# Feasible GLS estimator

I'm approaching for the first time GLS estimators.

Suppose that $$\operatorname{Var}(u|x)=\sigma^2 h(x)$$, where $$h(x)$$ is some function of the explanatory variables that determines the heteroscedasticity. Since variances must be positive, $$h(x)>0$$ for all possible values of the independent variables. Let's say, we don't know $$h(x)$$.

The book (Wooldridge, the introductory one) proposes the following way for modelling heteroscedasticity:

$$\operatorname{Var}(u|x)=\sigma^2 {\rm e}^{\delta_0 +\delta_1 x_1 + \cdots + \delta_k x_k}\tag 1$$

where $$\delta_j$$ are unknown parameters. Thus, $$h(x)={\rm e}^{\delta_0 +\delta_1 x_1 + ... + \delta_k x_k}$$

Then, the book says, under assumption $$[1]$$ I can write

$$u^2 =\sigma^2 ({\rm e}^{\delta_0 +\delta_1 x_1 + ... + \delta_k x_k}) \nu\tag 2$$

Where do equations $$[2]$$ and $$[8.31]$$ come from?

The assumptions of the generalized linear regression model is $$\mathbb E[\mathbf u\mid \mathbf X]=\mathbf 0$$ and $$\mathbb E[\mathbf u\mathbf u^\top\mid\mathbf X]=\sigma^2\mathbf\Omega.$$

If $$\bf\Omega$$ is unknown owing to its dependence on unknown parameter vector $$\boldsymbol\gamma,$$ then we seek a consistent estimator of $$\boldsymbol\gamma$$ eventually to get $$\boldsymbol\Omega\left(\hat{\boldsymbol\gamma}\right).$$

This is the essence of Feasible GLS.

Consider $$y_t=\mathbf X^\top\boldsymbol\beta+u_t, ~~~~\mathbb Eu^2=\exp(\mathbf Z_t^\top\boldsymbol\gamma),$$ where $$\mathbf Z_t$$ can be a function of $$\mathbf X_t$$ but more importantly it's based on all the exogenous variables of the information set on which conditioning is being done.

For consistent estimator of $$\boldsymbol\gamma,$$ we would calculate the OLS residuals $$\hat{u}_t$$ for $$\hat{\boldsymbol\beta}$$ and then we would run the auxiliary linear regression $$\ln{\hat{u}^2_t}=\mathbf Z_t^\top\boldsymbol\gamma+v_t.$$

Wooldridge is also following the same line of attack: take expectation of both sides of $$[2]$$ conditional on $$\mathbf x$$ and you would reach $$(8.30)$$ as $$\mathbb E[\nu\mid\mathbf x]=1.$$ And if $$\nu$$ is independent of $$\mathbf x,$$ the regression of $$\ln u^2$$ on $$x_i$$ follows suit from $$[2].$$

## Reference:

$$\rm [I]$$ Econometric Theory and Methods, Russell Davidson, James G. MacKinnon, Oxford University Press, $$2021,$$ sec. $$7.4.$$