Proving that the estimators of coefficients and variance in GLS model are independent I have come across this question in a textbook: I have a linear model $Y=Xb+u$ with for instance autocorrelation, in order to introduce GLS $Y^*=X^*b+u^*$ (with $Z^* = \Omega^{-1/2}Z$).
Then an additional question is to show that 


*

*$\hat{b}_{GLS} = (X^T\Omega^{-1}X)^{-1} X^T\Omega^{-1}Y$, and

*$\hat{\sigma}^2_{GLS} = \dfrac{\hat{u}^{*T}\hat{u}^*}{N-(k+1)}$


are independent.
And I have no precise idea to do it. I can show that their covariance is equal to zero, but unless I assume Normal distributions this does not help me.
Am I supposed to aim at joint density or is there something better? 
 A: Edit It's been a while and the OP has provided an answer, so here is a more detailed answer than was originally provided due to the homework-nature of the problem. I use a somewhat simplified notation where $Y = X\beta +u, u\sim N(0, \Omega)$.

We first notice that since $\hat{\sigma}^2$ is a function of $\hat{u}$, it suffices to prove that $\hat{u}$ is independent of $\hat{\beta}$.
Let $P = XH$ and $Q = I - P$, where $H = (X'\Omega^{-1}X)^{-1}X'\Omega^{-1}$. Then $\hat{\beta}=HY$, $\hat{\mu} = PY$ and $\hat{u}=QY$. 
Since $Y \sim N(X\beta, \Omega)$, we have $\hat{\beta} \sim N(\beta, H\Omega H')$ and $\hat{u}\sim N(0, Q\Omega Q')$. Thus, it suffices to prove that $\operatorname{Cov}(\hat{\beta}, \hat{u})=0$. Moreover, assuming $X$ is full column rank, we have $$\operatorname{Cov}(\hat{\mu}, \hat{u}) = \operatorname{Cov}(X\hat{\beta}, \hat{u}) = X\operatorname{Cov}(\hat{\beta}, \hat{u})=0 \iff \operatorname{Cov}(\hat{\beta}, \hat{u}) = 0$$
But 
\begin{align}\operatorname{Cov}(\hat{\mu}, \hat{u}) &= \operatorname{Cov}(PY, QY) \\
&= P\Omega Q' \\
&= P\Omega - P\Omega P' \\
&= X(X'\Omega^{-1}X)^{-1}X' - X(X'\Omega^{-1}X)^{-1}X'\Omega X(X'\Omega^{-1}X)^{-1}X' \\
&= X(X'\Omega^{-1}X)^{-1}X' - X(X'\Omega^{-1}X)^{-1}X'\\
&= 0,
\end{align}
 which by the above iff condition is equivalent to what we wanted to show.
A: Building on the 6th point nicely given by Student001, here is what I have come to : 
$cov(\widehat{b}_{MCG},\widehat{u}^*)  = E\left[\left(\widehat{u}^*-E[\widehat{u}^*]\right)
\left(\widehat{b}_{MCG}-E[\widehat{b}_{MCG}]\right)^T\right] \\
 = E\left[\widehat{u}^*\left(\widehat{b}_{MCG}-b\right)^T\right] \\
 = E\left[\widehat{u}^*   \widehat{b}^{\,T}_{MCG}\right] - E[\widehat{u}^*]  b^T \\
=E\left[M^*u^*\left((X^{*T}X^*)^{-1}X^{*T}u^*\right)^{T}\right] \\
=E\left[M^*u^*u^{*^T}X^*\left(X^{*^T}X^*\right)^{-1}\right] \\
=M^*E\left[u^*u^{*^T}\right] X^*(X^{*^T}X^*)^{-1}or\:E[u^*u^{*^T}]=\sigma_u^2 I_T \\
=\sigma^2_u M^*X^*(X^{*^T}X^*)^{-1} \\ %& &\text{ avec  } M^*=I_T-X^*(X^{*T}X^*)^{-1}X^{*T} \\
= \sigma^2_u [X^*(X^{*^T}X^*)^{-1} -X^*(X^{*^T}X^*)^{-1}X^{*^T}X^*(X^{*^T}X^*)^{-1}] \\
=\sigma_u^2[X^*(X^{*^T}X^*)^{-1} -X^*(X^{*^T}X^*)^{-1}] \\
 =0$
Then, as with the right assumption we have that $ \hat{u}^*$ is normal, and $\widehat{b}_{MCG}$ is too, they are independent. 
An last, as $\hat{\sigma}_{GLS}^2$ is a function of $ \hat{u}^*$, $\widehat{b}_{MCG}$ and $\hat{\sigma}_{GLS}^2$ are independent.
By the way, is that of any practical help ? 
