Feasible GLS (Generalized Least Squares)

GLS is applied when the residuals of OLS cannot satisfy two aspects of its classical assumptions: homoscedasticity and no auto-correlation, or rather,

$E(uu^{'})=\sigma ^{2}\Omega $

rather than

$E(uu^{'})=\sigma ^{2}I$

There are many forms of FGLS, and the following passage describes its basic theorem and one form of FGLS that could eliminate if there is only heteroscedasticity:

Theorem of GLS, WLS, and One Form of FGLS

There is also another model to eliminate only AR(1) pattern in the OLS residuals:

1) Estimate coefficient $\rho$ in the residuals that

$e_{t} = \rho e_{t-1} + \varepsilon$

2) Transfer

$y = X\beta +u $

to

$Py = PX\beta +Pu $

by

and 3) Estimate correspondingwith OLS estimation.

I will add more forms of FGLS in the following if possible.