Identification of Average Random Coefficients under Magnitude and Sign Restrictions on Confounding

This paper studies measuring the average effects of X on Y in a structural system with random coefficients and confounding. We do not require (conditionally) exogenous regressors or instruments. Using proxies W for the confounders U, we ask how do the average direct effects of U on Y compare in magnitude and sign to those of U on W. Exogeneity and equi- or proportional confounding are limit cases yielding full identification. Alternatively, the elements of beta-hat are partially identified in a sharp bounded interval if W is sufficiently sensitive to U, and sharp upper or lower bounds may obtain otherwise. We extend this analysis to accommodate conditioning on covariates and a semiparametric separable specification as well as a panel structure and proxies included in the Y equation. After studying estimation and inference, we apply this method to study the financial return to education and the black-white wage gap.