*J. Japan Statist. Soc.,* Vol. 34 (No. 1), pp. 101-109, 2004

## Exact Distributions of *R*^{2} and Adjusted *R*^{2} in a Linear Regression Model with Multivariate *t* Error Terms

Kazuhiro Ohtani

**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, ^{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, *n*_{0}) get small. The confidence intervals of *R*^{2} and ^{2} are examined, and it is shown that when the values of *n*_{0} and the parent coefficient of determination (say, Φ) are small, the upper confidence limits are very large, relative to the value of Φ.

*Key words and phrases*:
Adjusted *R*^{2}, exact distribution, interval estimation, multivariate *t* error terms, *R*^{2}.

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