Consider an OLS model with $n$ observations and $p$ explanatory variables (including an intercept term) $$y=X\beta + \epsilon$$ We may use a White test to (approximately) check for the presence of heteroscedasticity on an $\alpha$-significance level, by conducting an auxiliary regression: $$e = Y\gamma + \eta$$ where $e=y-X\widehat\beta$ ($\widehat\beta$ is the OLS estimator of $\beta$) and $Y$ is the matrix of all the auxiliary regression variables $1, X_i, X_i\odot X_j$. The Lagrange multiplier statistic $nR^2$ of this auxiliary regression is then compared against $\chi_\alpha^2(p)$. If $nR^2>\chi_\alpha^2(p)$, the null hypothesis that $\epsilon$ is homoscedastic is rejected.
Now suppose we believe $\epsilon$ is heteroscedastic with $\Bbb V(\epsilon\mid X) = \Lambda$ where $\Lambda$ is a positive diagonal matrix. Then we may use a WLS model to reestimate the original OLS by specifying the weight $W:=\Lambda^{-1}$, and premultiply the original model by $W^{1/2}$ to obtain $$W^{1/2}y=W^{1/2}X\beta+W^{1/2}\epsilon$$ and the WLS estimator of $\beta$ which is an OLS estimator of $\beta$ under the transformed model should be $$\widehat\beta_{W}=(X^TWX)^{-1}X^TWy.$$ Question: how do we use a White test to measure how accurately $W$ has captured the real heteroscedasticity structure?
A naive thought would be just using the WLS residual $e_W:=y-X\widehat\beta_W$ and the original $Y$ as the inputs for the White test. However, it is evident that $$\Bbb E(\widehat\beta_W)=(X^TWX)^{-1}X^TW\Bbb E(y)=(X^TWX)^{-1}X^TWX\beta=\beta=\Bbb E(\widehat\beta)$$ so $\widehat\beta_W$ usually shouldn't deviate too much from $\widehat\beta$, and thus $e_W$ won't be much different from $e$. Hence such a way of "White test" can tell little about how well $\Bbb V(\epsilon\mid X)$ is addressed by $W$.
Another more plausible thought is consider the transformed model instead of the original OLS model, i.e. use the weighted residual $W^{1/2}e_W$ for the auxiliary regression. But when it comes to choosing the explanatory variables we'll have a problem: should we choose the original $Y$ (spanned by zero, first and second degree terms of $X$)? It seems weird matching $W^{1/2}e_W$. So a more natural idea would be matching $e_W$ with $W^{1/2}X$. But wait, there's no longer an intercept term in $W^{1/2}X$, and there seems to be no way to naturally construct auxiliary variables from $W^{1/2}X$ without introducing a new dof..
So how is this done in reality?
