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Inference in Regression Models with Many Regressors

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  • Stanislav Anatolyev

    (New Economic School)

Abstract

We investigate the behavior of various standard and modified F, LR and LM tests in linear homoskedastic regressions, adapting an alternative asymptotic framework where the number of regressors and possibly restrictions grows proportionately to the sample size. When restrictions are not numerous, the rescaled classical test statistics are asymptotically chi-squared irrespective of whether there are many or few regressors. However, when restrictions are numerous, standard asymptotic versions of classical tests are invalid. We propose and analyze asymptotically valid versions of the classical tests, including those that are robust to the numerosity of regressors and restrictions. The local power of all asymptotically valid tests under consideration turns out to be equal. The "exact" F test that appeals to critical values of the F distribution is also asymptotically valid and robust to the numerosity of regressors and restrictions.

Suggested Citation

  • Stanislav Anatolyev, 2009. "Inference in Regression Models with Many Regressors," Working Papers w0125, New Economic School (NES).
    • Handle: RePEc:abo:neswpt:w0125
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    References listed on IDEAS

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    Cited by:

    1. Westerlund, Joakim, 2019. "Testing additive versus interactive effects in fixed-T panels," Economics Letters, Elsevier, vol. 174(C), pages 5-8.
    2. Hongchang Hu & Weifu Hu & Xinxin Yu, 2021. "Pseudo-maximum likelihood estimators in linear regression models with fractional time series," Statistical Papers, Springer, vol. 62(2), pages 639-659, April.
    3. Riccardo D'Adamo, 2018. "Cluster-Robust Standard Errors for Linear Regression Models with Many Controls," Papers 1806.07314, arXiv.org, revised Apr 2019.
    4. Ledoit, Olivier & Wolf, Michael, 2015. "Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 360-384.
    5. Martin, Ian W.R. & Nagel, Stefan, 2022. "Market efficiency in the age of big data," Journal of Financial Economics, Elsevier, vol. 145(1), pages 154-177.
    6. Patrick Kline & Raffaele Saggio & Mikkel Sølvsten, 2020. "Leave‐Out Estimation of Variance Components," Econometrica, Econometric Society, vol. 88(5), pages 1859-1898, September.
    7. Calhoun, Gray, 2011. "Hypothesis testing in linear regression when k/n is large," Journal of Econometrics, Elsevier, vol. 165(2), pages 163-174.
    8. Anatolyev, Stanislav & Mikusheva, Anna, 2022. "Factor models with many assets: Strong factors, weak factors, and the two-pass procedure," Journal of Econometrics, Elsevier, vol. 229(1), pages 103-126.
    9. Anatolyev, Stanislav & Sølvsten, Mikkel, 2023. "Testing many restrictions under heteroskedasticity," Journal of Econometrics, Elsevier, vol. 236(1).
    10. 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.
    11. 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.
    12. Mayer, Alexander, 2022. "On the local power of some tests of strict exogeneity in linear fixed effects models," Econometrics and Statistics, Elsevier, vol. 24(C), pages 49-74.
    13. Zhenhong Huang & Chen Wang & Jianfeng Yao, 2023. "The First-stage F Test with Many Weak Instruments," Papers 2302.14423, arXiv.org, revised Sep 2024.
    14. Laurini, Márcio Poletti & Sanvicente, Antônio Zoratto & Monteiro, Rogério da Costa, 2011. "Generalized Tests of Investment Fund Performance," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 31(2), December.
    15. Patrick Richard, 2014. "Bootstrap tests in linear models with many regressors," Cahiers de recherche 14-06, Departement d'économique de l'École de gestion à l'Université de Sherbrooke.
    16. Abhimanyu Gupta & Myung Hwan Seo, 2023. "Robust Inference on Infinite and Growing Dimensional Time‐Series Regression," Econometrica, Econometric Society, vol. 91(4), pages 1333-1361, July.
    17. Dante Amengual & Luca Repetto, 2014. "Testing a Large Number of Hypotheses in Approximate Factor Models," Working Papers wp2014_1410, CEMFI.
    18. Abhimanyu Gupta & Xi Qu, 2021. "Consistent specification testing under spatial dependence," Papers 2101.10255, arXiv.org, revised Aug 2022.
    19. Richard, Patrick, 2019. "Residual bootstrap tests in linear models with many regressors," Journal of Econometrics, Elsevier, vol. 208(2), pages 367-394.

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    More about this item

    Keywords

    Alternative asymptotic theory; linear regression; test size; test power; F test; Wald test; Likelihood Ratio test; Lagrange Multiplier test;
    All these keywords.

    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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