Identification in Nonlinear Difference in Difference Models with Multivalued Treatment Outcomes

We study the conditions to directly identify the joint distribution of outcomes for the treated group in absence of any treatment, avoiding to make assumptions that allow 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). We show that under a reasonable set of assumptions we can identify sharp bound for the counterfactual joint distribution of outcome variables. Moreover, we show identification power increases for copulas from the Archimedean family