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What’sNewinEconometrics?Lecture10JeffWooldridgeNBERSummerInstitute,20071.ReviewoftheBasicMethodology2.HowShouldWeViewUncertaintyinDD3.GeneralSettingsforDDAnalysis:MultipleGroupsandTimePeriods4.Individual-LevelPanelData5.SemiparametricandNonparametricApproaches6.SyntheticControlMethodsforComparativeCaseStudies ReviewoftheBasicMethodologyThestandardcase:outcomesareobservedfortwogroupsfortwotimeperiods.Oneofthegroupsisexposedtoatreatmentinthesecondperiodbutnotinthefirstperiod.Thesecondgroupisnotexposedtothetreatmentduringeitherperiod.Inthecasewherethesameunitswithinagroupareobservedineachtimeperiod(paneldata),theaveragegaininthesecond(control)groupissubstractedfromtheaveragegaininthefirst(treatment)group.Thisremovesbiasesinsecondperiodcomparisonsbetweenthetreatmentandcontrolgroupthatcouldbetheresultfrompermanentdifferencesbetweenthosegroups,aswellasbiasesfromcomparisonsovertimeinthe treatmentgroupthatcouldbetheresultoftrends.Withrepeatedcrosssections,letbethecontrolgroupandthetreatmentgroup.Write, istheoutcomeofinterest.Thedummycapturespossibledifferencesbetweenthetreatmentandcontrolgroupspriortothepolicychange.The2capturesaggregatefactorsthatwouldcausechangesinevenintheabsenseofapolicychange.ThecoefficientofinterestisThedifference-in-differencesestimateis. Inferencebasedonevenmoderatesamplesizesineachofthefourgroupsisstraightforward,andiseasilymaderobusttodifferentgroup/timeperiod variancesintheregressionframework.Moreconvincinganalysissometimesavailablebyrefiningthedefinitionoftreatmentandcontrolgroups.Example:changeinstatehealthcarepolicyaimedatelderly.Couldusedataonlyonpeopleinthestatewiththepolicychange,bothbeforeandafterthechange,withthecontrolgroupbeingpeople55to65(say)andandthetreatmentgroupbeingpeopleover65.ThisDDanalysisassumesthatthepathsofhealthoutcomesfortheyoungerandoldergroupswouldnotbesystematicallydifferentintheabsenseofintervention.Instead,mightusetheover-65populationfromanotherstateasanadditionalcontrol.Letbeadummyequaltooneforsomeoneover65. Thecoefficientofinterestis,thecoefficientonthetripleinteractionterm,.TheOLScanbeexpressedasfollows:wherethesubscriptmeansthestatenotimplementingthepolicyandtherepresentsthenon-elderly.Thisisthedifference-in-difference-in-differences(DDD)CanaddcovariatestoeithertheDDorDDDanalysisto(hopefully)controlforcompositionalCanusemultipletimeperiodsandgroups. HowShouldWeViewUncertaintyinDDStandardapproach:alluncertaintyininferenceentersthroughsamplingerrorinestimatingthemeansofeachgroup/timeperiodcombination.Longhistoryinanalysisofvariance.Recently,differentapproacheshavebeensuggestthatfocusondifferentkindsofuncertainty–perhapsinadditiontosamplingerrorinestimatingmeans.Bertrand,Duflo,andMullainathan(2004),DonaldandLang(2007),Hansen(2007a,b),andAbadie,Diamond,andHainmueller(2007)argueforadditionalsourcesofuncertainty.Infact,forthemostpart,theadditionaluncertaintyisassumedtoswampthesamplingerrorinestimatinggroup/timeperiodmeans.(SeeDL approachinclustersamplenotes,althoughwedidnotexplicitlyintroduceatimedimension.)OnewaytoviewtheuncertaintyintroducedintheDLframework–andaperspectiveexplicitlytakenbyADH–isthatouranalysisshouldbetterreflecttheuncertaintyinthequalityofthecontrolIssue:InthestandardDDandDDDcases,thepolicyeffectisjustidentifiedinthesensethatwedonothavemultipletreatmentorcontrolgroupsassumedtohavethesamemeanresponses.So,theDLapproachdoesnotallowinference.ExamplefromMeyer,Viscusi,andDurbin(1995)onestimatingtheeffectsofbenefitgenerosityonlengthoftimeaworkerspendsonworkers’compensation.MVDhavethestandard DDsetting:abeforeandafterperiod,wherethepolicychangewastoraisethecaponcoveredearnings;controlgroupislowearners.UsingKentuckyandatotalsamplesizeof5,626,theDDestimateofthepolicychangeisabout19.2%(longertimeonworkers’compensation)with2.76.UsingMichigan,withatotalsamplesizeof1,524,theDDestimateis19.1%with1.22.(Addingcontrolsdoesnothelpreducethestandarderror.)Thereseemstobeplentyofuncertaintyintheestimateevenwithaprettylargesamplesize.Shouldweconcludethatwereallyhavenousabledataforinference?GeneralSettingsforDDAnalysisGroupsandTimePeriodsTheDDandDDDmethodologiescanbeapplied tomorethantwotimeperiods.IntheDDcase,addafullsetoftimedummiestotheequation.Thisassumesthepolicyhasthesameeffectineveryyear;easilyrelaxed.InaDDDanalysis,afullsetofdummiesisincludedforeachofthetwokindsofgroupsandalltimeperiods,aswellasallpairwiseinteractions.Then,apolicydummy(orsometimesacontinuouspolicyvariable)measurestheeffectofthepolicy.SeeMeyer(1995)forapplications.Withmanytimeperiodsandgroups,ageneralframeworkconsideredbyBDM(2004)andHansen(2007b)isuseful.Theequationattheindividuallevelis1,...,indexesindividual,indexesgroup,and indexestime.Thismodelhasafullsetoftime,afullsetofgroupeffects,group/timeperiodcovariates,(thesearethepolicyvariables),individual-specificcovariates,,unobservedgroup/timeeffects,,andindividual-specificerrors,.WeareinterestedinAsinclustersamplecases,canwrite1,...,, (6)whichshowsamodelattheindividuallevelwhereboththeinterceptsandslopesareallowedtodifferacrossallpairs.Then,wethinkof. Wecanthinkof(7)asaregressionmodelatthegroup/timeperiodlevel. AsdiscussedbyBDM,acommonwaytoestimateandperforminferencein(5)istoignore,sotheindividual-levelobservationsaretreatedasindependent.Whenispresent,theresultinginferencecanbeverymisleading.BDMandHansen(2007b)allowserialcorrelationin1,2,...,butassumeindependenceacrossIfweview(7)asultimatelyofinterest,therearesimplewaystoproceed.Weobservehandledwithyeardummies,andjustrepresentsgroupdummies.Theproblem,then,isthatwedonotobserve.UseOLSontheindividual-leveldatatoestimatethe,assumingandthegroup/timeperiodsizes,,arereasonablylarge. Sometimesonewishestoimposesomehomogeneityintheslopes–say,oreven–inwhichcasepoolingcanbeusedtoimposesuchrestrictions.Inanycase,proceedasifarelargeenoughtoignoretheestimationerrorinthe;instead,theuncertaintycomesthroughin(7).TheMDapproachfromclustersamplenoteseffectivelyfrom(7)andviewsasasetofdeterministicrestrictionstobeimposed.InferenceusingtheefficientMDestimatorusesonlysamplingvariationinthe.Here,weproceedignoringestimationerror,andsoactasif(7)is,for1,...,1,..., WecanapplytheBDMfindingsandHansen(2007a)resultsdirectlytothisequation.Namely,ifweestimate(8)byOLS–whichmeansfullyearandgroupeffects,alongwith–thentheOLSestimatorhassatisfyingpropertiesasincrease,provided1,2,...,isaweaklydependenttimeseriesforall.ThesimulationsinBDMandHansen(2007a)indicatethatcluster-robustinference,whereeachclusterisasetoftimeperiods,workreasonablywellwhenfollowsastableAR(1)modelandmoderatelylarge.Hansen(2007b),notingthattheOLSestimator(thefixedeffectsestimator)appliedto(8)isinefficientwhenisseriallyuncorrelated,proposesfeasibleGLS.Whenissmall,estimating theparametersin,whereisthe1errorvectorforeach,isdifficultwhengroupeffectshavebeenremoved.EstimatesbasedontheFEresiduals,,disappearas,butcanbesubstantial.InAR(1)case,comesfrom2,...,1,...,. Onewaytoaccountforbiasin:usefullyrobustinference.But,asHansen(2007b)shows,thiscanbeveryinefficientrelativetohissuggestiontobias-adjusttheestimatorandthenusethebias-adjustedestimatorinfeasibleGLS.(HansencoversthegeneralHansenshowsthataniterativebias-adjustedprocedurehasthesameasymptoticdistributionasinthecaseshouldworkwell:tendingtoinfinity.Mostimportantlyforthe applicationtoDDproblems,thefeasibleGLSestimatorbasedontheiterativeprocedurehasthesameasymptoticdistributionastheinfeasibleGLSetsimatorwhenisfixed.Evenwhenarebothlarge,sothattheunadjustedARcoefficientsalsodeliverasymptoticefficiency,thebias-adustedestimatesdeliverhigher-orderimprovementsintheasymptoticOnelimitationofHansen’sresults:theyassume1,...,arestrictlyexogenous.IfwejustuseOLS,thatis,theusualfixedeffectsestimate–strictexogeneityisnotrequiredforconsistencyas.NothingnewthatGLSreliesonstrictexogeneityinserialcorrelationcases.Ininterventionanalyis,mightbeconcernedifthe policiescanswitchonandoffovertime.Withlargeandsmall,onecanestimateanunstrictedvariancematrixandproceedwithGLS–thisistheapproachsuggestedbyKiefer(1980)andstudiedmorerecentlybyHausmanandKuersteiner(2003).Worksprettywellwith10,butgetsubstantialsizedistortionsfor50andIfthearenotlarge,mightworryaboutignoringtheestimationerrorinthe.Caninsteadaggregatetheequationsoverindividuals,giving1,..,1,...,CanestimatethisbyFEandusefullyrobustinferencebecausethecompositeerror,,isweaklydependent. TheDonaldandLang(2007)approachappliesinthecurrentsettingbyusingfinitesampleanalysisappliedtothepooledregression(10).However,DLassumethattheerrorsareuncorrelatedacrosstime,andso,eventhoughforsmallitusessmalldegrees-of-freedominadistribution,itdoesnotaccountforuncertaintyduetoserialcorrelationLevelPanelDatabeabinaryindicator,whichisunityifparticipatesintheprogramattime.Consider1,2, 1if2andzerootherwise,isanobservedeffect,andaretheidiosyncraticerrors.Thecoefficientisthetreatmenteffect.Asimpleestimationprocedureistofirstdifferencetoremove . 0,thatis,thechangeintreatmentstatusisuncorrelatedwithchangesintheidiosyncraticerrors,thenOLSappliedto(13)is0forall,theOLSestimateis, whichisaDDestimateexceptthatwedifferentthemeansofthesameunitsovertime.Withmanytimeperiodsandarbitrarytreatmentpatterns,wecanuse1,...,, whichaccountsforaggregatetimeeffectsandallowsforcontrols,.EstimationbyFEorFDtoisstandard,providedthepolicy,isstrictlyexogenous:correlationforanyinconsistencyinbothestimators(withFEhavingsomeadvantagesforlargerisweaklyWhatifdesignationiscorrelatedwithunit-specifictrends?“Correlatedrandomtrend”isthetrendforunit.Ageneralanalysisallowsarbitrarycorrrelationbetween,whichrequiresatleast3.Ifwefirstdifference,weget,for2,..., . Candifferenceagainorestimate(17)byFE.Canderivestandardpaneldataapproachesusingthecounterfacturalframeworkfromthetreatmenteffectsliterature.Foreach,letdenotethecounterfactualoutcomes,andassumetherearenocovariates.Unconfoundedness,conditionalonunobservedheterogeneity,canbestatedasisthetimesequenceofalltreatments.Ifthegainfromtreatmentonlydependson andthen. Ifweassume, , anestimatingequationthatleadstoFEorFD(oftenIfaddstrictlyexogenouscovariates,andassumelinearityofconditionalexpectations,andallowthegainfromtreatmenttodependonandanadditiveunobservedeffect,getacorrelatedrandomcoefficientmodelbecausethe coefficienton.Caneliminate.Or,with,can“estimate”theandthen. SeeWooldridge(2002,Section11.2)forstandarderror,orbootstrapping.SemiparametricandNonparametricReturntothesettingwithtwogroupsandtwotimeperiods.AtheyandImbens(2006)generalizethestandardDDmodelinseveralways.Letthetwotimeperiodsbe0and1andlabelthetwo0and1.Letbethecounterfactualoutcomeintheabsenseofinterventionandthecounterfactualoutcomewithintervention.AI assumethat, isthetimeperiodandstrictlyincreasingin0,1 Therandomvariablerepresentsallunobservablecharacteristicsofindividual.Equation(26)incorporatestheideathattheoutcomeofanindividualwithwillbethesameinagiventimeperiod,irrespectiveofgroupmembership.Thedistributionofisallowedtovaryacrossgroups,butnotovertimewithingroups,sothat. ThestandardDDmodelcanbeexpressedinthisway,with although,becauseofthelinearity,wecangetbywiththemeanindependenceassumption0.Withconstanttreatmenteffect,, 0,theparametersin(31)canbeestimatedbyOLS(usualDDanalysis).AtheyandImbenscalltheextensionoftheusualDDmodelthe(CIC)model.Canrecover, undertheirassumptions(withanextrasupportcondition).Infact,ifthebecumulativedistributionfunctionof 1,2and1,2,andisthecdffortheobservedoutcomeconditionalon,then, istheinversefunctionofwhichexistsunderthestrictmonotonicityassumption.Because,wecanestimatetheentiredistributionsofbothcounterfactualsconditionalonintervention,Canapplytorepeatedcrosssectionsorpaneldata.Ofcourse,canalsoidentifytheaveragetreatmenteffect. Inparticular, . Otherapproacheswithpaneldata:AltonjiandMatzkin(2005)underexchaneabilityinHeckman,Ichimura,Smith,andTodd(1997)andAbadie(2005).Considerbasicsetupwithtwotimeperiods,notreatedunitsinfirsttimeperiod.Withoutanisthecounterfactualoutcomefortreatmentlevel0,1,attimeParameter:theaveragetreatmenteffectonthe. Remember,inthecurrentsetup,nounitsaretreatedintheinitialtimeperiod,so1meanstreatmentinthesecondtimeperiod. Keyunconfoundednessassumption:Alsoneed1 iscritical.Under(37)and(38), WpXY1Y01pX , 0,1aretheobservedoutcomes(forthesameunit)andisthepropensityscore.DehejiaandWahba(1999)derived(39)forthecross-sectionalcase.Allquantitiesareobservedor,inthecaseofthe,canbeestimated.AsinHirano,Imbens,andRidder(2003),aflexiblelogitmodelcanbeusedfor;thefractionofunitstreated wouldbeusedfor.Then WipXiYi1pXi . isconsistentand -asymptoticallynormal.HIRdiscussvarianceestimation.ImbensandWooldridge(2007)provideasimpleadjustmentavailableinthecasethatistreatedasaparametricmodel.SimilarapproachworksforRegressionversion:on1,1,...,ThecoefficientonistheestimatedATE.Requiressomefunctionalformrestrictions.Certainlypreferredtorunningtheregression.Thislatterregression requiresunconfoundednessinthelevels,andasdominatedbythebasicDDestimatefromRegressionadjustmentcanalsobeused,asinHIST(1997).SyntheticControlMethodsforComparativeCaseStudiesAbadie,Diamond,andHainmueller(2007)arguethatinpolicyanalysisattheaggregateleve,thereislittleornoestimationuncertainty:thegoalistodeterminetheeffectofapolicyonanentirepopulation,andtheaggregateismeasuredwithouterror(orverylittleerror).Application:California’stobaccocontrolprogramonstate-widesmokingADHfocusontheuncertaintywithchoosinga suitablecontrolforCaliforniaamongotherstates(thatdidnotimplementcomparablepoliciesoverthesameperiod).ADHsuggestusingmanypotentialcontrolgroups(38statesorso)tocreateasinglesyntheticcontrolgroup.Twotimeperiods:onebeforethepolicyandoneafter.Letbetheoutcomeforunitintime,with1thetreatedunit.Supposetherearecontrols,andindextheseas2,...,.Letobservedcovariatesforunitthatarenot(orwouldnotbe)affectedbythepolicy;maycontain2covariatesprovidedtheyarenotaffectedbythepolicy.Generally,wecanestimatetheeffectofthepolicyas , arenonnegativeweightsthatadduptoone.Howtochoosetheweightstobestestimatetheinterventioneffect?ADHproposechoosingtheweightssoastominimizethedistancebetween,say.Thatis,functionsofthepre-treatmentoutcomesandthepredictorsofpost-treatmentoutcomes.ADHproposepermutationmethodsforinference,whichrequireestimatingaplacebotreatmenteffectforeachregion,usingthesamesyntheticcontrolmethodasfortheregionthatunderwentthe