Identification of treatment response with social interactions

This paper develops a formal language for study of treatment response with social interactions, and uses it to obtain new findings on identification of potential outcome distributions. Defining a person's treatment response to be a function of the entire vector of treatments received by the population, I study identification when shape restrictions and distributional assumptions are placed on response functions. An early key result is that the traditional assumption of individualistic treatment response (ITR) is a polar case within the broad class of constant treatment response (CTR) assumptions, the other pole being unrestricted interactions. Important non-polar cases are interactions within reference groups and distributional interactions. I show that established findings on identification under assumption ITR extend to assumption CTR. These include identification with assumption CTR alone and when this shape restriction is strengthened to semi-monotone response. I next study distributional assumptions using instrumental variables. Findings obtained previously under assumption ITR extend when assumptions of statistical independence (SI) are posed in settings with social interactions, but with important caveats. The extended version of assumption SI has no power to identify counterfactual outcome distributions when social interactions are unrestricted. Random assignment of realized treatments has strong identifying power when reference groups are small, limited power when distributional interactions occur in large groups, and generically no power in settings with strong dependence. Finally, considering models of endogenous social interactions, I show that identification of structural equations differs from identification of outcome distributions under potential treatments. Analysis of familiar linear models illustrates this general point.