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Finite-Sample Instrumental Variables Inference using an Asymptotically Pivotal Statistic

Author

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  • Paul A. Bekker

    (University of Groningen)

  • Frank Kleibergen

    (University of Amsterdam)

Abstract

The paper considers the K-statistic, Kleibergen’s (2000) adaptation ofthe Anderson-Rubin (AR) statistic in instrumental variables regression.Compared to the AR-statistic this K-statistic shows improvedasymptotic efficiency in terms of degrees of freedom in overidentifiedmodels and yet it shares, asymptotically, the pivotal property of theAR statistic. That is, asymptotically it has a chi-square distributionwhether or not the model is identified. This pivotal property is veryrelevant for size distortions in finite-sample tests. Whereas Kleibergen(2000) focuses especially on the asymptotic behavior of the statistic,the present paper concentrates on finite-sample properties in a Gaussianframework. In that case the AR statistic has an F-distribution.However, the K-statistic is not exactly pivotal. Its finite-sample distributionis affected by nuisance parameters. Here we consider the twoextreme cases, which provide tight bounds for the exact distribution.The first case amounts to perfect identification—which is similar tothe asymptotic case—where the statistic has an F-distribution. Inthe other extreme case there is total underidentification. For the lattercase we show how to compute the exact distribution. Thus weprovide tight bounds for exact confidence sets based on the efficientK-statistic. Asymptotically the two bounds converge, except whenthere is a large number of redundant instruments. This paper has resulted in a publication in Econometric Theory , 2003, 19, 744-53.

Suggested Citation

  • Paul A. Bekker & Frank Kleibergen, 2001. "Finite-Sample Instrumental Variables Inference using an Asymptotically Pivotal Statistic," Tinbergen Institute Discussion Papers 01-055/4, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20010055
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    References listed on IDEAS

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    1. Zivot, Eric & Startz, Richard & Nelson, Charles R, 1998. "Valid Confidence Intervals and Inference in the Presence of Weak Instruments," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 1119-1146, November.
    2. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-681, May.
    3. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
    4. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, vol. 70(5), pages 1781-1803, September.
    5. Nelson, Charles R & Startz, Richard, 1990. "Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 58(4), pages 967-976, July.
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    Citations

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    Cited by:

    1. Whitney K. Newey & Frank Windmeijer, 2005. "GMM with many weak moment conditions," CeMMAP working papers CWP18/05, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Frank Kleibergen, 2004. "Expansions of GMM statistics that indicate their properties under weak and/or many instruments and the bootstrap," Econometric Society 2004 North American Summer Meetings 408, Econometric Society.
    3. Johannes W. Ligtenberg, 2023. "Inference in clustered IV models with many and weak instruments," Papers 2306.08559, arXiv.org, revised Oct 2025.
    4. Tom Boot & Didier Nibbering, 2024. "Inference on LATEs with covariates," Papers 2402.12607, arXiv.org, revised Dec 2024.
    5. Frank Kleibergen & Lingwei Kong & Zhaoguo Zhan, 2023. "Identification Robust Testing of Risk Premia in Finite Samples," Journal of Financial Econometrics, Oxford University Press, vol. 21(2), pages 263-297.
    6. D. S. Poskitt & C. L. Skeels, 2005. "Small Concentration Asymptotics and Instrumental Variables Inference," Monash Econometrics and Business Statistics Working Papers 4/05, Monash University, Department of Econometrics and Business Statistics.
    7. Elise Coudin & Jean-Marie Dufour, 2010. "Finite and Large Sample Distribution-Free Inference in Median Regressions with Instrumental Variables," Working Papers 2010-56, Center for Research in Economics and Statistics.
    8. Jean-Marie Dufour, 2003. "Identification, weak instruments, and statistical inference in econometrics," Canadian Journal of Economics, Canadian Economics Association, vol. 36(4), pages 767-808, November.
    9. Jean‐Marie Dufour, 2003. "Identification, weak instruments, and statistical inference in econometrics," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 36(4), pages 767-808, November.
    10. Bekker, Paul A. & Lawford, Steve, 2008. "Symmetry-based inference in an instrumental variable setting," Journal of Econometrics, Elsevier, vol. 142(1), pages 28-49, January.
    11. Phillips, Peter C.B. & Gao, Wayne Yuan, 2017. "Structural inference from reduced forms with many instruments," Journal of Econometrics, Elsevier, vol. 199(2), pages 96-116.
    12. Dufour, Jean-Marie & Taamouti, Mohamed, 2007. "Further results on projection-based inference in IV regressions with weak, collinear or missing instruments," Journal of Econometrics, Elsevier, vol. 139(1), pages 133-153, July.
    13. Tom Boot & Johannes W. Ligtenberg, 2023. "Identification- and many moment-robust inference via invariant moment conditions," Papers 2303.07822, arXiv.org, revised Oct 2025.
    14. Federico Crudu & Giovanni Mellace & Zsolt S'andor, 2026. "Jackknife Instrumental Variable Inference," Papers 2604.15437, arXiv.org.
    15. Boot, Tom & Ligtenberg, Johannes W., 2025. "Identification- and many moment-robust inference via invariant moment conditions," Journal of Econometrics, Elsevier, vol. 252(PA).

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