Difference-in-Differences Estimators When No Unit Remains Untreated

Abstract

We study treatment-effect estimation in two-period panels where all units are untreated initially and receive strictly positive treatment doses in the second period. With quasi-untreated units receiving doses local to zero, we show that, under parallel trends, a weighted average of potential-outcome slopes is identified by a difference-in-differences estimand using quasi-untreated units as controls, and we propose a nonparametric estimator based on regression-discontinuity methods. We then develop estimators for settings without quasi-untreated units and propose a test of the homogeneous-effect assumption underlying two-way fixed-effects regressions.

Online appendix starts at p.45