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Neglected heterogeneity, Simpson’s paradox, and the anatomy of least squares


Winkelmann, Rainer (2023). Neglected heterogeneity, Simpson’s paradox, and the anatomy of least squares. Working paper series / Department of Economics 426, University of Zurich.

Abstract

When a sample combines data from two or more groups, multivariate regression yields a matrix-weighted average of the group-specific coefficient vectors. However, it is possible that the weighted average of a specific coefficient falls outside the range of the group-specific coefficients, and it may even have a different sign compared to both group-level coefficients, a manifestation of Simpson's paradox. The result of the combined regression is then prone to misinterpretation. The purpose of this paper is to raise awareness of this problem and to state conditions under which such non-convex weighting or sign reversal can arise, for a model with two regressors and two groups. Two illustrative examples, an investment equation estimated with panel data, and a cross-sectional earnings equation for men and women, highlight the relevance of these findings for applied work.

Abstract

When a sample combines data from two or more groups, multivariate regression yields a matrix-weighted average of the group-specific coefficient vectors. However, it is possible that the weighted average of a specific coefficient falls outside the range of the group-specific coefficients, and it may even have a different sign compared to both group-level coefficients, a manifestation of Simpson's paradox. The result of the combined regression is then prone to misinterpretation. The purpose of this paper is to raise awareness of this problem and to state conditions under which such non-convex weighting or sign reversal can arise, for a model with two regressors and two groups. Two illustrative examples, an investment equation estimated with panel data, and a cross-sectional earnings equation for men and women, highlight the relevance of these findings for applied work.

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Additional indexing

Item Type:Working Paper
Communities & Collections:03 Faculty of Economics > Department of Economics
Working Paper Series > Department of Economics
Dewey Decimal Classification:330 Economics
JEL Classification:C21
Uncontrolled Keywords:Covariance-weighting, heterogeneity spillover, non-convex average, average treatment effect
Scope:Discipline-based scholarship (basic research)
Language:English
Date:July 2023
Deposited On:31 Jan 2023 11:53
Last Modified:09 Feb 2024 05:30
Series Name:Working paper series / Department of Economics
Number of Pages:20
ISSN:1664-7041
Additional Information:Revised version ; Former title: Neglected heterogeneity and the algebra of least squares
OA Status:Green
  • Content: Updated Version
  • Language: English
  • Permission: Download for registered users
  • Description: Revised version May 2023
  • Content: Updated Version
  • Language: English
  • Description: Revised version July 2023