Difference-in-Differences Estimators When No Unit Remains Untreated

TL;DR

This paper develops new difference-in-differences estimators for two-period panel data where all units are treated in the second period, including cases with no untreated units, and introduces tests for homogeneous effects.

Contribution

It introduces novel estimators for scenarios without untreated units and a test for the homogeneous-effect assumption in two-way fixed effects models.

Findings

Weighted difference-in-differences estimand identified under parallel trends.

A nonparametric estimator leveraging regression discontinuity insights.

Proposed tests for the homogeneous-effect assumption.

Abstract

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.