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

When one studies the effects of taxes, tariffs, or prices using panel data, the treatment is often continuously distributed in every period. We propose difference-in-differences (DID) estimators for such cases. 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, 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 consider two weighted averages of switchers' slopes, and discuss their respective advantages. For each weighted average, we propose a doubly-robust, nonparametric, and $\sqrt{n}$-consistent estimator. We generalize our results to the instrumental-variable case. We apply our method to estimate the price-elasticity of gasoline consumption.