  Difference-in-differences (DID) is a method to evaluate the effect of a
treatment. In its basic version, a "control group" is untreated at two dates,
whereas a "treatment group" becomes fully treated at the second date. However,
in many applications of the DID method, the treatment rate only increases more
in the treatment group. In such fuzzy designs, a popular estimator of treatment
effects is the DID of the outcome divided by the DID of the treatment. We show
that this ratio identifies a local average treatment effect only if two
homogeneous treatment effect assumptions are satisfied. We then propose two
alternative estimands that do not rely on any assumption on treatment effects,
and that can be used when the treatment rate does not change over time in the
control group. We prove that the corresponding estimators are asymptotically
normal. Finally, we use our results to revisit Duflo (2001).
