  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.
Regardless of whether the market is centralized or decentralized, assuming that
the observed matching is stable, we show nonparametric identification of
preferences of both sides under certain exclusion restrictions. To take our
results to the data, we use Monte Carlo simulations to evaluate different
estimators, including the ones that are directly constructed from the
identification. We find that a parametric Bayesian approach with a Gibbs
sampler works well in realistically sized problems. Finally, we illustrate our
methodology in decentralized admissions to public and private schools in Chile
and conduct a counterfactual analysis of an affirmative action policy.
