1、1Difference in Difference ModelsBill EvansSpring 20192Difference 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 3Problem set up Cross-secti
2、onal 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 changes4timeYt1t2YaYbYt1Yt2True effect = Yt2-Yt1Estimated effect = Yb-Yati5 Intervention occurs at
3、 time period t1 True effect of law Ya Yb Only have data at t1 and t2 If using time series, estimate Yt1 Yt2 Solution?6Difference 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
4、 absence of the intervention Key concept: can control for the fact that the intervention is more likely in some types of states7Three different presentations Tabular Graphical Regression equation8Difference in DifferenceBeforeChangeAfterChangeDifferenceGroup 1(Treat)Yt1Yt2Yt = Yt2-Yt1Group 2(Control
5、)Yc1Yc2Yc=Yc2-Yc1DifferenceYYt Yc9timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2Treatment effect=(Yt2-Yt1) (Yc2-Yc1)10Key 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 No
6、te that underlying levels of outcomes are not important (return to this in the regression equation)11timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2Treatment effect=(Yt2-Yt1) (Yc2-Yc1)TreatmentEffect12 In contrast, what is key is that the time trends in the absence of the intervention are the same in both gro
7、ups 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 Y13timeYt1t2Yt1Yt2treatmentcontrolYc1Yc2True treatment effectEstimated treatmentTrueTreatmentEffect14Basic Econometric
8、 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 - inter
9、action term, treatment states after the intervention Yit = 0 + 1Tit + 2Ait + 3TitAit + it16Yit = 0 + 1Tit + 2Ait + 3TitAit + itBeforeChangeAfterChangeDifferenceGroup 1(Treat)0+ 10+ 1+ 2+ 3Yt = 2+ 3Group 2(Control)00+ 2Yc= 2DifferenceY = 317More general model Data varies by state (i) time (t) Outcome
10、 is Yit Many periods Intervention will occur in a group of states but at a variety of times18 ui is a state effect vt is a complete set of year (time) effects Analysis of covariance model Yit = 0 + 3 TitAit + ui + t + it19What is nice about the model Suppose interventions are not random but systemat
11、ic 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 effects allows covariance between ui and TitAit t and TitAit20 Group effects Capture differences across groups that are constant over time Year effects Captur
12、e differences over time that are common to all groups21Meyer 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/ cap22 Typi
13、cal benefits schedule Min( pY,C) P=percent replacement Y = earnings C = cap e.g., 65% of earnings up to $400/month23 Concern: Moral hazard. Benefits will discourage return to work Empirical question: duration/benefits gradient Previous estimates Regress duration (y) on replaced wages (x) Problem: gi
14、ven progressive nature of benefits, replaced wages reveal a lot about the workers Replacement rates higher in higher wage states24 Yi = 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 l
15、onger duration and longer R (overstate)25Solution 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
16、 262728Model 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 32930Questions 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?
