  In a setting of many-to-one two-sided matching with non-transferable
utilities, e.g., college admissions, we study conditions under which
preferences of both sides are identified with data on one single market. The
main challenge is that every agent's actual choice set is unobservable to the
researcher. Assuming that the observed matching is stable, we show
nonparametric and semiparametric identification of preferences of both sides
under appropriate exclusion restrictions. Our identification arguments are
constructive and thus directly provide a semiparametric estimator. In Monte
Carlo simulations, the estimator can perform well but suffers from the curse of
dimensionality. We thus adopt a parametric model and estimate it by a Bayesian
approach with a Gibbs sampler, which works well in simulations. Finally, we
apply our method to school admissions in Chile and conduct a counterfactual
analysis of an affirmative action policy.
