Feasible Generalized Least Squares

Definition

FGLS estimates the structure of the conditional variance and corrects the heteroskedasticity in the data.

Algorithm

Assume that the variance of the data is following the structure ${\sigma }_i^2={\sigma }^{2}h(z_{i2},\dots ,z_{iS})$ and $\log ({\sigma }_i^2)={\alpha }_1+{\alpha }_2{z}_{i2}+\cdots +{\alpha }_S{z}_{iS}$

1. Estimate the original model $y_i={\beta }_1+{\beta }_2{x}_{i2}+\cdots +{\beta }_K{x}_{iK}+e_i$ by OLS, and save the residuals ${\hat{e}}_i$
2. Use the least squares residuals and the variables $z_{i2},\dots ,z_{iS}$ to estimate ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_S$
3. Calculate the estimated skedastic function $\hat{h}(z_{i2},\dots ,z_{iS})$ where ${\sigma }_i^2={\sigma }^{2}h(z_{i2},\dots ,z_{iS})$
4. Divide each observation by $\sqrt{\hat{h}(z_{i2},\dots ,z_{iS})}$ and apply OLS to the transformed data, or use WLS regression with weighting factor $1/\hat{h}(z_{i2},\dots ,z_{iS})$