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Finite-Sample Instrumental Variables Inference Using An Asymptotically Pivotal Statistic

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  • Bekker, Paul
  • Kleibergen, Frank

Abstract

We consider the K-statistic, Kleibergen's (2002, Econometrica 70, 1781–1803) adaptation of the Anderson–Rubin (AR) statistic in instrumental variables regression. Whereas Kleibergen (2002) especially analyzes the asymptotic behavior of the statistic, we focus on finite-sample properties in a Gaussian framework. The AR statistic then has an F-distribution. The finite-sample distribution of the K-statistic is, however, affected by nuisance parameters. We consider two extreme cases for the nuisance parameters, which provide tight bounds for the exact distribution. The first case amounts to perfect identification—which is similar to the asymptotic case—where the statistic has an F-distribution. In the other extreme case there is total underidentification. For the latter case we show how to compute the exact distribution. We thus provide tight bounds for exact confidence sets based on the K-statistic. Asymptotically the two bounds converge, except when there is a large number of redundant instruments.The authors' research documented in this paper has been funded by the NWO Vernieuwingsimpuls research grant “Empirical Comparison of Economic Models.”

Suggested Citation

  • Bekker, Paul & Kleibergen, Frank, 2003. "Finite-Sample Instrumental Variables Inference Using An Asymptotically Pivotal Statistic," Econometric Theory, Cambridge University Press, vol. 19(5), pages 744-753, October.
    • Handle: RePEc:cup:etheor:v:19:y:2003:i:05:p:744-753_19
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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 Nov 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.

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