An exact, unified distributional characterization of statistics used to test linear hypotheses in simple regression models

The Wald, likelihood ratio and Lagrange multiplier test statistics are commonly used to test linear restrictions in regression models. It is shown that for testing these restrictions in the classical regression model, the exact densities of these test statistics are special cases of the generalized beta distribution introduced by McDonald (1984); McDonald and Xu (1995a). This unified derivation provides a method by which one can derive small sample critical values for each test. These results may be indicative of the behavior of such test statistics in more general settings, and are useful in visualizing how each statistic changes with different parameter values in the simple regression model. For example, the results suggest that Wald tests may severely underreject the null hypothesis when the sample size is small or a large number of restrictions are tested.