Regressions, Short and Long
We study the problem of identification of the long regression E(y|x,z) when the short conditional distributions P(y|x) and P(z|x) are known but the long conditional distribution P(y|x,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 identification region containing feasible values of the long regression, and show that this region forms a sharp bound on the long regression. The identification region can be calculated precisely when y has finite support. When y has infinite support we characterize two sets, one that contains the identification 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 identification 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 different demographic composition, ceteris paribus?
|Date of creation:||01 Aug 2000|
|Date of revision:|
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- Horowitz, Joel L & Manski, Charles F, 1995. "Identification and Robustness with Contaminated and Corrupted Data," Econometrica, Econometric Society, vol. 63(2), pages 281-302, March.
- A. Meltzer & Peter Ordeshook & Thomas Romer, 1983. "Introduction," Public Choice, Springer, vol. 41(1), pages 1-5, January.
- Charles F. Manski & John V. Pepper, 2000.
"Monotone Instrumental Variables, with an Application to the Returns to Schooling,"
Econometric Society, vol. 68(4), pages 997-1012, July.
- Charles F. Manski & John V. Pepper, 1998. "Monotone Instrumental Variables: With an Application to the Returns to Schooling," Virginia Economics Online Papers 308, University of Virginia, Department of Economics.
- Charles F. Manski & John V. Pepper, 1998. "Monotone Instrumental Variables with an Application to the Returns to Schooling," NBER Technical Working Papers 0224, National Bureau of Economic Research, Inc.
- A. P. Thirlwall, 1983. "Introduction," Journal of Post Keynesian Economics, M.E. Sharpe, Inc., vol. 5(3), pages 341-344, April.
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