IDEAS home Printed from https://ideas.repec.org/a/bpj/jecome/v13y2024i1p131-144n7.html
   My bibliography  Save this article

Neglected Heterogeneity, Simpson’s Paradox, and the Anatomy of Least Squares

Author

Listed:
  • Winkelmann Rainer

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

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

  • 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.
    • Handle: RePEc:bpj:jecome:v:13:y:2024:i:1:p:131-144:n:7
    DOI: 10.1515/jem-2023-0028
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/jem-2023-0028
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/jem-2023-0028?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Breitung, Jörg & Salish, Nazarii, 2021. "Estimation of heterogeneous panels with systematic slope variations," Journal of Econometrics, Elsevier, vol. 220(2), pages 399-415.
    2. Graham, Bryan S. & Pinto, Cristine Campos de Xavier, 2022. "Semiparametrically efficient estimation of the average linear regression function," Journal of Econometrics, Elsevier, vol. 226(1), pages 115-138.
    3. Judea Pearl, 2014. "Comment: Understanding Simpson's Paradox," The American Statistician, Taylor & Francis Journals, vol. 68(1), pages 8-13, February.
    4. Philip Oreopoulos, 2006. "Estimating Average and Local Average Treatment Effects of Education when Compulsory Schooling Laws Really Matter," American Economic Review, American Economic Association, vol. 96(1), pages 152-175, March.
    5. Paul Goldsmith-Pinkham & Peter Hull & Michal Kolesár, 2024. "Contamination Bias in Linear Regressions," American Economic Review, American Economic Association, vol. 114(12), pages 4015-4051, December.
    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.
    7. Guido W. Imbens & Jeffrey M. Wooldridge, 2009. "Recent Developments in the Econometrics of Program Evaluation," Journal of Economic Literature, American Economic Association, vol. 47(1), pages 5-86, March.
    8. 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.
    9. Murillo Campello & Antonio F. Galvao & Ted Juhl, 2019. "Testing for Slope Heterogeneity Bias in Panel Data Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(4), pages 749-760, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Julius Schaper, 2025. "Residualised Treatment Intensity and the Estimation of Average Partial Effects," Papers 2502.10301, arXiv.org.
    3. Sloczynski, Tymon, 2018. "A General Weighted Average Representation of the Ordinary and Two-Stage Least Squares Estimands," IZA Discussion Papers 11866, Institute of Labor Economics (IZA).
    4. Tymon S{l}oczy'nski, 2018. "Interpreting OLS Estimands When Treatment Effects Are Heterogeneous: Smaller Groups Get Larger Weights," Papers 1810.01576, arXiv.org, revised May 2020.
    5. Tymon Słoczyński, 2022. "Interpreting OLS Estimands When Treatment Effects Are Heterogeneous: Smaller Groups Get Larger Weights," The Review of Economics and Statistics, MIT Press, vol. 104(3), pages 501-509, May.
    6. Jiafeng Chen, 2021. "Nonparametric Treatment Effect Identification in School Choice," Papers 2112.03872, arXiv.org, revised Jul 2025.
    7. Słoczyński, Tymon, 2012. "New Evidence on Linear Regression and Treatment Effect Heterogeneity," MPRA Paper 39524, University Library of Munich, Germany.
    8. W K Newey & S Stouli, 2022. "Heterogeneous coefficients, control variables and identification of multiple treatment effects [Multivalued treatments and decomposition analysis: An application to the WIA program]," Biometrika, Biometrika Trust, vol. 109(3), pages 865-872.
    9. Dirk Czarnitzki & Cindy Lopes-Bento, 2014. "Innovation Subsidies: Does the Funding Source Matter for Innovation Intensity and Performance? Empirical Evidence from Germany," Industry and Innovation, Taylor & Francis Journals, vol. 21(5), pages 380-409, July.
    10. Lihua Lei, 2024. "Causal Interpretation of Regressions With Ranks," Papers 2406.05548, arXiv.org.
    11. Czarnitzki, Dirk & Lopes-Bento, Cindy, 2013. "Value for money? New microeconometric evidence on public R&D grants in Flanders," Research Policy, Elsevier, vol. 42(1), pages 76-89.
    12. Peter Hull & Michal Kolesár & Christopher Walters, 2022. "Labor by design: contributions of David Card, Joshua Angrist, and Guido Imbens," Scandinavian Journal of Economics, Wiley Blackwell, vol. 124(3), pages 603-645, July.
    13. Jeffrey Smith & Arthur Sweetman, 2016. "Viewpoint: Estimating the causal effects of policies and programs," Canadian Journal of Economics, Canadian Economics Association, vol. 49(3), pages 871-905, August.
    14. Tamini, Lota D., 2011. "A nonparametric analysis of the impact of agri-environmental advisory activities on best management practice adoption: A case study of Québec," Ecological Economics, Elsevier, vol. 70(7), pages 1363-1374, May.
    15. Joshua D. Angrist & Jörn-Steffen Pischke, 2017. "Undergraduate Econometrics Instruction: Through Our Classes, Darkly," Journal of Economic Perspectives, American Economic Association, vol. 31(2), pages 125-144, Spring.
    16. Robert A. HartBy & Mirko Moro & J. Elizabeth Roberts, 2017. "Who gained from the introduction of free universal secondary education in England and Wales?," Oxford Economic Papers, Oxford University Press, vol. 69(3), pages 707-733.
    17. Ugur, Mehmet & Trushin, Eshref & Solomon, Edna, 2015. "UK and EU subsidies and private R&D investment: Is there input additionality?," MPRA Paper 68009, University Library of Munich, Germany, revised 16 Nov 2015.
    18. Eric Maurin & Theodora Xenogiani, 2007. "Demand for Education and Labor Market Outcomes: Lessons from the Abolition of Compulsory Conscription in France," Journal of Human Resources, University of Wisconsin Press, vol. 42(4).
    19. Czarnitzki, Dirk & Hanel, Petr & Rosa, Julio Miguel, 2011. "Evaluating the impact of R&D tax credits on innovation: A microeconometric study on Canadian firms," Research Policy, Elsevier, vol. 40(2), pages 217-229, March.
    20. Beck, Mathias & Lopes-Bento, Cindy & Schenker-Wicki, Andrea, 2016. "Radical or incremental: Where does R&D policy hit?," Research Policy, Elsevier, vol. 45(4), pages 869-883.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    JEL classification:

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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bpj:jecome:v:13:y:2024:i:1:p:131-144:n:7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Peter Golla (email available below). General contact details of provider: https://www.degruyterbrill.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.