1、1,Difference in Difference Models,Bill Evans Spring 2008,2,Difference in difference models,Maybe the most popular identification strategy in applied work today Attempts to mimic random assignment with treatment and “comparison” sample Application of two-way fixed effects model,3,Problem set up,Cross

2、-sectional and time series data One group is treated with intervention Have pre-post data for group receiving intervention Can 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,Int

3、ervention occurs at time period t1 True effect of law Ya Yb Only have data at t1 and t2 If using time series, estimate Yt1 Yt2 Solution?,6,Difference in difference models,Basic two-way fixed effects model Cross section and time fixed effects Use time series of untreated group to establish what would

4、 have occurred in the absence of the intervention Key concept: can control for the fact that the intervention is more likely in some types of states,7,Three different presentations,Tabular Graphical Regression equation,8,Difference in Difference,9,time,Y,t1,t2,Yt1,Yt2,treatment,control,Yc1,Yc2,Treat

5、ment effect= (Yt2-Yt1) (Yc2-Yc1),10,Key Assumption,Control group identifies the time path of outcomes that would have happened in the absence of the treatment In this example, Y falls by Yc2-Yc1 even without the intervention Note that underlying levels of outcomes are not important (return to this i

6、n the regression equation),11,time,Y,t1,t2,Yt1,Yt2,treatment,control,Yc1,Yc2,Treatment effect= (Yt2-Yt1) (Yc2-Yc1),Treatment Effect,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 differen

7、t trend, will under/over state the treatment effect In 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,True Treatment Effect,14,Basic Econometric Model,Data varies by state (i) time (t

8、) Outcome is Yit Only two periods Intervention will occur in a group of observations (e.g. states, firms, etc.),15,Three key variables Tit =1 if obs i belongs in the state that will eventually be treated Ait =1 in the periods when treatment occurs TitAit - interaction term, treatment states after th

9、e intervention Yit = 0 + 1Tit + 2Ait + 3TitAit + it,16,Yit = 0 + 1Tit + 2Ait + 3TitAit + it,17,More general model,Data varies by state (i) time (t) Outcome is Yit Many periods Intervention will occur in a group of states but at a variety of times,18,ui is a state effect vt is a complete set of year

10、(time) effects Analysis of covariance model Yit = 0 + 3 TitAit + ui + t + it,19,What is nice about the model,Suppose interventions are not random but systematic Occur in states with higher or lower average Y Occur in time periods with different Ys This is captured by the inclusion of the state/time

11、effects allows covariance between ui and TitAit t and TitAit,20,Group effects Capture differences across groups that are constant over time Year effects Capture differences over time that are common to all groups,21,Meyer et al.,Workers compensation State run insurance program Compensate workers for

12、 medical expenses and lost work due to on the job accident Premiums Paid by firms Function of previous claims and wages paid Benefits - % of income w/ cap,22,Typical benefits schedule Min( pY,C) P=percent replacement Y = earnings C = cap e.g., 65% of earnings up to $400/month,23,Concern: Moral hazar

13、d. Benefits will discourage return to work Empirical question: duration/benefits gradient Previous estimates Regress duration (y) on replaced wages (x) Problem: given progressive nature of benefits, replaced wages reveal a lot about the workers Replacement rates higher in higher wage states,24,Yi =

14、Xi + Ri + i Y (duration) R (replacement rate) Expect 0 Expect Cov(Ri, i) Higher wage workers have lower R and higher duration (understate) Higher wage states have longer duration and longer R (overstate),25,Solution,Quasi experiment in KY and MI Increased the earnings cap Increased benefit for high-

15、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 groups,26,27,28,Model,Yit = duration of spell on WC Ait = period after benefits hike Hit = high earnings group (IncomeE3) Yit = 0 + 1Hit + 2Ait + 3AitHit + 4Xit + it Diff-in-diff estimate is 3,29,30,Questions to ask?,What parameter is