On the properties of the t- and F-ratios in linear regressions with nonnormal errors

In this paper we derive the necessary and sufficient conditions for the t-ratio to be Student's t distributed. In particular, it is demonstrated for a special case that under conditions of nonnormality characterized by elliptical symmetry, the t-ratio remains Student's t distributed provided that the random vector forming the t-ratio has a diagonal covariance structure. Our results also show that the findings of Magnus (2002, in A. Ullah, A.T.K. Wan, & A. Chaturvedi (eds.), Handbook of Applied Econometrics and Statistical Inference, 277-285) on the sensitivity of the t-ratio remain invariant in the elliptically symmetric distribution setting. Extension to the linear model is considered. Exact results giving finite sample justification for the t-statistic under nonnormal error terms are derived. Furthermore, the distribution of the F-ratio assuming elliptical errors is examined. Our results reject the argument by Zaman (1996, Statistical Foundations for Econometric Techniques) that nonnormality of disturbances has in general no effect on the F-statistic.