The Commensurability Problem: Conceptual Difficulties in Estimating the Effect of Behavior on Behavior

We pose the commensurability problem: When do the estimates generated by actual research designs correspond to quantities of theoretical interest? We study this question in settings where both treatment and outcome are behavior and the treatment effect of interest is decomposable into direct and informational channels. We establish two results. First, the quantity estimated by an actual research design is only commensurate with the total effect in the ideal experiment if treatment status in the research design is a sufficient statistic for the decision-makers' information. Second, a research design corresponding to a nonideal experiment isolates just the direct effect in the ideal experiment if two conditions hold: (i) there is no information effect in the nonideal experiment and (ii) the decision-maker's response function is additively separable in treatment and information. We apply our results to three substantive literatures: the efficacy of protest, the empowerment of female candidates, and indiscriminate violence in counterinsurgency.