Difference in Difference Models

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-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. Y True effect = Yt2-Yt1 Estimated effect = Yb-Ya Yt1 Ya Yb Yt2 ti t1 t2 time

5. Intervention 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 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. Y Treatment effect= (Yt2-Yt1) – (Yc2-Yc1) Yc1 Yt1 Yc2 Yt2 control treatment t1 t2 time

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 in the regression equation)

11. Y Yc1 Treatment effect= (Yt2-Yt1) – (Yc2-Yc1) Yc2 Yt1 control Treatment Effect Yt2 treatment t1 t2 time

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 treatment effect • In this example, suppose intervention occurs in area with faster falling Y

13. Y Estimated treatment Yc1 Yt1 Yc2 control True treatment effect Yt2 True Treatment Effect treatment t1 t2 time

14. Basic Econometric Model • Data varies by • state (i) • time (t) • 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 the intervention • Yit = β0 + β1Tit + β2Ait + β3TitAit + εit

16. Yit = β0 + β1Tit + β2Ait + β3TitAit + εit

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 (time) effects • Analysis of covariance model • Yit = β0 + β3 TitAit + ui + λt + εit

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 Y’s • This is captured by the inclusion of the state/time effects – allows covariance between • ui and TitAit • λt 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 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 hazard. 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 = Xiβ + αRi + εi • Y (duration) • R (replacement rate) • Expect α > 0 • Expect Cov(Ri, εi) • 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-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

28. Model • Yit = duration of spell on WC • Ait = period after benefits hike • Hit = high earnings group (Income>E3) • Yit = β0 + β1Hit + β2Ait + β3AitHit + β4Xit’ + εit • Diff-in-diff estimate is β3

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 β3? • Do the authors provide any evidence supporting these assumptions?