Chapter 7 Generalized Least Squares and Related Topics Introduction If the parameters of a regression model are to be estimated efficiently by least squares, the error terms must be uncorrelated and have the same variance. \ | Chapter 7 Generalized Least Squares and Related Topics Introduction If the parameters of a regression model are to be estimated efficiently by least squares the error terms must be uncorrelated and have the same variance. These assumptions are needed to prove the Gauss-Markov Theorem and to show that the nonlinear least squares estimator is asymptotically efficient see Sections and . Moreover the usual estimators of the covariance matrices of the OLS and NLS estimators are not valid when these assumptions do not hold although alternative sandwich covariance matrix estimators that are asymptotically valid may be available see Sections and . Thus it is clear that we need new estimation methods to handle regression models with error terms that are heteroskedastic serially correlated or both. We develop some of these methods in this chapter. Since heteroskedasticity and serial correlation affect both linear and nonlinear regression models in the same way there is no harm in limiting our attention to the simpler linear case. We will be concerned with the model y X3 u E uwT where the covariance matrix of the error terms is a positive definite n x n matrix. If is equal to a2I then is just the linear regression model with error terms that are uncorrelated and homoskedastic. If is diagonal with nonconstant diagonal elements then the error terms are still uncorrelated but they are heteroskedastic. If is not diagonal then ui and Uj are correlated whenever j the ijth element of is nonzero. In econometrics covariance matrices that are not diagonal are most commonly encountered with time-series data and the correlations are usually highest for observations that are close in time. In the next section we obtain an efficient estimator for the vector 3 in the model by transforming the regression so that it satisfies the conditions of the Gauss-Markov theorem. This efficient estimator is called the generalized least squares or GLS estimator.
