We consider the estimation of the effect of a treatment, using panel data where groups of units are exposed to different doses of the treatment at different times. We consider two sets of parameters of interest. The first are the average effects of having changed treatment for the first time...

In many applications of the differences-in-differences (DID) method, the treatment increases more in the treatment group, but some units are also treated in the control group. In such fuzzy designs, a popular estimator of treatment effects is the DID of the outcome divided by the DID of the...

We study regressions with period and group fixed effects and several treatment variables. Under a parallel trends assumption, the coefficient on each treatment identifies the sum of two terms. The first term is a weighted sum of the effect of that treatment in each group and period, with weights...