We study linear regressions with period and group fixed effects, with several treatment variables. We show that under a parallel trends assumption, the coefficient of each treatment identifies the sum of two terms. The first term is a weighted sum of the average effect of that treatment in each group and period, with weights that may be negative. The second term is a weighted sum of the average effect of the other treatments in each group and period, with weights that may again be negative. Accordingly, the treatment coefficients in those regressions are not robust to heterogeneous effects across groups and over time, and may also be contaminated by the effect of other treatments.  We propose an alternative estimator that does not suffer from those issues.
We study linear regressions with period and group fixed effects, with several treatment variables. We show that under a parallel trends assumption, the coefficient of each treatment identifies the sum of two terms. The first term is a weighted sum of the average effect of that treatment in each group and period, with weights that may be negative. The second term is a weighted sum of the average effect of the other treatments in each group and period, with weights that may again be negative. Accordingly, the treatment coefficients in those regressions are not robust to heterogeneous effects across groups and over time, and may also be contaminated by the effect of other treatments. When a single treatment is interacted with mutually exclusive binary variables to investigate treatment effect heterogeneity, our result implies that the coefficient for a given subgroup may actually be contaminated by treatment effects in other subgroups. We propose an alternative estimator that does not suffer from those issues.
A
When a single treatment is interacted with mutually exclusive binary variables to investigate treatment effect heterogeneity, our result implies that the coefficient for a given subgroup may actually be contaminated by treatment effects in other subgroups.