Difference-in-Differences Estimators for Treatments Continuously Distributed at Every Period
This paper addresses causal effect estimation under continuous, time-varying treatments (e.g., taxes, tariffs, prices), extending the canonical difference-in-differences (DID) framework. Methodologically, it introduces a longitudinal comparison identification strategy anchored at baseline treatment levels to identify a weighted average of the treatment effect slope; constructs a doubly robust, √n-consistent, and asymptotically normal nonparametric estimator; and rigorously generalizes DID to settings with continuous treatment in every period—including an extension to instrumental variable settings. The approach avoids strong parametric assumptions on the treatment function and preserves testability of the parallel trends assumption. Empirically, the method successfully estimates the price elasticity of gasoline demand, demonstrating its validity and robustness in real-world economic policy evaluation.