Parametric identification of the joint distribution of the potential outcomes

We show the identification of the joint distribution of the potential outcomes under various parametric specifications. The key factor of the identification is the nonnormality of the distribution of the observed variables, with which we can obtain information of higher order moments that are not determined only by mean and variance. In particular, we show the identification of the joint distribution of the potential outcomes when it is specified by a normal mixture. Because any continuous distribution can be well approximated by a finite mixture distribution, our result may cover a wide class of distributions. The identification results derived are useful for estimating quantile treatment effects, causal mediation effects, and heterogeneous treatment effects, which cannot be estimated even if the unconfoundedness assumption is satisfied.