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

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  • Rainer Winkelmann

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.

Suggested Citation

  • Rainer Winkelmann, 2023. "Neglected heterogeneity, Simpson’s paradox, and the anatomy of least squares," ECON - Working Papers 426, Department of Economics - University of Zurich, revised Jul 2023.
    • Handle: RePEc:zur:econwp:426
    as

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    File URL: https://www.zora.uzh.ch/id/eprint/229123/13/econwp426.pdf
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    References listed on IDEAS

    as
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    More about this item

    Keywords

    Covariance-weighting; heterogeneity spillover; non-convex average; average treatment effect;
    All these keywords.

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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