Before revision:  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.
After revision:  We characterize sharp bounds on the joint distribution of potential outcomes and testable implications of the Roy self-selection model under an instrumental constraint on the joint distribution of potential outcomes we call stochastically monotone instrumental variable (SMIV). We show that testing the Roy model selection is equivalent to testing stochastic monotonicity of observed outcomes relative to the instrument. We apply our sharp bounds to the derivation of a measure of departure from Roy self-selection to identify values of observable characteristics that induce the most costly misallocation of talent and sector and are therefore prime targets for intervention . Special emphasis is put on the case of binary outcomes, which has received little attention in the literature to date.