Structural bounds on group differences in treatment effects

Abstract I analyze the use of group variation in the probability of receiving a treatment to recover information about the causal effects of that treatment, within the framework of a structural model of counterfactual outcomes and enrollment into the treatment. Specifically, I study the conditions under which a simple group difference in differences—that is, subtracting the difference in mean outcomes between treated and untreated units belonging to a group with a low treatment rate from that same difference for units belonging to a group with a high rate—identifies a lower bound on the difference in average treatment effects between the high- and low-rate groups. While the group difference in treatment effects is directly informative about inequality and heterogeneity in treatment effects, when theory or prior empirical evidence implies that the effect of the treatment is nonnegative, this group difference in differences also identifies a lower bound on the average treatment effect itself for the high-rate group. Although the conditions required for the lower-bound argument are not directly verifiable, I suggest falsification tests for whether they are consistent with the data. I also present several examples, illustrating the applicability of the identification results and the effectiveness of the falsification tests.