Generalized Least Squares

Abstract

This chapter considers a more general variance covariance matrix for the disturbances. In other words, u ~ (0, s2In) is relaxed so that u ~ (0, σ²Ω) where Ω is a positive definite matrix of dimension (n×n). First Ω is assumed known and the BLUE for β is derived. This estimator turns out to be different from $\hat{\beta}_{OLS}$, and is denoted by $\hat{\beta}_{GLS}$, the Generalized Least Squares estimator of β. Next, we study the properties of $\hat{\beta}_{OLS}$ under this nonspherical form of the disturbances. It turns out that the OLS estimates are still unbiased and consistent, but their standard errors as computed by standard regression packages are biased and inconsistent and lead to misleading inference. Section 9.3 studies some special forms of Ω and derive the corresponding BLUE for β. It turns out that heteroskedasticity and serial correlation studied in Chapter 5 are special cases of Ω. Section 9.4 introduces normality and derives the maximum likelihood estimator. Sections 9.5 and 9.6 study the way in which test of hypotheses and prediction get affected by this general variance-covariance assumption on the disturbances. Section 9.7 studies the properties of this BLUE for Ω when Ω is unknown, and is replaced by a consistent estimator. Section 9.8 studies what happens to the W, LR and LM statistics when u ~ N(0, σ²Ω). Section 9.9 gives another application of GLS to spatial autocorrelation.