes

57、sCondition:Ifweseezerocorrelationbetweentwovariables,thereasonweseeitisbecausethereisnoedgebetweenthesevariablesandnotcancellationofstructuralparameters.58CausalSufficiencyWewanttomeasuretheeffectofyonz(writethisasy)andwehavex,yandzinourstudy,butweleaveanothervariable,w,outofthestudy.Theworldisgener

58、atedbythegraph:wxwy0yz59CausalSufficiencyContinuedIfwefailtoincludewinoursamplewewillendupwiththefollowinggraph(letbyandbxrepresentourmeasuredeffectsbasedonx,yandz):wxyE(bx)0E(by)yzThekeytocausalsufficiencyisthatwedonthavetohaveeveryvariablethatcauseszinourstudy.Butwedoneedallvariablesthatcausetwoor

59、morevariablesinourstudy.(HereE(by)istheexpectedvalueoformeasureoftheeffectofyonz).60FaithfulnessHereweassumethatifwemeasurethecorrelationbetweentwovariables,sayxandy,aszero,itiszerobecausethereisnoedgebetweenxandyinthe“true”model.Itisnotzerobecauseofcancellationofdeeperparameters.61FaithfulnessConti

60、nuedSaywehavethefollowingtruemodel:xyxyxzyzzThe”true”parametersconnectingthesevariablesaregivenbythebetas(xyxzyz).62FaithfulnessContinuedIfitsohappensthatintherealworld:xz=-xyyzthenthecorrelationbetweenxandzwillequalzero.PCalgorithmwillremovetheedgebetweenxandz,eventhoughthetruemodelhassuchanedge.63

61、Example:TrafficFatalitiesVariablesandDatatakenfromPeltzmanJournalofPoliticalScience1977EightvariablesfortheU.S.1947-1974data:numberoftrafficfatalities;numberofyoungdrivers(15yy1y3-y1y3-y4y1ismoneysupplyy2isincomey3ispricelevely4iswheatprice93CorrelationandP-valuesonremovededgesypw-thisedgeisremoveda

62、sthecorr(y,pw)=.07;p=.68yp-thisedgeisremovedasthecorr(y,p)=-.05;p=.75m2pw-thisedgeisremovedasthecorr(m2,pw|p)=.12;p=.4494FinalDirectedAcyclicGraphpm2ypwMoney(M2)isendogenousincontemporaneoustime;aresultthatdoesnotagreewithBordoandSchwartz.Whatresultfromslide94allowedustodirecttheedgesonp,m2andyasani

63、nvertedfork?95StockMarketIntegrationArestockmarketmovementsfromvariouscountriesindependent?Iftheansweris“no”,whereisinformationcreated?Lookatindexesfromninecountries:Australia,Japan,KongKong,Germany,Switzerland,France,UnitedKingdom,UnitedStates,Canada.Firstweremovedanylaggedrelationshipbetweenthenin

64、eseries.Wenextstudytheirdailyco-movements.96StockMarketIntegration(standardclockdaybeginsat180degreeswestofGreenwichEngland)AustraliaJapanHongKongGermanySwitzerlandFranceUnitedKingdomUnitedStatesCanada97StockMarketIntegration(thedaybeginsat,60owestofGreenwich,England)AustraliaJapanHongKongGermanySwi

65、tzerlandFranceUnitedKingdomUnitedStatesCanada98ImplicationsofstockmarketresultsCanadaisaninformationsink.JapanisaninformationsinkinAsia.HongKongistheprimesourceofinformationmovementfromAsiatoEurope.YesterdayscloseintheU.S.movesAsiatodaythroughHongKongandAustralia.99TheLiteratureonSuchCausalStructureshasbeenAdvancedintheLastDecadeUndertheLabelofArtificialIntelligencePearl,Biometrika,1995Pearl,Causality,CambridgePress,2000Spirtes,GlymourandScheines,Causation,PredictionandSearch,Springer-Verlag,1993GlymourandCooper,Computation,CausationandDiscovery,MITPress,1999.100