On the Predictive Distributions of Outcome Gains in the Presence of an Unidentified Parameter

In this article we describe methods for obtaining the predictive distributions of outcome gains in the framework of a standard latent variable selection model. Although most previous work has focused on estimation of mean treatment parameters as the method for characterizing outcome gains from program participation, we show how the entire distributions associated with these gains can be obtained in certain situations. Although the out-of sample outcome gain distributions depend on an unidentified parameter, we use the results of Koop and Poirier to show that learning can take place about this parameter through information contained in the identified parameters via a positive definiteness restriction on the covariance matrix.