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1、1,Difference in Difference Models,Bill EvansSpring 2008,2,Difference in difference models,Maybe the most popular identification strategy in applied work todayAttempts to mimic random assignment with treatment and“comparison”sampleApplication of two-way fixed effects model,3,Problem set up,Cross-sect
2、ional and 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 changes,4,time,Y,t1,t2,Ya,Yb,Yt1,Yt2,True effect=Yt2-Yt1,Estimated effect=Yb-Ya,ti,5,Intervention occ
3、urs at time period t1True effect of lawYa YbOnly have data at t1 and t2If using time series,estimate Yt1 Yt2Solution?,6,Difference in difference models,Basic two-way fixed effects modelCross section and time fixed effectsUse time series of untreated group to establish what would have occurred in the
4、 absence of the interventionKey concept:can control for the fact that the intervention is more likely in some types of states,7,Three different presentations,TabularGraphicalRegression equation,8,Difference in Difference,9,time,Y,t1,t2,Yt1,Yt2,treatment,control,Yc1,Yc2,Treatment effect=(Yt2-Yt1)(Yc2
5、-Yc1),10,Key Assumption,Control 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 are not important(return to this in the regression equation),11,t
6、ime,Y,t1,t2,Yt1,Yt2,treatment,control,Yc1,Yc2,Treatment effect=(Yt2-Yt1)(Yc2-Yc1),TreatmentEffect,12,In 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 with a different trend,will under/over state the t
7、reatment effectIn this example,suppose intervention occurs in area with faster falling Y,13,time,Y,t1,t2,Yt1,Yt2,treatment,control,Yc1,Yc2,True treatment effect,Estimated treatment,TrueTreatmentEffect,14,Basic Econometric Model,Data varies by state(i)time(t)Outcome is YitOnly two periodsIntervention
8、 will occur in a group of observations(e.g.states,firms,etc.),15,Three 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+2Ait+3TitAit+it,16,Yit=0+1Tit+2Ai
9、t+3TitAit+it,17,More general model,Data varies by state(i)time(t)Outcome is YitMany periodsIntervention will occur in a group of states but at a variety of times,18,ui is a state effectvt is a complete set of year(time)effectsAnalysis of covariance modelYit=0+3 TitAit+ui+t+it,19,What is nice about t
10、he model,Suppose interventions are not random but systematicOccur in states with higher or lower average YOccur in time periods with different YsThis is captured by the inclusion of the state/time effects allows covariance between ui and TitAitt and TitAit,20,Group effects Capture differences across
11、 groups that are constant over timeYear effectsCapture differences over time that are common to all groups,21,Meyer et al.,Workers compensationState run insurance programCompensate workers for medical expenses and lost work due to on the job accidentPremiumsPaid by firmsFunction of previous claims a
12、nd wages paidBenefits-%of income w/cap,22,Typical benefits scheduleMin(pY,C)P=percent replacementY=earningsC=cape.g.,65%of earnings up to$400/month,23,Concern:Moral hazard.Benefits will discourage return to workEmpirical question:duration/benefits gradientPrevious estimatesRegress duration(y)on repl
13、aced wages(x)Problem:given progressive nature of benefits,replaced wages reveal a lot about the workersReplacement rates higher in higher wage states,24,Yi=Xi+Ri+iY(duration)R(replacement rate)Expect 0Expect Cov(Ri,i)Higher wage workers have lower R and higher duration(understate)Higher wage states
14、have longer duration and longer R(overstate),25,Solution,Quasi experiment in KY and MIIncreased the earnings capIncreased benefit for high-wage workers(Treatment)Did nothing to those already below original cap(comparison)Compare change in duration of spell before and after change for these two group
15、s,26,27,28,Model,Yit=duration of spell on WCAit=period after benefits hikeHit=high earnings group(IncomeE3)Yit=0+1Hit+2Ait+3AitHit+4Xit+itDiff-in-diff estimate is 3,29,30,Questions 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?,
