Nonparametric difference-in-differences in repeated cross-sections with continuous treatments

This paper studies the identification of causal effects of a continuous treatment using a new difference-in-difference strategy. Our approach allows for endogeneity of the treatment, and employs repeated cross-sections. It requires an exogenous change over time which affects the treatment in a heterogeneous way, stationarity of the distribution of unobservables and a rank invariance condition on the time trend. On the other hand, we do not impose any functional form restrictions or an additive time trend, and we are invariant to the scaling of the dependent variable. Under our conditions, the time trend can be identified using a control group, as in the binary difference-in-differences literature. In our scenario, however, this control group is defined by the data. We then identify average and quantile treatment effect parameters. We develop corresponding nonparametric estimators and study their asymptotic properties. Finally, we apply our results to the effect of disposable income on consumption.