Identification based on Difference-in-Differences Approaches with Multiple Treatments

This paper discusses identification based on difference-in-differences (DiD) approaches with multiple treatments. It shows that an appropriate adaptation of the common trend assumption underlying the DiD strategy for the comparison of two treatments restricts the possibility of effect heterogeneity for at least one of the treatments. The required assumption of effect homogeneity is likely to be violated because of non-random assignment to treatment based on both observables and unobservables. However, this paper shows that, under certain conditions, the DiD estimate comparing two treatments identifies a lower bound in absolute values on the average treatment effect on the treated compared to the unobserved non-treatment state, even if effect homogeneity is violated. This is possible if the treatments have ordered treatment effects, that is, in expectation, the effects of both treatments compared to no treatment have the same sign, and one treatment has a stronger effect than the other treatment on the respective recipients. Such assumptions are plausible if treatments are ordered or vary in intensity.