PRACTITIONERS CORNER: Systems of Axioms for the Estimators of the Parameters in the Standard Linear Model
| Published date | 01 February 1989 |
| Date | 01 February 1989 |
| DOI | http://doi.org/10.1111/j.1468-0084.1989.mp51001007.x |
91
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 51, 1(1989)
0305-9049 $3.00
Systems of Axioms for the Estimators of the
Parameters in the Standard Linear Model
R. W Farebrother
I. INTRODUCTION
This paper is concerned with the estimation of the p X 1 matrix of parameters
ß and the scalar parameter u2 in the standard linear model
y=Xß+e E(e)=O E(ee')=a21 (1.1)
where y is an n X 1 matrix of observations on the dependent variable, Xis an
n xp matrix of observations on the p regressors, and e is an n X 1 matrix of
disturbances, where X has full column rank p.
The conventional estimators of ß and u2
ß(XIX)_1X'y (1.2)
and
ô2 = (y Xß)'(y Xß)/m (1.3)
are usually obtained by applying the maximum likelihood (m = n) or least
squares (m = n p) principles in the context of model (1.1). In this paper we
shall adopt an alternative approach based on the following three transforma-
tions of model (1.1)
y+XdX(ß+d)+eE(e)=O E(ee')u21 (1.4)
sy=X(sß)+seE(se)=O E(s2ee')s2u21 (1.5)
Q'y=Xß+Q'eE(Q'e)() E(Q'ee'Q)=u21 (1.6)
where dis a p X 1 matrix, s is a nonzero scalar and Q is an n X n orthonormal
matrix satisfying Q'X =Xand Q'Q = I.
Now (1.1) is a linear model and linear models are unique under non-
singular transformations. Thus, comparing these models with model (1.1), we
find that the estimator of ¡3 + d in model (1.4) should be equal to d plus the
estimator of fi in model (1.1), the estimator of sß in model (1.5) should be
equal to s times the estimator of fi in model (1.1), and the estimator of fi in
model (1.6) should be identical to the estimator of fi in model (1.1). Further,
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