IDEAS home Printed from
   My bibliography  Save this paper

Bartlett Adjustments for Hypothesis Testing in Linear Models with General Error Covariance Matrices


  • Masahiro Kojima

    (Graduate School of Economics, University of Tokyo)

  • Tatsuya Kubokawa

    (Faculty of Economics, University of Tokyo)


Consider the problem of testing a linear hypothesis of regression coefficients in a general linear regression model with an error term having a covariance matrix involving several nuisance parameters. Three typical test statistics of Wald, Score and Likelihood Ratio (LR) and their Bartlett adjustments have been derived in the literature when the unknown nuisance parameters are estimated by maximum likelihood (ML). On the other hand, statistical inference in linear mixed models has been studied actively and extensively in recent years with applications to smallarea estimation. The marginal distribution of the linear mixed model is included in the framework of the general linear regression model, and the nuisance parameters correspond to the variance components and others in the linear mixed model. Although the restricted ML (REML), minimum norm quadratic unbiased estimator (MINQUE) and other specific estimators are available for estimating the variance components, the Bartlett adjustments given in the literature are not correct for those estimators other than ML. In this paper, using the Taylor series expansion, we derive the Bartlett adjustments of the Wald, Score and modified LR tests for general consistent estimators of the unknown nuisance parameters. These analytical results may be harder to calculate for a model with a complicate structure of the covariance matrix. Thus, we propose the simple parametric bootstrap methods for estimating the Bartlett adjustments and show that they have the second order accuracy. Finally, it is shown that both Bartlett adjustments work well through simulation experiments in the nested error regression model.

Suggested Citation

  • Masahiro Kojima & Tatsuya Kubokawa, 2013. "Bartlett Adjustments for Hypothesis Testing in Linear Models with General Error Covariance Matrices," CIRJE F-Series CIRJE-F-884, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2013cf884

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Gauri Sankar Datta & J. N. K. Rao & David Daniel Smith, 2005. "On measuring the variability of small area estimators under a basic area level model," Biometrika, Biometrika Trust, vol. 92(1), pages 183-196, March.
    2. Srivastava, Muni S. & Kubokawa, Tatsuya, 2010. "Conditional information criteria for selecting variables in linear mixed models," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1970-1980, October.
    3. 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.
    4. Florin Vaida & Suzette Blanchard, 2005. "Conditional Akaike information for mixed-effects models," Biometrika, Biometrika Trust, vol. 92(2), pages 351-370, June.
    5. Rayner, Robert K., 1990. "Bartlett's correction and the bootstrap in normal linear regression models," Economics Letters, Elsevier, vol. 33(3), pages 255-258, July.
    6. Kubokawa, Tatsuya, 2011. "Conditional and unconditional methods for selecting variables in linear mixed models," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 641-660, March.
    Full references (including those not matched with items on IDEAS)

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:tky:fseres:2013cf884. 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: (CIRJE administrative office). General contact details of provider: .

    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.