Clément de Chaisemartin (Sciences Po)
“Two-way Fixed Effects and Differences-in-Differences Estimators with Several Treatments”
Abstract
We study two-way-fixed-effects regressions with several treatment variables. Under a parallel trends assumption, we show that 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. Therefore, those coefficients are not robust to heterogeneous effects and may be contaminated by other treatments’ effects. We propose an alternative difference-in-differences estimator, robust to heterogeneous effects and immune to the contamination problem. To estimate, say, the first treatment’s effect, our estimator compares 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 as the switching group.