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

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.