Feasible GLS estimator

Question

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?

Answer 1

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 $[2].$

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Reference:

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