In this paper we describe methods for predicting distributions of outcome gains in the framework of a latent variable selection model. We describe such procedures for Student-t selection models and a finite mixture of Gaussian selection models. Importantly, our algorithms for fitting these models are simple to implement in practice, and also permit learning to take place about the non-identified cross-regime correlation parameter. Using data from High School and Beyond, we apply our methods to determine the impact of dropping out of high school on a math test score taken at the senior year of high school. Our results show that selection bias is an important feature of this data, that our beliefs about this non-identified correlation are updated from the data, and that generalized models of selectivity offer an improvement over the textbook Gaussian model. Further, our results indicate that on average dropping out of high school has a large negative impact on senior-year test scores. However, for those individuals who actually drop out of high school, the act of dropping out of high school does not have a significantly negative impact on test scores. This suggests that policies aimed at keeping students in school may not be as beneficial as first thought, since those individuals who must be induced to stay in school are not the ones who benefit significantly (in terms of test scores) from staying in school.
