Distribution of Linear Transformations of Internally Studentized Least Squared Residuals

Abstract

Ordinary least squares regression residuals have a distribution that is dependent on a scale parameter. The term 'Studentization' is commonly used to describe a scale parameter dependent quantity U by a scale estimate S such that the resulting ratio, U/S, has a distribution that is free of from the nuisance unknown scale parameter. {\it External} Studentization refers to a ratio in which the nominator and denominator are independent, while internal Studentization refers to a ratio in which these are dependent. The advantage of the internal Studentization is that typically one can use a single common scale estimator, while in the external Studentization every single residual is scaled by different scale estimator to gain the independence. With normal regression errors the joint distribution of an arbitrary (linearly independent) subset of internally Studentized residuals is well documented. However, in some applications a linear combination of internally Studentized residuals may be useful. The boundedness of them is well documented, but the distribution seems not be derived in the literature. This paper contributes to the existing literature by deriving the joint distribution of an arbitrary linear transformation of internally Studentized residuals from ordinary least squares regression with spherical error distribution. All major versions of commonly utilized internally Studentized regression residuals in literature are obtained as special cases of the linear transformation.

Keywords: Borel transformation of Studentized residuals; normed residuals; spherical distribution; elliptical distribution;

JEL Classification: C40, C46, C50