Difference-in-Differences Estimators When No Unit Remains Untreated

We consider treatment-effect estimation with a two-periods panel, where units are untreated at period one, and receive strictly positive doses at period two. First, we consider designs with some quasi-untreated units, with a period-two dose local to zero. We show that under a parallel-trends assumption, a weighted average of slopes of units' potential outcomes is identified by a difference-in-difference estimand using quasi-untreated units as the control group. We leverage results from the regression-discontinuity-design literature to propose a nonparametric estimator. Then, we propose estimators for designs without quasi-untreated units. Finally, we propose a test of the homogeneous-effect assumption underlying two-way-fixed-effects regressions.