On the geometry of F, Wald, LR, and LM tests in linear regression models

Abstract

In this article, we examine F, Wald, LR, and LM test statistics in the linear regression model using vector geometry. These four statistics are expressed as a function of one random variable-the angle between the vectors of unrestricted and restricted residuals. The exact and nominal sampling distributions of this angle are derived to illuminate some facts about the four statistics. Alternatively, we offer that the angle itself can be used as a test statistic. A Mathematica program is also written to carry out the approach.