## 5.4 Generalized Least Squares

Consider

$\mathbf{y = X\beta + \epsilon}$ where,

$var(\epsilon) = \mathbf{G} = \left( \begin{array} {cccc} g_{11} & g_{12} & ... & g_{1n} \\ g_{21} & g_{22} & ... & g_{2n} \\ . & . & . & . \\ g_{n1} & . & . & g_{nn}\\ \end{array} \right)$

The variances are heterogeneous, and the errors are correlated.

$\mathbf{\hat{b}_G = (X'G^{-1}X)^{-1}X'G^{-1}Y}$

if we know G, we can estimate $$\mathbf{b}$$ just like OLS. However, we do not know G. Hence, we model the structure of G.