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 study difference-in-differences designs in which all units are initially untreated and later receive heterogeneous, strictly positive treatment doses. When quasi-untreated units receive doses close to zero, a difference-in-difference estimand using those units as the control group identifies a weighted average treatment effect per unit of treatment, under a parallel-trends assumption. We develop a nonparametric estimator and bias-corrected confidence intervals for this parameter. We also show that the assumptions justifying the usual two-way fixed-effects estimator imply testable linearity restrictions, and propose specification tests that can guide its use.

Date: 2024-05, Revised 2026-10
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