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THREE RANK FORMULAS ASSOCIATED WITH THE COVARIANCE MATRICES OF THE BLUE AND THE OLSE IN THE GENERAL LINEAR MODEL

Published online by Cambridge University Press:  22 April 2005

Simo Puntanen ,
George P.H. Styan  and
Yongge Tian
Simo Puntanen
Affiliation:
University of Tampere
George P.H. Styan
Affiliation:
McGill University
Yongge Tian
Affiliation:
University of Alberta
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Abstract

In this paper we consider the estimation of the expectation vector Xβ under the general linear model {y,Xβ,σ2V}. We introduce a new handy representation for the rank of the difference of the covariance matrices of the ordinary least squares estimator OLSE(Xβ) (= Hy, say) and the best linear unbiased estimator BLUE(Xβ) (= Gy, say). From this formula some well-known conditions for the equality between Hy and Gy follow at once. We recall that the equality between Hy and Gy can be characterized by the rank-subtractivity ordering between the covariance matrices of y and Hy. This rank characterization suggests a particular presentation for the rank of the difference of the covariance matrices of Hy and Gy. We show, however, that this presentation is valid if and only if the model is connected.

Type
NOTES AND PROBLEMS
Information
Econometric Theory , Volume 21 , Issue 3 , June 2005 , pp. 659 - 663
Copyright
© 2005 Cambridge University Press

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References

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