  This paper analyzes difference-in-differences designs with a continuous
treatment. We show that treatment effect on the treated-type parameters can be
identified under a generalized parallel trends assumption that is similar to
the binary treatment setup. However, interpreting differences in these
parameters across different values of the treatment can be particularly
challenging due to treatment effect heterogeneity. We discuss alternative,
typically stronger, assumptions that alleviate these challenges. We also
provide a variety of treatment effect decomposition results, highlighting that
parameters associated with popular linear two-way fixed-effect (TWFE)
specifications can be hard to interpret, \emph{even} when there are only two
time periods. We introduce alternative estimation procedures that do not suffer
from these TWFE drawbacks, and show in an application that they can lead to
different conclusions.
