双向固定效应和双重差分

Two-wayfixed-effectmodels

Differenceindifference1.Two-wayfixedeffectsBalancedpanelsi=1,2,3….Ngroupst=1,2,3….Tobservations/groupEasiesttothinkofdataasvaryingacrossstates/timeWritemodelassingleobservationYit=α+Xitβ+ui+vt+εitXitis(1xk)vector2.Three-parterrorstructureui–groupfixed-effects.Controlforpermanentdifferencesbetweengroupsvt–timefixedeffects.Impactscommontoallgroupsbutvarybyyearεit--idiosyncraticerror3.ExcisestaxesonpoorhealthAlcoholandcigarettesaretaxedatthefederal,stateandlocallevelSomestatessellliquorratherthantaxit(VA,PA,etc.)Mostofthesetaxesareexcisetaxes--thetaxisperunitRatesdifferbytypeofalcohol,alcoholcontentNearlyallcigarettestaxedthesame4.CurrentexcisetaxratesCigarettesLow:SC($0.07),MO($0.17),VA($0.30)High:RI($3.46),NY($2.75);NJ($2.70)Averageof$1.32acrossstatesAverageintobaccoproducingstates:$0.40Averageinnon-tobaccostates,$1.44Averagepriceperpackis$5.12BeerLow(WY,$0.02/gallon)High(SC,$0.77/gallon)5.6.FederaltaxesCigarettes,$1.01/packWine$0.21/750mlbottlefor14%alcoholorless$0.31/750mlbottlefor14–21%alcoholBeer,$0.02acanLiquor,$13.50per100proofgallon(50%alcohol),or,$2.14/750mlbottleof80proofliquorTotaltaxesoncigarettesaresuchthatinNYC,youspendmoreintaxesbuyingonecaseofcigarettesthanifyoubuy33casesofwine.7.Dotaxesreduceconsumption?LawofdemandFundamentalresultofmicroeconomictheoryConsumptionshouldfallaspricesriseGeneratedfromatheoreticalmodelofconsumerchoiceThoughtbyeconomiststobefairlyuniversalinapplicationMedical/psychologicalview–certaingoodsnotsubjecttotheselaws8.Startingin1970s,severalauthorsbegantoexaminelinkbetweencigarettepricesandconsumptionSimpleresearchdesignPricestypicallychangedduetostate/federaltaxhikesStateswithchangesare‘treatment’Stateswithoutchangesarecontrol9.Nearuniversalagreementinresults10%increaseinpricereducesdemandby4%ChangeinsmokingevenlysplitbetweenReductionsinnumberofsmokersReductionsincigs/dayamongremainingsmokersResultshavebeenreplicatedinothercountries/timeperiods,varietyofstatisticalmodels,subgroupsForotheraddictivegoods:alcohol,cocaine,marijuana,heroin,gambling10.TaxesnowanintegralpartofantismokingcampaignsKeycomponentof‘MasterSettlement’SurgeonGeneral’sreport“raisingtobaccoexcisetaxesiswidelyregardedasoneofthemosteffectivetobaccopreventionandcontrolstrategies.〞Taxhikesarenowdesignedtoreducesmoking11.12.13.14.15.Currentexcisetaxrates:///publications/show/245.htmlStatetaxes:Low:KY($0.30/pack),VA($0.30),SC($0.07)High:RI($2.46),NJ($2.58)Averageof$1.07acrossstatesFederaltaxes:39cents/pack16.CautionInbalancedpanel,two-wayfixed-effectsequivalenttosubtractingWithingroupmeansWithintimemeansAddingsamplemeanOnlytrueinbalancedpanelsIfunbalanced,needtodothefollowing17.Cansubtractoffmeansononedimension(iort)Butneedtoaddthedummiesfortheotherdimension18.*generaterealtaxesgens_f_rtax=(state_tax+federal_tax)/cpilabelvars_f_rtax"state+federalrealtaxoncigs,cents/pack"

*realpercapitaincomegenln_pcir=ln(pci/cpi)labelvarln_pcir"lnofrealrealpercapitaincome"

*generatelnpacks_pcgenln_packs_pc=ln(packs_pc)

*constructstateandyeareffectsxii.statei.year19.*runtwowayfixedeffectmodelbybruteforce*covariatesarerealtaxandlnpercapitaincomeregln_packs_pc_I*ln_pcirs_f_rtax

*nowbemoreeleganttakeoutthestateeffectsbyaregaregln_packs_pc_Iyear*ln_pcirs_f_rtax,absorb(state)

*forsimplicity,redefinevariablesasyx1(ln_pcir)*x2(s-f_rtax)

geny=ln_packs_pcgenx1=ln_pcirgenx2=s_f_rtax20.*sortdatabystate,thengetmeansofwithinstatevariablessortstatebystate:egeny_state=mean(y)bystate:egenx1_state=mean(x1)bystate:egenx2_state=mean(x2)

*sortdatabystate,thengetmeansofwithinstatevariablessortyearbyyear:egeny_year=mean(y)byyear:egenx1_year=mean(x1)byyear:egenx2_year=mean(x2)21.*getsamplemeansegeny_sample=mean(y)egenx1_sample=mean(x1)egenx2_sample=mean(x2)

*generatethedevaitionsfrommeansgeny_tilda=y-y_state-y_year+y_samplegenx1_tilda=x1-x1_state-x1_year+x1_samplegenx2_tilda=x2-x2_state-x2_year+x2_sample

*themeansshouldbemachingzerosumy_tildax1_tildax2_tilda22.*runtheregressionondifferencedvalues*sincemeansarezero,youshouldhavenoconstant*noticethatthestandarderrorsareincorrect*becausethemodelisnotcountingthe51statedummies*and19yeardummies.TherecordedDOFare*1020-2=1018butitshouldbe1020-2-51-19=948*multiplythestandarderrorsbysqrt(1018/948)=1.036262regy_tildax1_tildax2_tilda,noconstant23..*runtwowayfixedeffectmodelbybruteforce.*covariatesarerealtaxandlnpercapitaincome.regln_packs_pc_I*ln_pcirs_f_rtax

Source|SSdfMSNumberofobs=1020-------------+------------------------------F(71,948)=226.24Model|73.7119499711.03819648Prob>F=0.0000Residual|4.35024662948.004588868R-squared=0.9443-------------+------------------------------AdjR-squared=0.9401Total|78.06219651019.07660667RootMSE=.06774

------------------------------------------------------------------------------ln_packs_pc|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------_Istate_2|.0926469.03211222.890.004.0296277.155666_Istate_3|.245017.03424147.160.000.1778192.3122147

Deleteresults

_Iyear_1998|-.3249588.0226916-14.320.000-.3694904-.2804272_Iyear_1999|-.3664177.0232861-15.740.000-.412116-.3207194_Iyear_2000|-.373204.0255011-14.630.000-.4232492-.3231589ln_pcir|.2818674.05857994.810.000.1669061.3968287s_f_rtax|-.0062409.0002227-28.030.000-.0066779-.0058039_cons|2.294338.59667983.850.0001.1233723.465304------------------------------------------------------------------------------24.Source|SSdfMSNumberofobs=1020-------------+------------------------------F(2,1018)=466.93Model|3.9907057521.99535287Prob>F=0.0000Residual|4.350246621018.004273327R-squared=0.4784-------------+------------------------------AdjR-squared=0.4774Total|8.340952371020.008177404RootMSE=.06537

------------------------------------------------------------------------------y_tilda|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------x1_tilda|.2818674.056534.990.000.1709387.3927961x2_tilda|-.0062409.0002149-29.040.000-.0066626-.0058193------------------------------------------------------------------------------

SEonX10.05653*1.036262=0.05858SEonX20.0002149*1.036262=0.000222725.DifferenceindifferencemodelsMaybethemostpopularidentificationstrategyinappliedworktodayAttemptstomimicrandomassignmentwithtreatmentand“comparison〞sampleApplicationoftwo-wayfixedeffectsmodel26.ProblemsetupCross-sectionalandtimeseriesdataOnegroupis‘treated’withinterventionHavepre-postdataforgroupreceivinginterventionCanexaminetime-serieschangesbut,unsurehowmuchofthechangeisduetosecularchanges27.timeYt1t2YaYbYt1Yt2Trueeffect=Yt2-Yt1Estimatedeffect=Yb-Yati28.Interventionoccursattimeperiodt1TrueeffectoflawYa–YbOnlyhavedataatt1andt2Ifusingtimeseries,estimateYt1–Yt2Solution?29.DifferenceindifferencemodelsBasictwo-wayfixedeffectsmodelCrosssectionandtimefixedeffectsUsetimeseriesofuntreatedgrouptoestablishwhatwouldhaveoccurredintheabsenceoftheinterventionKeyconcept:cancontrolforthefactthattheinterventionismorelikelyinsometypesofstates30.ThreedifferentpresentationsTabularGraphicalRegressionequation31.DifferenceinDifferenceBeforeChangeAfterChangeDifferenceGroup1(Treat)Yt1Yt2ΔYt=Yt2-Yt1Group2(Control)Yc1Yc2ΔYc=Yc2-Yc1DifferenceΔΔYΔYt–ΔYc32.timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2Treatmenteffect=(Yt2-Yt1)–(Yc2-Yc1)33.KeyAssumptionControlgroupidentifiesthetimepathofoutcomesthatwouldhavehappenedintheabsenceofthetreatmentInthisexample,YfallsbyYc2-Yc1evenwithouttheinterventionNotethatunderlying‘levels’ofoutcomesarenotimportant(returntothisintheregressionequation)34.timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2Treatmenteffect=(Yt2-Yt1)–(Yc2-Yc1)TreatmentEffect35.Incontrast,whatiskeyisthatthetimetrendsintheabsenceoftheinterventionarethesameinbothgroupsIftheinterventionoccursinanareawithadifferenttrend,willunder/overstatethetreatmenteffectInthisexample,supposeinterventionoccursinareawithfasterfallingY36.timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2TruetreatmenteffectEstimatedtreatmentTrueTreatmentEffect37.BasicEconometricModelDatavariesbystate(i)time(t)OutcomeisYitOnlytwoperiodsInterventionwilloccurinagroupofobservations(e.g.states,firms,etc.)38.ThreekeyvariablesTit=1ifobsibelongsinthestatethatwilleventuallybetreatedAit=1intheperiodswhentreatmentoccursTitAit--interactionterm,treatmentstatesaftertheinterventionYit=β0+β1Tit+β2Ait+β3TitAit+εit39.Yit=β0+β1Tit+β2Ait+β3TitAit+εitBeforeChangeAfterChangeDifferenceGroup1(Treat)β0+β1β0+β1+β2+β3ΔYt

=β2+β3Group2(Control)β0β0+β2ΔYc=β2DifferenceΔΔY=β340.MoregeneralmodelDatavariesbystate(i)time(t)OutcomeisYitManyperiodsInterventionwilloccurinagroupofstatesbutatavarietyoftimes41.uiisastateeffectvtisacompletesetofyear(time)effectsAnalysisofcovariancemodelYit=β0+β3TitAit+ui+vt+εit42.WhatisniceaboutthemodelSupposeinterventionsarenotrandombutsystematicOccurinstateswithhigherorloweraverageYOccurintimeperiodswithdifferentY’sThisiscapturedbytheinclusionofthestate/timeeffects–allowscovariancebetweenuiandTitAitvtandTitAit43.GroupeffectsCapturedifferencesacrossgroupsthatareconstantovertimeYeareffectsCapturedifferencesovertimethatarecommontoallgroups44.Meyeretal.Workers’compensationStateruninsuranceprogramCompensateworkersformedicalexpensesandlostworkduetoonthejobaccidentPremiumsPaidbyfirmsFunctionofpreviousclaimsandwagespaidBenefits--%ofincomew/cap45.TypicalbenefitsscheduleMin(pY,C)P=percentreplacementY=earningsC=cape.g.,65%ofearningsupto$400/month46.Concern:Moralhazard.BenefitswilldiscouragereturntoworkEmpiricalquestion:duration/benefitsgradientPreviousestimatesRegressduration(y)onreplacedwages(x)Problem:givenprogressivenatureofbenefits,replacedwagesrevealalotabouttheworkersReplacementrateshigherinhigherwagestates47.Yi=Xiβ+αRi+εiY(duration)R(replacementrate)Expectα>0ExpectCov(Ri,εi)HigherwageworkershavelowerRandhigherduration(understate)HigherwagestateshavelongerdurationandlongerR(overstate)48.SolutionQuasiexperimentinKYandMIIncreasedtheearningscapIncreasedbenefitforhigh-wageworkers(Treatment)Didnothingtothosealreadybeloworiginalcap(comparison)Comparechangeindurationofspellbeforeandafterchangeforthesetwogroups49.50.51.ModelYit=durationofspellonWCAit=periodafterbenefitshikeHit=highearningsgroup(Income>E3)Yit=β0+β1Hit+β2Ait+β3AitHit+β4Xit’+εitDiff-in-diffestimateisβ352.53.Questionstoask?Whatparameterisidentifiedbythequasi-experiment?Isthisaneconomicallymeaningfulparameter?Whatassumptionsmustbetrueinorderforthemodeltoprovideandunbiasedestimateofβ3?Dotheauthorsprovideanyevidencesupportingtheseassumptions?54.MoregeneralmodelManywithingroupestimatorsthatdonothavethenicediscretetreatmentsoutlinedabovearealsocalleddifferenceindifferencemodelsCookandTauchen.ExamineimpactofalcoholtaxesonheavydrinkingStatestaxalcoholExamineimpactonconsumptionandresultsofheavyconsumptiondeathduetolivercirrhosis55.Yit=β0+β1INCit+β2INCit-1

+β1TAXit+β2TAXit-1+ui+vt+εitiisstate,tisyearYitispercapitaalcoholconsumptionINCispercapitaincomeTAXistaxpaidpergallonofalcohol56.SomeKeysModelrequiresthatuntreatedgroupsprovideestimateofbaselinetrendwouldhavebeenintheabsenceofinterventionKey–findadequatecomparisonsIftrendsarenotaligned,cov(TitAit,εit)≠0OmittedvariablesbiasHowdoyouknowyouhaveadequatecomparisonsample?57.Dothepre-treatmentsampleslooksimilarTricky.D-in-Dmodeldoesnotrequiremeansmatch–onlytrends.Ifmeansmatch,noguaranteetrendswillHowever,ifmeansdiffer,aren’tyoususpiciousthattrendswillaswell?58.DevelopteststhatcanfalsifymodelYit=β0+β3TitAit+ui+vt+εitWillprovideunbiasedestimatesolongascov(TitAit,εit)=0Concern:supposethattheinterventionismorelikelyinastatewithadifferenttrendIftrue,coefficientmay‘showup’priortotheintervention59.Add“leads〞tothemodelforthetreatmentInterventionshouldnotchangeoutcomesbeforeitappearsIfitdoes,thensuspiciousthatcovariancebetweentrendsandintervention60.Yit=β0+β3TitAit+α1TitAit+1+α2TitAit+2+α3TitAit+3+ui+vt+εitThree“leads〞Testnull:Ho:α1=α2=α3=061.Pickcontrolgroupsthathavesimilarpre-treatmenttrendsMoststudiespickalluntreateddataascontrolsExample:Somestatesraisecigarettetaxes.UsestatesthatdonotchangetaxesascontrolsExample:SomestatesadoptwelfarereformpriortoTANF.Useallnon-reformstatesascontrolsIntuitivebutnotlikelycorrect62.CanuseeconometricproceduretopickcontrolsAppealingifinterventionsarediscreteandfewinnumberEasytoidentifypre-post63.CardandSullivanExaminetheimpactofjobtrainingSomemenaretreatedwithjobskills,othersarenotMostarelowskillmen,highunemployment,frequentmovementinandoutofworkEightquartersofpre-treatmentdatafortreatmentandcontrols64.LetYit=1if“i〞workedintimetThereisthenaneightdigitsequenceofoutcomes“11110000〞or“10100111〞Menwithsame8digitpre-treatmentsequencewillformcontrolforthetreatedPeoplewithsamepre-treatmenttimeseriesare‘matched’65.IntuitivelyappealingandsimpleprocedureDoesnotguaranteethatposttreatmenttrendswouldbethesamebut,thisisthebestyouhave.66.MoresystematicmodelDatavariesbyindividual(i),state(s),timeInterventionisinaparticularstateYist=β0+Xistβ2+β3TstAst+us+vt+εistManystatesavailabletobecontrolsHowdoyoupickthem?67.Restrictsampletopre-treatmentperiodState1isthetreatedstateStatekisapotentialcontrolRundatawithonlythesetwostatesEstimateseparateyeareffectsforthetreatmentstateIfyoucannotrejectnullthattheyeareffectsarethesame,useascontrol68.UnrestrictedmodelPretreatmentyearssoTstAstnotinmodelMpre-treatmentyearsLetWt=1ifobsfromyeartYist=α0+Xistα2+Σt=2γtWt+Σt=2

λtTiWt+us+εistHo:λ2=λ3=…λm=069.Tyleretal.ImpactofGEDonwagesGeneraleducationdevelopmentdegreeEarnaHSdegreebypassinganexamExampassratesvarybystateIntroducedin1942asawayforveteranstoearnaHSdegreeHasexpandedtothegeneralpublic70.In1996,760KdropoutsattemptedtheexamLittlehumancapitalgeneratedbystudyingfortheexamReallymeasuresstockofknowledgeHowever,passingmay‘signal’somethingaboutability71.IdentificationstrategyUsevariationacrossstatesinpassratestoidentifybenefitofaGEDHighscoringpeoplewouldhavepassedtheexamregardlessofwhatstatetheylivedinLowscoringpeoplearesimilaracrossstates,butonisgrantedaGEDandtheotherisnot72.NYCTABDCEFIncreasingscoresPassingScoresCTPassingscoreNY73.GroupsAandBpassineitherstateGroupDpassesinCTbutnotinNYGroupClookssimilartoDexceptitdoesnotpass74.WhatisimpactofpassingtheGEDYis=earningsofpersoniinstatesLis=earnedalowscoreCTis=1ifliveinastatewithagenerouspassingscoreYis=β0+Lisβ1+CTβ2+LisCTisβ3+εis75.DifferenceinDifferenceCTNYDifferenceTestscoreislowDC(D-C)TestscoreishighBA(B-A)Difference(D-C)–(B-A)76.HowdoyougetthedataFromETS(testingagency)getsocialsecuritynumbers(SSN)oftesttakes,somedemographicdata,state,andtestscoreGiveSocialSecurityAdmin.alistofSSNsbygroup(lowscoreinCT,highscoreinNY)SSNgivesyoubackmean,std.dev.#obspercell77.78.79.AcemogluandAngristada_jpe.doada_jpe.log80.AmericanswithDisabilityActRequiresthatemployersaccommodatedisabledworkersOutlawsdiscriminationbasedondisabilitiesPassesinJuly1990,effectiveJuly1992MaydiscourageemploymentofdisabledCostsofaccommodationsMaybemoredifficulttofiredisabled81.EconometricmodelDifferenceindifferenceHavedatabefore/afterlawgoesintoeffectTreatedgroup–disabledControl–non-disabledTreatmentvariableisinteractionDiabled*1992andafter82.Yit=Xitπ+Diδ+Yeartγt+YeartDitαt+εitYit=labormarketoutcome,personiyeartXitvectorofindividualcharacteristicsDit=1ifdisableldYeart=yeareffectYeartDit=completesetofyearxdisabilityinteractions

83.Coefonαi’sshouldbezerobeforethelawMaybenonzeroforyears>=199284.85.86.DataMarchCPSAsksallparticipantsemployment/incomedataforthepreviousyearEarnings,weeksworked,usualhours/weekDatafrom1988-1997MarchCPSDataforcalendaryears1987-1996Menandwomen,aged21-58Generateresultsforvarioussubsamples87.ConstructssetsofdummiesForyear,regionandageGenerateyearxDisabilityinteractions88.Table2ADAnotineffectEffectiveyearsofADA89.ModelwithfewcontrolsAfteraddingextensivelistOfcontrols,resultschangelittle90.regwkswork1_Iy*disabledd_y*;Includeallvariablesthatbeginwith_lyIncludeallvariablesthatbeginwithd_y91.NeedtodeleteoneyeareffectSinceconstantisinmodelDisabilitymaineffectDisabilitylawinteractions#obsclosetowhatisReportedinpaper92.RundifferentmodelOnetreatmentvariable:Disabledxafter1991.genada=yearw>=1992;.gentreatment=ada*disabled;Addyeareffectstomodel,disabled,themADAxdisabledinteraction93.ADAreducedworkbyalmost2weeks/yearRegressionstatement94.Shouldyoucluster?Interventionvariesbyyear/disabilityShouldbewithin-yearcorrelationinerrorsPeopleareinthesampletwoyearsinarowsothereshouldbesomecorrelationovertimeCannotclusteronyearssince#groupstoosmall95.NeedlargersetthatmakessenseTwooptions(manymore)ClusteronstateClusteronstate/disability96..gendisabled_state=100*disabled+statefip;regwkswork1_Ia*_Iy*_Ir*whiteblackhispaniclthshsgradsomecoldisabledtreatment,cluster(statefip);.regwkswork1_Ia*_Iy*_Ir*whiteblackhispaniclthshsgradsomecoldisabledtreatment,cluster(disabl