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Regressions, Short and Long

Author

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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?

Suggested Citation

  • Philip Cross & Charles F. Manski, 2000. "Regressions, Short and Long," Econometric Society World Congress 2000 Contributed Papers 0385, Econometric Society.
  • Handle: RePEc:ecm:wc2000:0385
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    Cited by:

    1. Francesca Molinari, 2020. "Microeconometrics with Partial Identi?cation," CeMMAP working papers CWP15/20, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Tatiana Komarova & Denis Nekipelov & Evgeny Yakovlev, 2018. "Identification, data combination, and the risk of disclosure," Quantitative Economics, Econometric Society, vol. 9(1), pages 395-440, March.
    3. 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.
    4. Cheti Nicoletti, 2010. "Poverty analysis with missing data: alternative estimators compared," Empirical Economics, Springer, vol. 38(1), pages 1-22, February.
    5. Matthew A. Masten & Alexandre Poirier, 2018. "Identification of Treatment Effects Under Conditional Partial Independence," Econometrica, Econometric Society, vol. 86(1), pages 317-351, January.
    6. Sung Jae Jun & Sokbae Lee, 2023. "Identifying the Effect of Persuasion," Journal of Political Economy, University of Chicago Press, vol. 131(8), pages 2032-2058.
    7. Xavier D'Haultfoeuille & Christophe Gaillac & Arnaud Maurel, 2021. "Rationalizing rational expectations: Characterizations and tests," Quantitative Economics, Econometric Society, vol. 12(3), pages 817-842, July.
    8. Charles F. Manski & John Mullahy & Atheendar S. Venkataramani, 2025. "Prediction with Differential Covariate Classification: Illustrated by Racial/Ethnic Classification in Medical Risk Assessment," Papers 2501.02318, arXiv.org.
    9. Francesca Molinari, 2019. "Econometrics with Partial Identification," CeMMAP working papers CWP25/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    10. 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.
    11. D'Haultfoeuille, Xavier & Gaillac, Christophe & Maurel, Arnaud, 2024. "Linear Regressions with Combined Data," TSE Working Papers 24-1602, Toulouse School of Economics (TSE).
    12. Xavier D’Haultfoeuille & Christophe Gaillac & Arnaud Maurel, 2025. "Linear Regressions with Combined Data," Working Papers 2025-04, Center for Research in Economics and Statistics.
    13. Charles F. Manski, 2003. "Identification Problems in the Social Sciences and Everyday Life," Southern Economic Journal, John Wiley & Sons, vol. 70(1), pages 11-21, July.
    14. Molinari, Francesca & Peski, Marcin, 2006. "Generalization Of A Result On “Regressions, Short And Long”," Econometric Theory, Cambridge University Press, vol. 22(1), pages 159-163, February.
    15. Sarah Moon, 2024. "Partial Identification of Individual-Level Parameters Using Aggregate Data in a Nonparametric Model," Papers 2403.07236, arXiv.org, revised Feb 2025.
    16. Thomas F. Crossley & Peter Levell & Stavros Poupakis, 2022. "Regression with an imputed dependent variable," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(7), pages 1277-1294, November.
    17. Yanqin Fan & Carlos A. Manzanares, 2017. "Partial identification of average treatment effects on the treated through difference-in-differences," Econometric Reviews, Taylor & Francis Journals, vol. 36(6-9), pages 1057-1080, October.
    18. Peter Sandholt Jensen & Allan H. Würtz, 2006. "On determining the importance of a regressor with small and undersized samples," Economics Working Papers 2006-08, Department of Economics and Business Economics, Aarhus University.
    19. Nathan Kallus, 2022. "What's the Harm? Sharp Bounds on the Fraction Negatively Affected by Treatment," Papers 2205.10327, arXiv.org, revised Nov 2022.
    20. Matthew Masten & Alexandre Poirier, 2016. "Partial independence in nonseparable models," CeMMAP working papers 26/16, Institute for Fiscal Studies.
    21. 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.
    22. Obradović, Filip, 2024. "Measuring diagnostic test performance using imperfect reference tests: A partial identification approach," Journal of Econometrics, Elsevier, vol. 244(1).
    23. Fan, Yanqin & Shi, Xuetao & Tao, Jing, 2023. "Partial identification and inference in moment models with incomplete data," Journal of Econometrics, Elsevier, vol. 235(2), pages 418-443.
    24. Dang, Hai-Anh & Lanjouw, Peter & Luoto, Jill & McKenzie, David, 2014. "Using repeated cross-sections to explore movements into and out of poverty," Journal of Development Economics, Elsevier, vol. 107(C), pages 112-128.
    25. Borowczyk-Martins, Daniel & Pacini, David, 2024. "Measuring labor market transitions with time series of cross sections," Economics Letters, Elsevier, vol. 237(C).

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