Moments of regression F-statistics under non-normality

Abstract

Assuming the errors to be non-normal with finite moments, the mean, variance, skewness and kurtosis of the null distribution of the F-statistics in a linear regression model (while testing a subset, or all the coefficients to be equal to zero) are derived to the order O(1/n²), where n is the sample size. It is found that non-normality of errors has insignificant effect on the mean. The variance is sensitive to kurtosis but insensitive to skewness of the error distribution. The extent of his sensitivity depends upon the kurtosis-type measure of non-normality of the “test” regressors.The test regressors are the ones for whose coefficients are tested to be zeroes. In contrast, both skewness and kurtosis of the F-statistics are found to be sensitive to both skewness and kurtosis of the error distributions. Again, the extent of this sensitivity depends upon the skewness and kurtosis-type measures of non-normality of the test regressors. These effects are all of order O(l/n), so that they are negligible in large samples.

Keywords:

• regression model
• non-nonncrl errors
• F-statistics
• robustness to non-normality