Difference-in-Differences Estimators for Treatments Continuously Distributed at Every Period

Abstract: We propose difference-in-differences (DID) estimators in designs where the treatment is continuously distributed in every period, as is often the case when one studies the effects of taxes, tariffs, or prices. We assume that between consecutive periods, the treatment of some units, the switchers, changes, while the treatment of other units, the stayers, remains constant. We show that under a parallel-trends assumption, weighted averages of the slopes of switchers' potential outcomes are nonparametrically identified by difference-in-differences estimands comparing the outcome evolutions of switchers and stayers with the same baseline treatment. Controlling for the baseline treatment ensures that our estimands remain valid if the treatment's effect changes over time. We highlight two possible ways of weighting switcher's slopes, and discuss their respective advantages. For each weighted average of slopes, we propose a doubly-robust, nonparametric, $\sqrt{n}$-consistent, and asymptotically normal estimator. We generalize our results to the instrumental-variable case. Finally, we apply our method to estimate the price-elasticity of gasoline consumption.