Feasible generalized least squares

Feasible generalized least squares (FGLS or Feasible GLS) is a regression technique. It is similar to generalized least squares except that it uses an estimated variance-covariance matrix since the true matrix is not known directly.

The following description follows loosely the references presented in Heteroscedasticity-consistent_standard_errors.

The dataset is assumed to be represented by:

where "X" is the design matrix and β is a column vector of parameters to be estimated. The residuals in the vector "u", are not assumed to have equal variances: instead the assumptions are that they are uncorrelated but with different unknown variances. These assumptions together are represented by the assumption that the residaul vector has a diagonal covariance matrix Ω.

Ordinary Least Squares estimation can be applied to a linear system with heteroskedastic errors, butOLS in this case is not Best Linear Unbiased Estimator (BLUE).To estimate the error variance-covariance $Omega$, the following process can be iterated:

The ordinary least squares (OLS) estimator is calculated as usual by

:

and estimates of the residuals $widehat\{u\}\_j$are constructed.

Construct $widehat\{Omega\}\_\{OLS\}$:

:$widehat\{Omega\}\_\{OLS\}\; =\; operatorname\{diag\}(widehat\{u\}^2\_1,\; widehat\{u\}^2\_2,\; dots\; ,\; widehat\{u\}^2\_n).$

Estimate using $widehat\{Omega\}\_\{OLS\}$ using weighted least squares

:

:

:$widehat\{Omega\}\_\{FGLS1\}\; =\; operatorname\{diag\}(widehat\{u\}^2\_\{FGLS1,1\},\; widehat\{u\}^2\_\{FGLS1,2\},\; dots\; ,\; widehat\{u\}^2\_\{FGLS1,n\})$

:

This estimation of $widehat\{Omega\}$ can be iterated to convergence given that the assumptions outlined in White and Halbert hold.

Estimations from WLS and FGLS are as follows

:

:

References

Citation
last = White
first =
last2 = Halbert
first2 =
title = A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity
journal = Econometrica
volume = 48
number = 4
pages = 817--838
url = http://www.jstor.org/stable/1912934
year = 1980