Explicit Formulas to Determine the E¢ ciency of OLS in the Presence of First Order Autoregressive Disturbances

Abstract

In problems concerning time series, it is often the case that the distur-
bances are, in fact, correlated. It is known that the ordinary least squares
(OLS) may not be optimal in this context. We have proved that the rela-
tive e¢ ciency of the variance of the generalized least squares (GLS) to that
of OLS is invariant to scaling and shifting of the design vectors. We have
derived explicit formulas for the relative e¢ ciencies of the GLS estimator
to that of OLS estimator in some important special cases. We consider
both linear and quadratic design vectors in the presence of AR(1) distur-
bances with and without an intercept term included in the design and use
these formulas to show some asymptotic properties of the estimators.