Exact Distributions of R 2 and Adjusted R 2 in a Linear Regression Model with Multivariate t Error Terms

Abstract

In this paper we consider a linear regression model when error terms obey a multivariate t distribution, and examine the effects of departure from normality of error terms on the exact distributions of the coefficient of determination (say, R 2) and adjusted R 2 (say, R 2). We derive the exact formulas for the density function, distribution function and m-th moment, and perform numerical analysis based on the exact formulas. It is shown that the upward bias of R 2 gets serious and the standard error of R 2 gets large as the degrees of freedom of the multivariate t error distribution (say, ν0) get small. The confidence intervals of R 2 and R 2 are examined, and it is shown that when the values of ν0 and the parent coefficient of determination (say, Φ) are small, the upper confidence limits are very large, relative to the value of Φ.