Difference-in-Differences Estimators When No Unit Remains Untreated

Cl\'ement de Chaisemartin, Diego Ciccia, Xavier D'Haultf{\oe}uille and Felix Knau

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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.

Date: 2024-05, Revised 2025-08
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