Regressions, Short and Long

We study the problem of identi cation of the long regression E(y jx; z) when the short conditional distributions P (y jx) and P (z jx) are known but the long conditional distribution P (y jx; z) is not known. This problem often arises when a researcher utilizes data from two separate data sets. (A leading example is the ecological inference problem of political science, where voting behavior across electoral districts is observed from administrative records, the demographic composition of voters within a district is observed from census data, and the researcher wants to infer voting behavior conditional on district and demographic attributes.) We isolate an identi cation region containing feasible values of the long regression, and show that this region forms a sharp bound on the long regression. The identi cation region can be calculated precisely when y has nite support. When y has in nite support we characterize two sets, one that contains the identi cation region, and one that is contained by it. Following this completely nonparametric analysis, we examine the identifying power yielded by exclusion restrictions across distinct covariate values. Such restrictions cause the identi cation region to shrink, in many cases to a single point. To illustrate the theory, we pose and address this hypothetical question: What would be the outcome if the 1996 U.S. presidential election were re-enacted in a population of di erent demographic composition, ceteris paribus? We have bene tted from the opportunity to present this research in seminars at Northwestern University and the University of Wisconsin Madison. This research was supported in part by National Science Foundation Grant SBR-9722846.