The Efficiency of OLS in the Presence of Auto-correlated Errors

It is known that the ordinary least squares (OLS) estimates in the regression model are efficient when the errors have mean zero, constant variance and are uncorrelated. In time series, it is often the errors are 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 formulae for the relative efficiencies of the GLS to that of OLS in some important special cases. We consider both linear and quadratic design vectors in the presence of AR(1) errors and show some asymptotic properties of the estimators. Additionally, using computer simulations, we consider the robustness of various estimators, including estimated GLS. We found that if the error structure is autoregressive and the dependent variable is nonstochastic and linear or quadratic, the OLS performs nearly as well asits competitors. For other forms of the dependent variable, we have developed rules of thumb to guide practitioners in their choice of estimators.