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

When studying the effects of taxes, tariffs, or prices using panel data, the treatment is often continuously distributed in every period. We develop difference-in-differences (DID) estimators for such settings. We partition units into switchers, whose treatment changes between consecutive periods, and stayers, whose treatment remains constant. Under a parallel-trends assumption, we show that the slopes of switchers' potential outcomes with respect to the treatment are nonparametrically identified by DID comparisons between switchers and stayers sharing the same baseline treatment level. Conditioning on the baseline treatment is key, as it ensures that the underlying parallel-trends assumption accommodates time-varying treatment effects. We then study two weighted averages of these slopes, discuss their respective advantages, and propose for each a doubly robust, semiparametrically efficient, and $\sqrt{n}$-consistent estimator. Finally, we extend our framework to instrumental variables and illustrate it by estimating the effects of gasoline taxes on prices and fuel consumption.