Estimation of random functions proxying for unobservables

This article considers the model Y=M(X, U) where U is an unobservable continuously distributed scalar and M is monotonic with respect to U. It is assumed there is an observable scalar W satisfying the restriction TFW|X, U=TFW|U almost surely where T is a known functional and FA|B denotes the distribution of A|B. This article shows that M can be identified under a mild monotonicity condition. This result requires neither statistical independence between X and U nor X to be continuously distributed. The estimation problem is treated when TFA|B≡E[A|B]. The proposed pointwise estimator of M is asymptotically normally distributed under weak technical conditions. Furthermore, the rate of convergence in probability is equal to n−r/(2r+d+1) where d denotes the dimension of the continuously distributed components of X and r is a positive integer which relates to the smoothness of certain functions. A Monte Carlo experiment is conducted and reveals the benefits of the estimator in the presence of endogeneity. We apply our estimator to estimate the returns to schooling.