  We study two-way-fixed-effects regressions (TWFE) with several treatment
variables. Under a parallel trends assumption, we show that the coefficient on
each treatment identifies a weighted sum of that treatment's effect, with
possibly negative weights, plus a weighted sum of the effects of the other
treatments. Thus, those estimators are not robust to heterogeneous effects and
may be contaminated by other treatments' effects. We further show that omitting
a treatment from the regression can actually reduce the estimator's bias,
unlike what would happen under constant treatment effects. We propose an
alternative difference-in-differences estimator, robust to heterogeneous
effects and immune to the contamination problem. In the application we
consider, the TWFE regression identifies a highly non-convex combination of
effects, with large contamination weights, and one of its coefficients
significantly differs from our heterogeneity-robust estimator.
