A large class of two-sided matching models that include both transferable and non-transferable utility result in positive assortative matching along a latent index. Data from matching markets, however, may not exhibit perfect assortativity due to the presence of unobserved characteristics. This paper studies the identification and estimation of such models. We show that the distribution of the latent index is not identified when data from one-to-one matches are observed. Remarkably, the model is nonparametrically identified using data in a single large market when each agent on one side has at least two matched partners. The additional empirical content in many-to-one matches is demonstrated using simulations and stylized examples. We then derive asymptotic properties of a minimum distance estimator as the size of the market increases, allowing estimation using dependent data from a single large matching market. The nature of the dependence requires modification of existing empirical process techniques to obtain a limit theorem.

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We study estimation and non-parametric identi…cation of preferences in two-sided matching markets using data from a single market with many agents. We consider a model in which preferences of each side of the market are vertical, utility is nontransferable and the observed matches are pairwise stable. We show that preferences are not identi…ed with data on one-to-one matches but are non-parametrically iden-ti…ed when data from many-to-one matches are observed. The additional empirical content in many-to-one matches is illustrated by comparing two simulated objective functions, one that does and the other that does not use information available in manyto-one matching. We also prove consistency of a method of moments estimator for a parametric model under a data generating process in which the size of the matching market increases, but data only on one market is observed. Since matches in a single market are interdependent, our proof of consistency cannot rely on observations of independent matches. Finally, we present Monte Carlo studies of a simulation based estimator.

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