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Two-stage least squares as minimum distance

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  • Frank Windmeijer

Abstract

SummaryThe two-stage least-squares (2SLS) 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 generalized method of moments 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, 2019. "Two-stage least squares as minimum distance," The Econometrics Journal, Royal Economic Society, vol. 22(1), pages 1-9.
    • Handle: RePEc:oup:emjrnl:v:22:y:2019:i:1:p:1-9.
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    File URL: http://hdl.handle.net/10.1111/ectj.12115
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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. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769.
    6. 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.
    7. 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.
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    Cited by:

    1. Kiviet, Jan F., 2020. "Testing the impossible: Identifying exclusion restrictions," Journal of Econometrics, Elsevier, vol. 218(2), pages 294-316.
    2. David T. Frazier & Eric Renault & Lina Zhang & Xueyan Zhao, 2020. "Weak Identification in Discrete Choice Models," Papers 2011.06753, arXiv.org, revised Jan 2021.
    3. 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.
    4. Nicolas Apfel & Frank Windmeijer, 2022. "The Falsification Adaptive Set in Linear Models with Instrumental Variables that Violate the Exogeneity or Exclusion Restriction," Papers 2212.04814, arXiv.org.
    5. 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.
    6. Frank Windmeijer, 2023. "The Robust F-Statistic as a Test for Weak Instruments," Papers 2309.01637, arXiv.org.
    7. 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.

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    More about this item

    Keywords

    Instrumental variables; Minimum distance; Overidentification test; Two-stage least squares;
    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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