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

TL;DR

This paper analyzes the limitations of two-way fixed effects regressions with multiple treatments, showing they can produce biased estimates due to heterogeneous effects and proposing a robust alternative estimator.

Contribution

It reveals the contamination problem in TWFE estimators with multiple treatments and introduces a new difference-in-differences estimator that is robust to heterogeneity.

Findings

TWFE coefficients can be negatively weighted sums of treatment effects.

Omitting a treatment can reduce bias under heterogeneity.

The proposed estimator avoids contamination and yields different effect estimates.

Abstract

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.