1. MOTIVATION AND RESULTS

Applied economists often face the problem that no single data set contains all the variables that are necessary to conduct inference on a population of interest. When this is the case, they need to integrate the information contained in different samples; for example, they might need to combine survey data with administrative data (see Ridder and Moffitt, in press, for a survey of the econometrics of data combination). From a methodological perspective, the problem is that although the samples being combined might contain some common variables, other variables belong only to one of the samples. When the data are collected at the same aggregation level (e.g., individual level, household level, etc.), if the common variables include a unique (and correctly recorded) identifier of the units constituting each sample, and there is a substantial overlap of units across all samples, then exact matching of the data sets is relatively straightforward, and the combined data set provides all the relevant information to iden- tify features of the population of interest. However, it is rather common that there is a limited overlap in the units constituting each sample, or that variables that allow identification of units are not available in one or more of the input files, or that one sample provides information at the individual or household level (e.g., survey data) whereas the second sample provides information at a more aggregate level (e.g., administrative data providing information at the precinct or district level).

This empirical problem can be formalized in the following way. Suppose that each member l of a population of interest L has an outcome y_l in

and covariates (x_l, z_l) in a space X × Z. Let the random variables

have distribution Q(y,x,z). Here X is a finite-dimensional real space, and Z is a J-element finite set. The identification problem arises from the fact that the joint realizations of (y,x,z) are not observable. All that the researcher can observe are realizations from two separate sampling processes, one that draws persons at random from L and collects realizations of (x,z) but not y, and the other that draws persons at random from L and collects realizations of (y,x) but not z. Abstracting from statistical considerations, these sampling processes reveal the conditional probabilities Pr(z = j|x), j ∈ Z, and the conditional distributions P(y|x), but not the conditional distributions P(y|x,z = j), j ∈ Z. Features of these conditional distributions, for example, conditional mean, conditional quantiles, and so on, are the objects that the researcher would like to identify and estimate; the empirical evidence alone, however, does not allow for point identification and point estimation.

The literature on statistical matching has aimed at using the common variable(s) x as a bridge to create synthetic records containing (y,x,z). As Sims (1972) pointed out,¹

the inherent assumption at the base of statistical matching is that, conditional on x, y and z are independent. This conditional independence assumption is very strong and untestable. Although it does guarantee point identification of features of the conditional distributions P(y|x,z = j), j ∈ Z, it often finds very little justification in practice. Cross and Manski (2002) have recently proposed an alternative approach to the problem of data combination when exact matching is not possible. They suggested a method to learn features of the conditional expectations E(y|x,·) ≡ [E(y|x,z = j), j ∈ Z], using the empirical evidence alone and without maintaining any untestable assumption.²

Cross and Manski (2002) showed that the vector of conditional expectations E(y|x,·) belongs to an identification region (i.e., a set of values of E(y|x,·) that are feasible given the maintained assumptions and the empirical evidence) given by a bounded and convex set. They derived the extreme points of such an identification region, showing that they are the expectations of certain J-vectors of distributions. When the support of P(y|x) is finite or J = 2, Cross and Manski were able to characterize the identification region fully, as the convex hull of its extreme points. However, for the case in which P(y|x) has infinite support and J ≥ 3, Cross and Manski could not fully characterize the topology of the identification region (they did show, however, that the identification region contains the convex hull of its extreme points and is contained in another convex polytope). This implies that for empirical problems in which the variable y is continuous (e.g., y is given by capital gains, income, etc.), the results of Cross and Manski provide the extreme points of the identification region but are not conclusive as to whether the identification region itself is given by the convex hull of these extreme points.

The purpose of this paper is to fully characterize the topology of the identification region in Cross and Manski (2002). A full characterization allows empirical researchers to precisely calculate the identification region of interest and to estimate it when only a finite sample is available by replacing population parameters with sample analogs. We show that the identification region is given exactly by the convex hull of its extreme points irrespective of the support of P(y|x). The paper is organized as follows. Section 2 formally states the problem; Proposition 1 presents the main result of the paper, showing that the identification region of E(y|x,·) is closed even when P(y|x) has infinite support and J ≥ 3. Corollary 2 uses well-known results of convex analysis to argue that compactness of the identification region (established through Proposition 1) implies that the identification region is, in fact, given by the convex hull of its extreme points.