Feasible generalized least squares

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[edit] 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, but OLS in this case is not Best Linear Unbiased Estimator (BLUE). To estimate the error variance-covariance Ω, the following process can be iterated:

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

and estimates of the residuals are constructed.

Construct :

Estimate β_FGLS1 using using weighted least squares

This estimation of can be iterated to convergence given that the assumptions outlined in White and Halbert hold.

Estimations from WLS and FGLS are as follows

[edit] References

White & Halbert (1980), “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity”, Econometrica 48 (4): 817--838, <http://www.jstor.org/stable/1912934>