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1、Difference in Difference ModelsBill EvansSpring 20081Difference in difference modelsMaybe the most popular identification strategy in applied work todayAttempts to mimic random assignment with treatment and“comparison”sampleApplication of two-way fixed effects model 2Problem set upCross-sectional an
2、d time series dataOne group is treated with interventionHave pre-post data for group receiving interventionCan examine time-series changes but,unsure how much of the change is due to secular changes3timeYt1t2YaYbYt1Yt2True effect=Yt2-Yt1Estimated effect=Yb-Yati4Intervention occurs at time period t1T
3、rue effect of lawYa YbOnly have data at t1 and t2If using time series,estimate Yt1 Yt2Solution?5Difference in difference modelsBasic two-way fixed effects modelCross section and time fixed effectsUse time series of untreated group to establish what would have occurred in the absence of the intervent
4、ionKey concept:can control for the fact that the intervention is more likely in some types of states6Three different presentationsTabularGraphicalRegression equation7Difference in DifferenceBeforeChangeAfterChangeDifferenceGroup 1(Treat)Yt1Yt2Yt=Yt2-Yt1Group 2(Control)Yc1Yc2Yc=Yc2-Yc1DifferenceYYt Y
5、c8timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2Treatment effect=(Yt2-Yt1)(Yc2-Yc1)9Key AssumptionControl group identifies the time path of outcomes that would have happened in the absence of the treatmentIn this example,Y falls by Yc2-Yc1 even without the interventionNote that underlying levels of outcomes
6、are not important(return to this in the regression equation)10timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2Treatment effect=(Yt2-Yt1)(Yc2-Yc1)TreatmentEffect11In contrast,what is key is that the time trends in the absence of the intervention are the same in both groups If the intervention occurs in an area
7、with a different trend,will under/over state the treatment effectIn this example,suppose intervention occurs in area with faster falling Y12timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2True treatment effectEstimated treatmentTrueTreatmentEffect13Basic Econometric ModelData varies by state(i)time(t)Outcome i
8、s YitOnly two periodsIntervention will occur in a group of observations(e.g.states,firms,etc.)14Three key variablesTit=1 if obs i belongs in the state that will eventually be treatedAit=1 in the periods when treatment occursTitAit -interaction term,treatment states after the interventionYit=0+1Tit+2
9、Ait+3TitAit+it15Yit=0+1Tit+2Ait+3TitAit+itBeforeChangeAfterChangeDifferenceGroup 1(Treat)0+10+1+2+3Yt=2+3Group 2(Control)00+2Yc=2DifferenceY=316More general modelData varies by state(i)time(t)Outcome is YitMany periodsIntervention will occur in a group of states but at a variety of times17ui is a st
10、ate effectvt is a complete set of year(time)effectsAnalysis of covariance modelYit=0+3 TitAit +ui+t+it18What is nice about the modelSuppose interventions are not random but systematicOccur in states with higher or lower average YOccur in time periods with different YsThis is captured by the inclusio
11、n of the state/time effects allows covariance between ui and TitAitt and TitAit19Group effects Capture differences across groups that are constant over timeYear effectsCapture differences over time that are common to all groups20Meyer et al.Workers compensationState run insurance programCompensate w
12、orkers for medical expenses and lost work due to on the job accidentPremiumsPaid by firmsFunction of previous claims and wages paidBenefits-%of income w/cap21Typical benefits scheduleMin(pY,C)P=percent replacementY=earningsC=cape.g.,65%of earnings up to$400/month22Concern:Moral hazard.Benefits will
13、discourage return to workEmpirical question:duration/benefits gradientPrevious estimatesRegress duration(y)on replaced wages(x)Problem:given progressive nature of benefits,replaced wages reveal a lot about the workersReplacement rates higher in higher wage states23Yi=Xi+Ri+iY(duration)R(replacement
14、rate)Expect 0Expect Cov(Ri,i)Higher wage workers have lower R and higher duration(understate)Higher wage states have longer duration and longer R(overstate)24SolutionQuasi experiment in KY and MIIncreased the earnings capIncreased benefit for high-wage workers(Treatment)Did nothing to those already
15、below original cap(comparison)Compare change in duration of spell before and after change for these two groups 252627ModelYit=duration of spell on WCAit=period after benefits hikeHit=high earnings group(IncomeE3)Yit=0+1Hit+2Ait+3AitHit+4Xit+itDiff-in-diff estimate is 32829Questions to ask?What parameter is identified by the quasi-experiment?Is this an economically meaningful parameter?What assumptions must be true in order for the model to provide and unbiased estimate of 3?Do the authors provide any evidence supporting these assumptions?30