Two-way Fixed Effects and Differences-in-Differences Estimators with Several Treatments

We study regressions with period and group fixed effects and several treatment variables. Under a parallel trends assumption, the coefficient on each treatment identifies the sum of two terms. The first term is a weighted sum of the effect of that treatment in each group and period, with weights that may be negative and sum to one. The second term is a sum of the effects of the other treatments, with weights summing to zero. Accordingly, coefficients in those regressions are not robust to heterogeneous effects, and may be contaminated by the effect of other treatments. We propose alternative differences-in-differences estimators. To estimate, say, the effect of the first treatment, our estimators compare the outcome evolution of a group whose first treatment changes while its other treatments remain unchanged, to control groups whose treatments all remain unchanged, and with the same baseline treatments or treatments' history as the switching group. Those carefully selected comparisons are robust to heterogeneous effects, and do not suffer from the contamination problem