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Sorting and spillovers can create correlation in individual outcomes. In this situation, standard discrete choice estimators cannot consistently estimate the probability of joint and conditional events, and alternative estimators can yield incoherent statistical models or intractable estimators. I propose a random effects estimator that models the dependence among the unobserved heterogeneity of individuals in the same cluster using a parametric copula. This estimator makes it possible to compute joint and conditional probabilities of the outcome variable, and is statistically coherent. I describe its properties, establishing its efficiency relative to standard random effects estimators, and propose a specification test for the copula. The likelihood function for each cluster is an integral whose dimension equals the size of the cluster, which may require high-dimensional numerical integration. To overcome the curse of dimensionality from which methods like Monte Carlo integration suffer, I propose an algorithm that works for Archimedean copulas. I illustrate this approach by analysing labour supply in married couples.

I address the decomposition of the differences between the distribution of outcomes of two groups when individuals self-select themselves into participation. I differentiate between the decomposition for participants and the entire population, highlighting how the primitive components of the model affect each of the distributions of outcomes. Additionally, I introduce two ancillary decompositions that help uncover the sources of differences in the distribution of unobservables and participation between the two groups. The estimation is done using existing quantile regression methods, for which I show how to perform uniformly valid inference. I illustrate these methods by revisiting the gender wage gap, finding that changes in female participation and self-selection have been the main drivers for reducing the gap.

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