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

y = X \beta + u, \,

where X is the design matrix and ? 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 ?.

Ordinary Least Squares estimation can be applied to a linear system with heteroskedastic errors, but OLS 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

\widehat \beta_{OLS} = (X' X)^{-1} X' y

and estimates of the residuals \widehat{u}_jare constructed.

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

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

Estimate \beta_{FGLS1} using \widehat{\Omega}_{OLS} using weighted least squares

\widehat \beta_{FGLS1} = (X'\widehat{\Omega}^{-1}_{OLS} X)^{-1} X' \widehat{\Omega}^{-1}_{OLS} y

\widehat{u}_{FGLS1} = Y - X \widehat \beta_{FGLS1}

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

\widehat \beta_{FGLS2} = (X'\widehat{\Omega}^{-1}_{FGLS1} X)^{-1} X' \widehat{\Omega}^{-1}_{FGLS1} y

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

\widehat \beta_{WLS} ~\sim N(\beta , (X'\Omega^{-1}X)^{-1})

\widehat \beta_{FGLS} ~\sim N(\beta , (X'\widehat{\Omega}_{OLS}^{-1}X)^{-1}(X'\widehat{\Omega}_{OLS}^{-1}\Omega\widehat{\Omega}_{OLS}^{-1}X)(X'\widehat{\Omega}_{OLS}^{-1}X)^{-1})

References

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