Fuzzy Differences in Differences

Difference in differences (DID) require that the treatment rate is equal to 0% in the control group and during period 0 (no “always takers”) and to 100% in the treatment group in period 1 (no “never takers”). Sometimes, treatment rate increases more in the treatment than in the control group but there are never or always takers. This paper derives identification results applying to such settings. They only require one common trend assumption on the outcome of interest (Y) whereas the standard instrumental variable result usually invoked also requires common trend on treatment rate. When there are never takers but no or few always takers, common trend on Y is sufficient to identify exactly an ATT or at least its sign.