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Regression as Best Linear Prediction: The Case of Discrete Regressors

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  • Winkelmann Rainer

    (Department of Economics, 27217 University of Zurich , Zurich, Switzerland)

Abstract

This paper examines the properties of the ordinary least squares (OLS) estimator when applied to a model with a non-linear relationship between outcome and a discrete regressor. I investigate what parameters OLS estimates in such a case, focusing on both level and incremental effects. The analysis reveals that the OLS estimand is a convex average of incremental effects, but weights can be negative for level effects and in the presence of neglected heterogeneity. An empirical application to a wage equation demonstrates these issues, highlighting the importance of using unrestricted models or carefully considering the limitations of OLS estimates in similar situations.

Suggested Citation

  • Winkelmann Rainer, 2025. "Regression as Best Linear Prediction: The Case of Discrete Regressors," Journal of Econometric Methods, De Gruyter, vol. 14(2), pages 59-69.
    • Handle: RePEc:bpj:jecome:v:14:y:2025:i:2:p:59-69:n:1002
    DOI: 10.1515/jem-2025-0016
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    References listed on IDEAS

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    1. Kirill Borusyak & Xavier Jaravel & Jann Spiess, 2024. "Revisiting Event-Study Designs: Robust and Efficient Estimation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 91(6), pages 3253-3285.
    2. Winkelmann Rainer, 2024. "Neglected Heterogeneity, Simpson’s Paradox, and the Anatomy of Least Squares," Journal of Econometric Methods, De Gruyter, vol. 13(1), pages 131-144, January.
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    4. Brantly Callaway & Andrew Goodman-Bacon & Pedro H. C. Sant'Anna, 2021. "Difference-in-Differences with a Continuous Treatment," Papers 2107.02637, arXiv.org, revised Dec 2025.
    5. Yitzhaki, Shlomo, 1996. "On Using Linear Regressions in Welfare Economics," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(4), pages 478-486, October.
    6. Joshua D. Angrist, 1998. "Estimating the Labor Market Impact of Voluntary Military Service Using Social Security Data on Military Applicants," Econometrica, Econometric Society, vol. 66(2), pages 249-288, March.
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    Keywords

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    JEL classification:

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

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