The efficiency of OLS in the presence of auto -correlated disturbances in regression models

It is well known that the ordinary least squares (OLS) estimates in the regression model are efficient when the disturbances have mean zero, constant variance and are uncorrelated. In problems concerning time series, it is often the case that the disturbances are, in fact, correlated. It is known that OLS may not be optimal in this context. We have proved that the relative efficiency 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 efficiencies 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) disturbances with and without an intercept term included in the design and use these formulas to show some asymptotic properties of the estimators. Additionally, using computer simulations, we consider the robustness of various estimators, including estimated generalized least squares. We found that if the disturbance structure is autoregressive and the dependent variable is nonstochastic and linear or quadratic, the OLS performs nearly as well as its competitors. For other forms of the dependent variable, we have developed rules of thumb to guide practitioners in their choice of estimators.