For regression models with general unstable regressors having
characteristic roots on the unit circle and general stationary
errors independent of the regressors, sufficient conditions
are investigated under which the ordinary least squares estimator
(OLSE) is asymptotically efficient in that it has the same limiting
distribution as the generalized least squares estimator (GLSE)
under the same normalization. A key condition for the asymptotic
efficiency of the OLSE is that one multiplicity of a characteristic
root of the regressor process is strictly greater than the
multiplicities of the other roots. Under this condition, the
covariance matrix Γ of the errors and the regressor matrix
X are shown to satisfy a relationship
(ΓX = XC + V for some matrix
C) for V asymptotically dominated by X,
which is analogous to the condition (ΓX = XC
for some matrix C) for numerical equivalence of the
OLSE and the GLSE.
The Traditional Ordinary Least Squares Estimator under Collinearity
