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1 Difference in Difference Models Bill Evans Spring 2008

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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

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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

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4 time Y t1t1 t2t2 YaYa YbYb Y t1 Y t2 True effect = Y t2 -Y t1 Estimated effect = Y b -Y a titi

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5 Intervention occurs at time period t 1 True effect of law –Y a – Y b Only have data at t 1 and t 2 –If using time series, estimate Y t1 – Y t2 Solution?

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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

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7 Three different presentations Tabular Graphical Regression equation

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8 Difference in Difference Before Change After ChangeDifference Group 1 (Treat) Y t1 Y t2 ΔY t = Y t2 -Y t1 Group 2 (Control) Y c1 Y c2 ΔY c =Y c2 -Y c1 DifferenceΔΔY ΔY t – ΔY c

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9 time Y t1t1 t2t2 Y t1 Y t2 treatment control Y c1 Y c2 Treatment effect= (Y t2 -Y t1 ) – (Y c2 -Y c1 )

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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 Y c2 -Y c1 even without the intervention Note that underlying ‘levels’ of outcomes are not important (return to this in the regression equation)

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11 time Y t1t1 t2t2 Y t1 Y t2 treatment control Y c1 Y c2 Treatment effect= (Y t2 -Y t1 ) – (Y c2 -Y c1 ) Treatment Effect

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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

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13 time Y t1t1 t2t2 Y t1 Y t2 treatment control Y c1 Y c2 True treatment effect Estimated treatment True Treatment Effect

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14 Basic Econometric Model Data varies by –state (i) –time (t) –Outcome is Y it Only two periods Intervention will occur in a group of observations (e.g. states, firms, etc.)

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15 Three key variables –T it =1 if obs i belongs in the state that will eventually be treated –A it =1 in the periods when treatment occurs –T it A it -- interaction term, treatment states after the intervention Y it = β 0 + β 1 T it + β 2 A it + β 3 T it A it + ε it

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16 Y it = β 0 + β 1 T it + β 2 A it + β 3 T it A it + ε it Before Change After ChangeDifference Group 1 (Treat) β 0 + β 1 β 0 + β 1 + β 2 + β 3 ΔY t = β 2 + β 3 Group 2 (Control) β0β0 β 0 + β 2 ΔY c = β 2 DifferenceΔΔY = β 3

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17 More general model Data varies by –state (i) –time (t) –Outcome is Y it Many periods Intervention will occur in a group of states but at a variety of times

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18 u i is a state effect v t is a complete set of year (time) effects Analysis of covariance model Y it = β 0 + β 3 T it A it + u i + λ t + ε it

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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 –u i and T it A it –λ t and T it A it

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20 Group effects –Capture differences across groups that are constant over time Year effects –Capture differences over time that are common to all groups

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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

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22 Typical benefits schedule –Min( pY,C) –P=percent replacement –Y = earnings –C = cap –e.g., 65% of earnings up to $400/month

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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

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24 Y i = X i β + αR i + ε i Y (duration) R (replacement rate) Expect α > 0 Expect Cov(R i, ε i ) –Higher wage workers have lower R and higher duration (understate) –Higher wage states have longer duration and longer R (overstate)

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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

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28 Model Y it = duration of spell on WC A it = period after benefits hike H it = high earnings group (Income>E 3 ) Y it = β 0 + β 1 H it + β 2 A it + β 3 A it H it + β 4 X it ’ + ε it Diff-in-diff estimate is β 3

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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 β 3 ? Do the authors provide any evidence supporting these assumptions?

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