Content area
Full Text
Statistical Papers 42, 253-263 (2001) Statistical Papers
9 Springer-Verlag 2001
Butte Gotu
Department of Statistics, Addis Ababa University, P.O. Box 1176,
Addis Ababa, Ethiopia
Received: December 18, 1998; revised version: September 13, 1999
Necessary and sufficient conditions for the equality of ordinary least squares and generalized least squares estimators in the linear regression model with first-order spatial error processes are given.
Consider the linear regression model for spatial correlation
y=Xfl+u , u--Ce , (1)
where y is a T x 1 observable random vector, X is a T x k matrix of known constants with full column rank k, fl is a k 1 vector of unknown parameters, e is a T x 1 random vector with expectation zero and covariance matrix Coy(e) = a~I (I is the T-dimensional identity matrix and a~ an unknown positive scalar). C denotes a T x T nonsingular matrix.
The ordinary least squares (OLS) and the generalized least squares (GLS) estimators of the vector of unknown parameters fl in model (1) are given by fl -" (X'X)-lX'y and fl = (X'V,-1X)-lX'V,-ly, respectively with covariance matrices Cov(~) = a~(X'X)-IX'V,X(X 'X) -1
1This work was partly supported by the Deutsche Forschungsgemeinschaft
(DFG), Graduiertenkolleg "Angewandte Statistik'.
The equality of OLS and GLS estimatorsin the linear regression modelwhen the disturbances are spatially correlated
1 Introduction
254
and Cov(~)= a~(X'V,'X)-', where V , - CC'.
When the covariance of the disturbance vector u is not a scalar multiple of the identity matrix, that is Coy(u) ~ a2I as in model (1), it is well known that the GLS estimator provides the best linear unbiased estimator (BLUE) of/~ in contrast to OLS. Since Coy(u)
usually involves unknown parameters like spatial correlation coefficient, it is natural to ask when both estimators coincide so that the OLS estimator can be applied without loss of efficiency.
Usually, criteria developed for the purpose of checking the equality of least squares estimators are not operational because of the unknown parameters involved (see Puntanen and Styan, 1989).
In this paper, conditions under first-order spatial error processes which can be verified in practice by using spatial weight matrix with known nonnegative weights and the matrix X of known constants are developed. The first group of conditions is based on the invariance property...