IDEAS home Printed from https://ideas.repec.org/p/cfr/cefirw/w0125.html
   My bibliography  Save this paper

Inference in Regression Models with Many Regressors

Author

Listed:
  • 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, Center for Economic and Financial Research (CEFIR).
  • Handle: RePEc:cfr:cefirw:w0125
    as

    Download full text from publisher

    File URL: http://www.cefir.ru/papers/WP125.pdf
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    as
    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Evans, G B A & Savin, N E, 1982. "Conflict among the Criteria Revisited: The W, LR and LM Tests," Econometrica, Econometric Society, vol. 50(3), pages 737-748, May.
    3. Koenker, Roger & Machado, Jose A. F., 1999. "GMM inference when the number of moment conditions is large," Journal of Econometrics, Elsevier, vol. 93(2), pages 327-344, December.
    4. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    5. Peter Sandholt Jensen & Allan H. Würtz, 2006. "On determining the importance of a regressor with small and undersized samples," Economics Working Papers 2006-08, Department of Economics and Business Economics, Aarhus University.
    6. Burnside, Craig & Eichenbaum, Martin S, 1996. "Small-Sample Properties of GMM-Based Wald Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(3), pages 294-308, July.
    7. Rothenberg, Thomas J., 1984. "Approximating the distributions of econometric estimators and test statistics," Handbook of Econometrics,in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 15, pages 881-935 Elsevier.
    8. Berndt, Ernst R & Savin, N Eugene, 1977. "Conflict among Criteria for Testing Hypotheses in the Multivariate Linear Regression Model," Econometrica, Econometric Society, vol. 45(5), pages 1263-1277, July.
    9. Olivier Ledoit & Michael Wolf, 2001. "Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size," Economics Working Papers 575, Department of Economics and Business, Universitat Pompeu Fabra.
    10. John Galbraith & Victoria Zinde-Walsh, 2006. "Reduced-Dimension Control Regression," Departmental Working Papers 2006-17, McGill University, Department of Economics.
    11. de Jong, R.M. & Bierens, H.J., 1994. "On the Limit Behavior of a Chi-Square Type Test if the Number of Conditional Moments Tested Approaches Infinity," Econometric Theory, Cambridge University Press, vol. 10(01), pages 70-90, March.
    12. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-681, May.
    13. H. Kelejian, Harry & Prucha, Ingmar R., 2001. "On the asymptotic distribution of the Moran I test statistic with applications," Journal of Econometrics, Elsevier, vol. 104(2), pages 219-257, September.
    14. Hong, Yongmiao & White, Halbert, 1995. "Consistent Specification Testing via Nonparametric Series Regression," Econometrica, Econometric Society, vol. 63(5), pages 1133-1159, September.
    15. Trevor S. Breusch, 1986. "Hypothesis Testing in Unidentified Models," Review of Economic Studies, Oxford University Press, vol. 53(4), pages 635-651.
    16. Gérard Letac & Hélène Massam, 2004. "All Invariant Moments of the Wishart Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(2), pages 295-318.
    17. Rothernberg, Thomas J, 1984. "Hypothesis Testing in Linear Models When the Error Covariance Matrix Is Nonscalar," Econometrica, Econometric Society, vol. 52(4), pages 827-842, July.
    18. Donald, Stephen G. & Imbens, Guido W. & Newey, Whitney K., 2003. "Empirical likelihood estimation and consistent tests with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 117(1), pages 55-93, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Patrick Richard, 2014. "Bootstrap tests in linear models with many regressors," Cahiers de recherche 14-06, Departement d'Economique de l'École de gestion à l'Université de Sherbrooke.
    2. Calhoun, Gray, 2011. "Hypothesis testing in linear regression when k/n is large," Journal of Econometrics, Elsevier, vol. 165(2), pages 163-174.
    3. repec:sbe:breart:v:31:y:2011:i:2:a:7173 is not listed on IDEAS
    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. 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.
    6. 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.
    7. Dante Amengual & Luca Repetto, 2014. "Testing a Large Number of Hypotheses in Approximate Factor Models," Working Papers wp2014_1410, CEMFI.

    More about this item

    Keywords

    Alternative asymptotic theory; linear regression; test size; test power; F test; Wald test; Likelihood Ratio test; Lagrange Multiplier test;

    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cfr:cefirw:w0125. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Julia Babich). General contact details of provider: http://edirc.repec.org/data/cefirru.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.