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Instrumental variables estimation and inference in the presence of many exogenous regressors

Author

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  • Stanislav Anatolyev

Abstract

We consider a standard instrumental variables model contaminated by the presence of a large number of exogenous regressors. In an asymptotic framework where this number is proportional to the sample size, we study the impact of their ratio on the validity of existing estimators and tests. When the instruments are few, the inference using the conventional 2SLS estimator and associated t and J statistics, as well as the Anderson-Rubin and Kleibergen tests, is still valid. When the instruments are many, the LIML estimator remains consistent, but the presence of many exogenous regressors changes its asymptotic variance. Moreover, the conventional bias correction of the 2SLS estimator is no longer appropriate. We provide asymptotically correct versions of bias correction for the 2SLS estimator, derive its asymptotically correct variance estimator, extend the Hansen-Hausman-Newey LIML variance estimator to the case of many exogenous regressors, and propose asymptotically valid modi cations of the J overidenti cation tests based on the LIML and bias corrected 2SLS estimators.
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Suggested Citation

  • 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.
    • Handle: RePEc:wly:emjrnl:v:16:y:2013:i:1:p:27-72
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    Cited by:

    1. Alyssa G. Anderson & Wenxin Du & Bernd Schlusche, 2021. "Arbitrage Capital of Global Banks," Finance and Economics Discussion Series 2021-032, Board of Governors of the Federal Reserve System (U.S.).
    2. Mattia Filomena & Matteo Picchio & Alessia Lo Turco, 2024. "Trade exposure, immigrants and workplace injuries," Working Papers 488, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
    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. Tiziano Arduini & Eleonora Patacchini & Edoardo Rainone, 2014. "Identification and Estimation of Outcome Response with Heterogeneous Treatment Externalities," EIEF Working Papers Series 1407, Einaudi Institute for Economics and Finance (EIEF), revised Sep 2014.
    5. Daniel A. Broxterman & William D. Larson, 2020. "An empirical examination of shift‐share instruments," Journal of Regional Science, Wiley Blackwell, vol. 60(4), pages 677-711, September.
    6. Anatolyev, Stanislav & Sølvsten, Mikkel, 2023. "Testing many restrictions under heteroskedasticity," Journal of Econometrics, Elsevier, vol. 236(1).
    7. 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.
    8. Hongwei Shi & Xinyu Zhang & Xu Guo & Baihua He & Chenyang Wang, 2025. "Testing overidentifying restrictions on high-dimensional instruments and covariates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 77(2), pages 331-352, April.
    9. Zhenhong Huang & Chen Wang & Jianfeng Yao, 2023. "A specification test for the strength of instrumental variables," Papers 2302.14396, arXiv.org.
    10. Kirill S. Evdokimov & Michal Kolesár, 2018. "Inference in Instrumental Variable Regression Analysis with Heterogeneous Treatment Effects," Working Papers 2018-16, Princeton University. Economics Department..
    11. Muhammad Qasim, 2024. "A weighted average limited information maximum likelihood estimator," Statistical Papers, Springer, vol. 65(5), pages 2641-2666, July.
    12. Kolesár, Michal, 2018. "Minimum distance approach to inference with many instruments," Journal of Econometrics, Elsevier, vol. 204(1), pages 86-100.
    13. Paul Goldsmith-Pinkham & Isaac Sorkin & Henry Swift, 2020. "Bartik Instruments: What, When, Why, and How," American Economic Review, American Economic Association, vol. 110(8), pages 2586-2624, August.
    14. Eugenio Levi & Isabelle Sin & Steven Stillman, 2024. "The lasting impact of external shocks on political opinions and populist voting," Economic Inquiry, Western Economic Association International, vol. 62(1), pages 349-374, January.
    15. Helmut Farbmacher & Rebecca Groh & Michael Muhlegger & Gabriel Vollert, 2024. "Revisiting the Many Instruments Problem using Random Matrix Theory," Papers 2408.08580, arXiv.org, revised Aug 2025.
    16. Zou, Yanchi & Li, Kun & He, Lilong & Guo, Jiapei, 2025. "The employment effects of ICT investment: Evidence from the U.S. Commuting Zones," Economic Modelling, Elsevier, vol. 151(C).

    More about this item

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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