Identification in Nonlinear Difference in Difference Model with Multivalued Treatment Outcomes∗

We study the conditions to identify the joint distribution of outcomes for the treated group in absence of any treatment, avoiding to make assumptions that allow to identify each counterfactual marginal distribution. Our starting point is Athey & Imbens (2006)’s Changes-In-Changes Model, but we generalize it letting the treatment also affect the distribution of unobservables even within each group (e.g. treated and untreated). The key to achieve identification is replace any identifying assumption on the marginal distribution of unobservables with identifying assumptions on the copula between all unobservables. We show that under a reasonable set of assumptions we can identify sharp bound for the counterfactual joint distribution of outcome variables. Moreover, if we restrict ourselves to copulas from the Archimedean family we can even achieve point identification.