Difference-in-Differences with a Continuous Treatment
This paper addresses three key challenges in difference-in-differences (DID) estimation under continuous treatment: (i) selection bias due to non-random treatment assignment, (ii) incomparability of treatment effects across varying intensities, and (iii) ambiguous causal interpretation of conventional two-way fixed-effects (TWFE) estimators. We propose a generalized parallel trends assumption and establish the first rigorous identification framework for continuous-treatment DID. We formally prove that TWFE estimators—even in a two-period setting—lack clear causal interpretation under continuous treatment. To overcome this, we develop a bias-corrected, group-weighted estimator grounded in treatment-effect decomposition and augmented with selection-bias sensitivity analysis. Empirically, our method substantially revises policy effect estimates derived from standard TWFE, mitigating systematic misattribution. The proposed approach provides a robust, interpretable tool for causal inference in settings involving graded or intensity-varying interventions.