If we assumed that $y \sim N(X\beta,S)$ where
S= $\sigma^2\begin{bmatrix} 1 & \rho & \rho &...\\ \rho & 1 & \rho &...\\ \rho & \rho & 1 &...\\ \rho & \rho & \rho & ...\\ ... & ... & ... & ...\\ \end{bmatrix}$
Then the OLS approximation of $\beta$ would be the same as in the independent case:
$\hat{\beta} = (X'X)^{-1}X'y$.
However, the distribution of $\hat{\beta}$ seems to become harder to determine.
The variance of $\hat{\beta}$ would be $Var((X'X)^{-1}X'y)$ = $(X'X)^{-1}X'SX(X'X)^{-1}$. How can I further simplify this?