Using Monotonicity Restrictions to Identify Models with Partially Latent Covariates

This paper develops a new method for identifying econometric models with partially latent covariates. Such data structures arise in industrial organization and labor economics settings where data are collected using an input-based sampling strategy, e.g., if the sampling unit is one of multiple labor input factors. We show that the latent covariates can be nonparametrically identified, if they are functions of a common shock satisfying some plausible monotonicity assumptions. With the latent covariates identified, semiparametric estimation of the outcome equation proceeds within a standard IV framework that accounts for the endogeneity of the covariates. We illustrate the usefulness of our method using a new application that focuses on the production functions of pharmacies. We find that differences in technology between chains and independent pharmacies may partially explain the observed transformation of the industry structure.