Can you use MLE to estimate a model with correlated errors? For a simple linear model such as $y_i = X_i\beta + \epsilon $ for i = 1,2...
and we assume that 
$E[\epsilon_i | X] = 0$  and that  $E[\epsilon_i^2|X] = \sigma_i^2 I_T$ for i=1,2...,n 
and that 
$E[\epsilon_i,\epsilon_j'] = \sigma_{ij} I_T$ for $i \neq j$
Can I use MLE to estimate this model? Will the correlation between errors cause problems with the standard errors of the MLE estimates of $\beta$?
 A: The error (correlation) structure you assume simply needs to be incorporated in the likelihood function that you are trying to maximize. Remember that the logarithmic Gaussian likelihood function is generally written
$$
\ell(\mathbf{\beta},\mathbf{\Sigma}) = -\frac{n}{2} \log ( 2 \pi) - \frac{1}{2} \log \left| \mathbf{\Sigma} \right| -\frac{1}{2}\left( \mathbf{y} - \mathbf{X} \mathbf{\beta} \right)^{\mathsf{T}} \mathbf{\Sigma}^{-1} \left( \mathbf{y} - \mathbf{X} \mathbf{\beta} \right)
$$
where in the homoskedastic model we assume spherical errors, i.e. $\mathbf{\Sigma} = \sigma^{2} \mathbf{I}_{n}$, such that the variance-covariance matrix of the error has a constant $\sigma^{2}$ on the diagonal and zeros off-diagonally. But nothing prevents us from assuming (and estimating) a more complex variance-covariance matrix, including heteroskedasticity and/or serial correlation.
A: Yes, the solution is to use weighted least squares. The correlation structure is exchangeable, if I understand your notation. Generalized least squares simultaneously estimate the linear model and correlation structure with the EM algorithm. 
