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Exact Distribution of the F-statistic under Heteroskedasticity of Unknown Form for Improved Inference

Author

Listed:
  • Jianghao Chu

    (Ford Motor Co)

  • Tae-Hwy Lee

    (Department of Economics, University of California Riverside)

  • Aman Ullah

    (University of California Riverside)

  • Haifeng Xu

    (Xiamen University)

Abstract

The exact finite sample distribution of the $F$ statistic using the heteroskedasticity-consistent (HC) covariance matrix estimators of the regression parameter estimators is unknown. In this paper, we derive the exact finite sample distribution of the $F$ ($=t^2$) statistic for a single linear restriction on the regression parameters. We show that the $F$ statistic can be expressed as a ratio of quadratic forms, and therefore its exact cumulative distribution under the null hypothesis can be derived from the result of Imhof (1961). A numerical calculation is carried out for the exact distribution of the $F$ statistic using various HC covariance matrix estimators, and the rejection probability under the null hypothesis (size) based on the exact distribution is examined. The results show the exact finite sample distribution is remarkably reliable, while, in comparison, the use of the $F$-table leads to a serious over-rejection when the sample is not large or leveraged/unbalanced. An empirical application highlights that the use of the exact distribution of the $F$ statistic will increase the accuracy of inference in empirical research.

Suggested Citation

  • Jianghao Chu & Tae-Hwy Lee & Aman Ullah & Haifeng Xu, 2020. "Exact Distribution of the F-statistic under Heteroskedasticity of Unknown Form for Improved Inference," Working Papers 202027, University of California at Riverside, Department of Economics.
    • Handle: RePEc:ucr:wpaper:202027
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    File URL: https://economics.ucr.edu/repec/ucr/wpaper/202027.pdf
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    References listed on IDEAS

    as
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    More about this item

    Keywords

    Heteroskedastisity; Finite sample theory; Imhof distribution;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables

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