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 fac- tors. 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 ac- counts 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 inde- pendent pharmacies may partially explain the observed transformation of the industry structure.