Identification and Inference on the Correlation using Data from Two Independent Samples

Abstract

We examine what can be learned about the correlation between Y and X when data are available from two independent random samples; the first sample gives information on variables (Y;Z), while the second sample gives information on (X;Z). The variable Z has the same distribution in both samples, but the samples have no common observational units. A difficulty arises because neither sample has joint information on the variables (Y;X). This situation applies, for instance, to the ecological correlation problem or in the measurement of impact heterogeneity in program evaluation. Our first contribution is to sharply characterize the set of all possible values of the correlation of interest that are compatible with hypothetical knowledge of the distribution of (Y;Z) and of (X;Z) (the identification result). Unlike the existing literature, our characterization does not rely on assumptions, other than regularity conditions, on the joint distribution of (Y;X;Z). The second contribution is to propose a series-based estimator for the later set, which turns to be consistent and asymptotically normal and thereby permitting relatively easy inference in applications. We evaluate the small sample properties of the proposed estimator by means of Monte-Carlo experiments.