SELECTION BIAS: Difference-in-Differences

In this paper, we introduce conditional semiparametric and nonparametric versions of the difference-in-differences estimator that apply the method of matching to a panel or to repeated cross sections of persons. Differencing is done conditional on X. The critical identifying assumption in our proposed method is that conditional on X, the biases are the same on average in different time periods before and after the period of participation in the program so that differencing the differences between participants and nonparticipants eliminates the bias. To see how this estimator works, let t be a post-program period and t’ a preprogram
where Bt denotes the bias in time t, defined in (10). This method extends the method of matching because it does not require that the bias vanish for any X, just that it be the same across t and tf conditional on X. Notice further that (12) is implied by the conventional econometric selection estimator if E{Uot \P{X),D = 1) — E(UW |F(X), D = 1) is the same for different choices of t and In application, (12) is often assumed to hold for all t and t’ or for t and t’ defined symmetrically around t = 0, the date of participation in the program (i.e., t = — t’).
We now compare B(X) to the more conventional measure of bias used in the literature.

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Re-examining the Conventional Measure of Selection Bias