## Bruce E. Hansen

The Exact Distribution of the White t-ratio

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April 2021

Abstract:

This paper presents new expressions for the exact finite sample distribution of the White (1980) heteroskedasticity-robust t-ratio under the assumption of normal heteroskedastic errors. The first expression shows that the distribution function equals the expectation of a nonlinear function of a weighted sum of chi-square random variables, with the weights an explicit function of the regressor matrix and error variances. The second expression shows that the distribution function equals a mixture of student t distribution functions. These are the first expressions for the exact distribution of the White t-ratio allowing for heteroskedastic error variances, other than expressions based on the numerical inversion of the characteristic function.

Our exact distribution function is inconvenient to evaluate in practice, so we recommend a simple approximation with excellent computational and approximation properties. The motivation is the first result described above that the distribution function is completely determined by a specific weighted sum of chi-squares. Using results from the recent literature on approximation of the distribution function of weighted sums of chi-squares, we obtain a practical approximation to the distribution function of the White t-ratio which is computationally fast in small to moderate samples and is exceedingly accurate.

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Some of the above material is based upon work supported by the National Science Foundation under Grants No. SES-9022176, SES-9120576, SBR-9412339, and SBR-9807111.
Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the NSF.