Heteroskedasticity assumption in fGLS into linear form? I am following Chapter 8 ("Heteroskedasticity" p. 259) in the 6th edition of Woolridge Introductory Econometrics: A Modern Approach and I don't understand one piece of the transformation of our model.

For fGLS, we assume [1] $Var(u|\boldsymbol{x}) = \sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k)$, where $x_i, x_2,...,x_k$ are the independent variables appearing in the regression model and the $\delta_j$ are unknown parameters.

Then, we use the definition of conditional variance to say $Var(u|\boldsymbol{x}) = E(u^2|\boldsymbol{x}$), since our zero conditional mean assumption tells us that $(E(u))^2$ is zero.

Now, here is where I'm stuck: Wooldridge says that our assumption [1] above allows us to write $u^2=\sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k)v$, where $v$ has a mean equal to unity, conditional on $\boldsymbol{x}=x_1,x_2, ...,x_k$

Can someone please help me to develop an intuition for this last step that Wooldridge has taken?  Essentially, it seems like we've assumed that $E(u^2|\boldsymbol{x})=\frac{u^2}{v}$ and I don't understand why or the properties that allow us to do this.

I've found this paper https://www.econ.uzh.ch/dam/jcr:e3cddc1b-f89d-4fb4-9474-c2c380355d69/joe_2017.pdf to be useful (specifically, assumption #6 on p. 2), but it doesn't leave me with much intuition for what we're doing and why, especially because I'm fairly new to econometrics.

Thanks-
Maurus
I'm using the 4th edition, but the content should be the same.  It sounds like you are asking how he got from

$$
Var(u|\boldsymbol{x}) = \sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k)
$$

to

$$
u^2=\sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k)v
$$

As you correctly noted, $Var(u|\boldsymbol{x}) = E(u^2|\boldsymbol{x})$ under the zero conditional mean expectation assumption ($E(u|\boldsymbol{x}) = 0$).  So that allows us to replace the left-hand side of the equation, as Wooldridge did.  $E(u^2|\boldsymbol{x})$ turned into $u^2$ because $u^2 = E(u^2|\boldsymbol{x}) + \epsilon$, where $\epsilon$ is the component of $u^2$ that's uncorrelated with any function of $\boldsymbol{x}$.  That component could have been left in at this stage, but it can be removed because later on it would turn into a part of the error term anyway that does not affect the fitted values from the regression of $log(u^2)$ on $\boldsymbol{x}$ (i.e., part of $e$ in the next equation in the book).

On the right-hand side of the equation, Wooldridge adds $v$, which has a mean equal to unity, conditional on $\boldsymbol{x} = (x_1, x_2, ..., x_k)$.  That means that given some set of values for $\boldsymbol{x}$, $v$ is expected to have a multiplicative factor effect of $1$ on $u^2$ (in other words, no effect).  By introducing this term to the equation (and assuming that it is independent of $\boldsymbol{x}$), we can then convert it into the error term $e$ in the subsequent equation:

$$
log(u^2) = \alpha_0 + \delta_1x_1+\delta_2x_2+...+\delta_kx_k+e
$$

because taking a log of a variable that has a mean of $1$ will give us a variable that has a mean of $0$ plus some constant (equal to the negative of the mean of the log values of the original variable $v$).  That constant is moved to the constant term (that's why this last equation has $\alpha_0$ instead of $\delta_0$) and we are left with an equation that can be estimated using OLS.

P.S. The $v$ in this book is not the same $v$ that you found in Assumption 6 in the linked paper.  There it is a function, not a variable.