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Regressions, Short and Long

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  • Philip Cross

    (Georgetown University)

  • Charles F. Manski

    (Northwestern University)

Abstract

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?

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Bibliographic Info

Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 0385.

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Date of creation: 01 Aug 2000
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Handle: RePEc:ecm:wc2000:0385

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  1. Charles F. Manski & John V. Pepper, 2000. "Monotone Instrumental Variables, with an Application to the Returns to Schooling," Econometrica, Econometric Society, vol. 68(4), pages 997-1012, July.
  2. A. P. Thirlwall, 1983. "Introduction," Journal of Post Keynesian Economics, M.E. Sharpe, Inc., vol. 5(3), pages 341-344, April.
  3. A. Meltzer & Peter Ordeshook & Thomas Romer, 1983. "Introduction," Public Choice, Springer, vol. 41(1), pages 1-5, January.
  4. 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.
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Cited by:
  1. Galichon, Alfred & Henry, Marc, 2009. "A test of non-identifying restrictions and confidence regions for partially identified parameters," Journal of Econometrics, Elsevier, vol. 152(2), pages 186-196, October.
  2. Mullin, Charles H., 2006. "Identification and estimation with contaminated data: When do covariate data sharpen inference?," Journal of Econometrics, Elsevier, vol. 130(2), pages 253-272, February.
  3. Dang, Hai-Anh & Lanjouw, Peter & Luoto, Jill & McKenzie, David, 2011. "Using repeated cross-sections to explore movements in and out of poverty," Policy Research Working Paper Series 5550, The World Bank.
  4. Peter Sandholt Jensen & Allan H. W├╝rtz, 2006. "On determining the importance of a regressor with small and undersized samples," Economics Working Papers 2006-08, School of Economics and Management, University of Aarhus.
  5. Timothy G. Conley & Giorgio Topa, 2003. "Identification of local interaction models with imperfect location data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(5), pages 605-618.

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