I'm using Wooldridge's Introductory Econometrics (2013) and having some trouble understanding his derivation of the feasible generalized least squares (FGLS) method. He writes, assume that the variance of the error term is given by
$$\mathrm{Var}(u|x) = \sigma^2\exp(\delta_0 + \delta_1x_1 + ... + \delta_kx_k)$$
To transform this into a model that can be estimated using OLS, we rewrite it as
$$u^2 = \sigma^2\exp(\delta_0 + \delta_1x_1 + ... + \delta_kx_k)v$$
where $v$ has a mean equal to unity, conditional on $x = (x_1, ..., x_k)$ If we assume that $v$ is actually independent of $x$, we can write
$$\log(u^2) = a_0 + \delta_1x_1 + ... + \delta_kx_k + e$$
I think I understand that
$$\mathrm{Var}(u|x) = \mathbb{E}[u^2|x] - [\mathbb{E}[u|x]]^2 = \mathbb{E}[u^2|x]$$
and therefore
$$\mathbb{E}[u^2|x] = \sigma^2\exp(\delta_0 + \delta_1x_1 + ... + \delta_kx_k)$$
but how does the expected $\mathbb{E}[u^2|x]$ become $u^2$, and where does the $v$ come from? Additionally, I understand how
$$u^2 = \sigma^2\exp(\delta_0 + \delta_1x_1 + ... + \delta_kx_k) \implies \log(u^2) = a_0 + \delta_1x_1 + ... + \delta_kx_k + e$$
but what happens to the $v$?