Feasible GLS: estimate the variance, then weight

Question

You believe the error variance depends on the regressors but you do not know the coefficients of that relationship in advance.

Describe the feasible generalized least squares (FGLS) procedure that estimates the variance from the data and then weights by it, and note its main caveats.

Accepted Answer

The idea. If you knew each observation's variance you would weight by its inverse (GLS). You do not, so you estimate the variance model first, then weight. That two-step version is called feasible GLS. The steps. 1. Run ordinary least squares and save the residuals ei. 2. Model how the variance depends on the regressors. A common choice: regress \log(ei^2) on the regressors (logging keeps the fitted variances positive). This estimates how the spread grows with x. 3. Form fitted variances \hat\sigmai^2 from that auxiliary regression (exponentiate the fitted log values). 4. Re-run the original …

Feasible GLS: estimate the variance, then weight

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