Why Generalized Least Square Estimator?

It is known that heteroskedasticity affects the properties of the OLS estimatror (though still unbiased, but less efficient, namely larger variance). When you draw a scatter plot on raw data, the higher absolute values of the residuals to the right in the graph indicate that there is a positive relationship between the error variance and the independent variable. With this kind of error pattern, a few additional large positive errors near the right in this graph would tilt (make something move, into a position with one side or end higher than the other) the OLS regression line considerably. A few additional large negative errors would tilt it in the opposite direction considerably. In repeated sampling these unusual cases would average out, leaving the OLS estimator unbiased, but the variation of the OLS regression line around its mean will be greater - i. e., the variance of βOLS will be greater. The Generalized Least Square (GLS) technique pays less attention to the residuals associated with high-variance observations (by assigning them a low weight in the weighted sum of squared residuals it minimizes) since these observations give a less precise indication of where the true regression line lies. This avoids these large tilts, making the variance of βGLS smaller than that of βOLS.

In the case of that Durbin-Watson test indicates autocorrelated errors. It is typically concluded that estimation via Feasible GLS is called for. This is not always appropriate, however, the significant value of the Durbin-Watson statistic could result from an omitted explanatory variable, an incorrect functional form, or a dynamic misspecification. Only if a researcher is satisfied that none of these phenomena are responsible for the significant Durbin-Watson statistic value should estimation via feasible GLS proceed.