  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.
