We analyze the empirical content of the Roy model, stripped down to its essential features, namely sector specific unobserved heterogeneity and self-selection on the basis of potential outcomes. We characterize sharp bounds on the joint distribution of potential outcomes and the identifying power of exclusion restrictions. The latter include variables that affect market conditions only in one sector and variables that affect sector selection only. Special emphasis is put on the case of binary outcomes, which has received little attention in the literature to date. For richer sets of outcomes, we emphasize the distinction between pointwise sharp bounds and functional sharp bounds, and its importance, when constructing sharp bounds on functional features, such as inequality measures.
We analyze a Roy model of college major choice in Canada within this framework, and we take a new look at the under-representation of women in Science, Technology, Engineering or Mathematics (STEM).