Estimation of Heteroscedastic Variances in Linear Models

Abstract

Let Y=Xβ+e be a Gauss-Markoff linear model such that E(e) = 0 and D(e), the dispersion matrix of the error vector, is a diagonal matrix δ whose ith diagonal element is σi 2, the variance of the ith observation yi. Some of the σi 2 may be equal. The problem is to estimate all the different variances. In this article, a new method known as MINQUE (Minimum Norm Quadratic Unbiased Estimation) is introduced for the estimation of the heteroscedastic variances. This method satisfies some intuitive properties: (i) if S 1 is the MINQUE of Σ piσi 2 and S 2 that of Σqiσi 2 then S 1+S 2 is the MINQUE of σ(pi + qi )σi 2, (ii) it is invariant under orthogonal transformation, etc. Some sufficient conditions for the estimation of all linear functions of the σi 2 are given. The use of estimated variances in problems of inference on the β parameters is briefly indicated.