Identication of Average Random Coe¢ cients under Magnitude and Sign Restrictions on Confounding

This paper studies measuring the average e¤ects  of X on Y in a structural system with random coe¢ cients 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 e¤ects of U on Y compare in magnitude and sign to those of U on W . Exogeneity and equior proportional confounding are limit cases yielding full identi…cation. Alternatively, the elements of  are partially identi…ed in a sharp bounded interval if W is su¢ ciently 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 speci…cation 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 …nancial return to education and the black-white wage gap. Keywords: causality, confounding, endogeneity, omitted variable, partial identi…cation, proxy.