Least Square Linear Prediction with Two-Sample Data

This paper investigates the identification and estimation of the least square linear predictor for the conditional expectation of an outcome variable Y given covariates (X;Z0) from data consisting of two independent random samples; the first sample contains replications of the variables (Y;Z0) but not X, while the second sample contains replications of (X;Z0) but not Y . The contribution is to characterize the identified set of the least square linear predictor when no assumption on the joint distribution of (Y;X;Z0), except for the existence of second order moments, is imposed. We show that the identified set is not a singleton, so the least square linear predictor of interest is set identified. The characterization is used to construct a sample analog estimator of the identified set. The asymptotic properties of the estimator are established and its implementation is illustrated via Monte Carlo exercises.