Difference-in-Difference Estimators with Continuous Treatments and No Stayers

Many treatments, such as prices, taxes, or temperatures, are continuous in nature. Empirical researchers usually rely on two-way fixed effect regressions to estimate treatment effects in such cases. However, such estimators are not robust to heterogeneous treatment effects in general and rely on the linearity of treatment effects. We develop a difference-in-difference strategy for continuous treatments without imposing such restrictions when there are no stayers: treatment of all units changes between consecutive periods. We extend the nonparametric results of de Chaisemartin et al. (2023) to this setup and present a parametric approach that overcomes some limitations of the nonparametric approach.