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Many Instruments Asymptotic Approximations Under Nonnormal Error Distributions

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  • Hasselt, Martijn van

Abstract

In this paper we derive an alternative asymptotic approximation to the sampling distribution of the limited information maximum likelihood estimator and a bias-corrected version of the two-stage least squares estimator. The approximation is obtained by allowing the number of instruments and the concentration parameter to grow at the same rate as the sample size. More specifically, we allow for potentially nonnormal error distributions and obtain the conventional asymptotic distribution and the results of Bekker (1994, Econometrica 62, 657–681) and Bekker and Van der Ploeg (2005, Statistica Neerlandica 59, 139–267) as special cases. The results show that when the error distribution is not normal, in general both the properties of the instruments and the third and fourth moments of the errors affect the asymptotic variance. We compare our findings with those in the recent literature on many and weak instruments.

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  • Hasselt, Martijn van, 2010. "Many Instruments Asymptotic Approximations Under Nonnormal Error Distributions," Econometric Theory, Cambridge University Press, vol. 26(2), pages 633-645, April.
    • Handle: RePEc:cup:etheor:v:26:y:2010:i:02:p:633-645_10
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    Cited by:

    1. Liu, Xiaodong & Lee, Lung-fei, 2010. "GMM estimation of social interaction models with centrality," Journal of Econometrics, Elsevier, vol. 159(1), pages 99-115, November.
    2. Winkelried, D. & Smith, R.J., 2011. "Principal Components Instrumental Variable Estimation," Cambridge Working Papers in Economics 1119, Faculty of Economics, University of Cambridge.
    3. Eric Gautier & Christiern Rose, 2022. "Fast, Robust Inference for Linear Instrumental Variables Models using Self-Normalized Moments," Papers 2211.02249, arXiv.org, revised Nov 2022.
    4. Bekker, Paul A. & Crudu, Federico, 2015. "Jackknife instrumental variable estimation with heteroskedasticity," Journal of Econometrics, Elsevier, vol. 185(2), pages 332-342.
    5. Crudu, Federico & Mellace, Giovanni & Sándor, Zsolt, 2021. "Inference In Instrumental Variable Models With Heteroskedasticity And Many Instruments," Econometric Theory, Cambridge University Press, vol. 37(2), pages 281-310, April.
    6. Abutaliev, Albert & Anatolyev, Stanislav, 2013. "Asymptotic variance under many instruments: Numerical computations," Economics Letters, Elsevier, vol. 118(2), pages 272-274.
    7. Anatolyev, Stanislav & Smirnov, Maksim, 2024. "Off-diagonal elements of projection matrices and dimension asymptotics," Economics Letters, Elsevier, vol. 239(C).
    8. Guy Tchuente, 2021. "A Note on the Topology of the First Stage of 2SLS with Many Instruments," Papers 2106.15003, arXiv.org.
    9. Kolesár, Michal, 2018. "Minimum distance approach to inference with many instruments," Journal of Econometrics, Elsevier, vol. 204(1), pages 86-100.
    10. Matsushita, Yukitoshi & Otsu, Taisuke, 2023. "Second-order refinements for t-ratios with many instruments," Journal of Econometrics, Elsevier, vol. 232(2), pages 346-366.
    11. Lee, Yoonseok & Okui, Ryo, 2012. "Hahn–Hausman test as a specification test," Journal of Econometrics, Elsevier, vol. 167(1), pages 133-139.
    12. Michal Kolesár & Raj Chetty & John Friedman & Edward Glaeser & Guido W. Imbens, 2015. "Identification and Inference With Many Invalid Instruments," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(4), pages 474-484, October.
    13. Byunghoon Kang, 2018. "Higher Order Approximation of IV Estimators with Invalid Instruments," Working Papers 257105320, Lancaster University Management School, Economics Department.
    14. Wang, Wenjie & Kaffo, Maximilien, 2016. "Bootstrap inference for instrumental variable models with many weak instruments," Journal of Econometrics, Elsevier, vol. 192(1), pages 231-268.
    15. Stanislav Anatolyev, 2013. "Instrumental variables estimation and inference in the presence of many exogenous regressors," Econometrics Journal, Royal Economic Society, vol. 16(1), pages 27-72, February.
    16. Yoonseok Lee & Yu Zhou, 2015. "Averaged Instrumental Variables Estimators," Center for Policy Research Working Papers 180, Center for Policy Research, Maxwell School, Syracuse University.
    17. Guy Tchuente, 2016. "Estimation of social interaction models using regularization," Studies in Economics 1607, School of Economics, University of Kent.
    18. Yukitoshi Matsushita & Taisuke Otsu, 2020. "Second-order refinements for t-ratios with many instruments," STICERD - Econometrics Paper Series 612, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    19. Guy Tchuente, 2019. "Weak Identification and Estimation of Social Interaction Models," Papers 1902.06143, arXiv.org.
    20. Matsushita, Yukitoshi & Otsu, Taisuke, 2023. "Second-order refinements for t-ratios with many instruments," LSE Research Online Documents on Economics 111065, London School of Economics and Political Science, LSE Library.
    21. Liu, Xiaodong, 2012. "On the consistency of the LIML estimator of a spatial autoregressive model with many instruments," Economics Letters, Elsevier, vol. 116(3), pages 472-475.

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