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Two-Stage Least Squares as Minimum Distance

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  • Frank Windmeijer

Abstract

The Two-Stage Least Squares instrumental variables (IV) estimator for the parameters in linear models with a single endogenous variable is shown to be identical to an optimal Minimum Distance (MD) estimator based on the individual instrument specific IV estimators. The 2SLS estimator is a linear combination of the individual estimators, with the weights determined by their variances and covariances under conditional homoskedasticity. It is further shown that the Sargan test statistic for overidentifying restrictions is the same as the MD criterion test statistic. This provides an intuitive interpretation of the Sargan test. The equivalence results also apply to the efficient two-step GMM and robust optimal MD estimators and criterion functions, allowing for general forms of heteroskedasticity. It is further shown how these results extend to the linear overidentified IV model with multiple endogenous variables.

Suggested Citation

  • Frank Windmeijer, 2017. "Two-Stage Least Squares as Minimum Distance," Bristol Economics Discussion Papers 17/683, School of Economics, University of Bristol, UK, revised 13 Jun 2018.
    • Handle: RePEc:bri:uobdis:17/683
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    References listed on IDEAS

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    1. Frank Windmeijer, 2018. "Testing Over- and Underidentification in Linear Models, with Applications to Dynamic Panel Data and Asset-Pricing Models," Bristol Economics Discussion Papers 18/696, School of Economics, University of Bristol, UK.
    2. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    3. Goldberger, Arthur S & Olkin, Ingram, 1971. "A Minimum-Distance Interpretation of Limited-Information Estimation," Econometrica, Econometric Society, vol. 39(3), pages 635-639, May.
    4. Chen, Xiaohong & Jacho-Chávez, David T. & Linton, Oliver, 2016. "Averaging Of An Increasing Number Of Moment Condition Estimators," Econometric Theory, Cambridge University Press, vol. 32(1), pages 30-70, February.
    5. Parente, Paulo M.D.C. & Santos Silva, J.M.C., 2012. "A cautionary note on tests of overidentifying restrictions," Economics Letters, Elsevier, vol. 115(2), pages 314-317.
    6. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769.
    7. Angrist, Joshua D., 1991. "Grouped-data estimation and testing in simple labor-supply models," Journal of Econometrics, Elsevier, vol. 47(2-3), pages 243-266, February.
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    Citations

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    Cited by:

    1. David T. Frazier & Eric Renault & Lina Zhang & Xueyan Zhao, 2020. "Weak Identification in Discrete Choice Models," Papers 2011.06753, arXiv.org, revised Jan 2021.
    2. Gibbons, Stephen & Heblich, Stephan & Pinchbeck, Edward W., 2024. "The spatial impacts of a massive rail disinvestment program: The Beeching Axe," Journal of Urban Economics, Elsevier, vol. 143(C).
    3. Frank Windmeijer, 2022. "Weak Instruments, First-Stage Heteroskedasticity, the Robust F-Test and a GMM Estimator with the Weight Matrix Based on First-Stage Residuals," Papers 2208.01967, arXiv.org.
    4. Kiviet, Jan F., 2020. "Testing the impossible: Identifying exclusion restrictions," Journal of Econometrics, Elsevier, vol. 218(2), pages 294-316.
    5. Windmeijer, Frank, 2025. "The robust F-statistic as a test for weak instruments," Journal of Econometrics, Elsevier, vol. 247(C).
    6. Frank Windmeijer & Xiaoran Liang & Fernando P. Hartwig & Jack Bowden, 2021. "The confidence interval method for selecting valid instrumental variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(4), pages 752-776, September.
    7. Nicolas Apfel & Frank Windmeijer, 2022. "The Falsification Adaptive Set in Linear Models with Instrumental Variables that Violate the Exclusion or Conditional Exogeneity Restriction," Papers 2212.04814, arXiv.org, revised Apr 2024.
    8. Mogstad, Magne & Torgovitsky, Alexander, 2024. "Instrumental variables with unobserved heterogeneity in treatment effects," Handbook of Labor Economics,, Elsevier.
    9. Frank Windmeijer, 2019. "Weak Instruments, First-Stage Heteroskedasticity and the Robust F-test," Bristol Economics Discussion Papers 19/708, School of Economics, University of Bristol, UK.
    10. Gibbons, Stephen & Heblich, Stephan & Pinchbeck, Edward W., 2024. "The spatial impacts of a massive rail disinvestment program: the Beeching axe," LSE Research Online Documents on Economics 124531, London School of Economics and Political Science, LSE Library.
    11. Domguia, Edmond Noubissi & Ngounou, Borice Augustin & Pondie, Thierry Messie & Bitoto, Fabrice Ewolo, 2024. "Environmental tax and energy poverty: An economic approach for an environmental and social solution," Energy, Elsevier, vol. 308(C).

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    More about this item

    Keywords

    Instrumental Variables; Two-Stage Least Squares; Minimum Distance; Overidentification Test.;
    All these keywords.

    JEL classification:

    • C26 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Instrumental Variables (IV) Estimation
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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