Correlated error restrictions and OLS I have a VAR model of the form
$$
Y_t = \beta Y_{t-1} + \varepsilon_t
$$
Where $Y_t$ and $\varepsilon_t$ are $n\times 1$ vectors, and $\beta$ is an $n \times n$ matrix.
The residuals $\varepsilon_{t,i}$ ($i=1,2,...,n$) are each normally distributed with a mean of zero, but they may be correlated with one-another (i.e. the variance-covariance matrix $\Sigma$ is non-diagonal). There is no serial correlation, and $\Sigma$ does not change over time.
I have a matrix of data $D$, which has $m$ observation (i.e. $D$ is a $n \times m$ matrix). I wish to estimate the elements of $\beta$ and $\Sigma$ using OLS estimation (and/or maximum likelihood, if that would be different).
How I estimate this model, when errors are known to be correlated?
I've looked at other posts on OLS with correlated errors, but these don't match the constraints of my problem (i.e. errors correlated with one-another, but not serially correlated), and tend to focus on heteroscedasticity.

The correlation structure $\Sigma$ of the error terms has an impact on the standard error estimate of the regression coefficients $\beta$s. That is, you can still use OLS to estimate $\beta$s but the estimate of covariance of $\hat\beta$s underestimates the true covariance because of the non-diagonal $\Sigma$, and hence you will draw a wrong inference. The easiest solution would be to use a sandwich-estimator of the covariance of $\hat\beta$s, where $\hat\beta$s are the OLS estimates of $\beta$s (under the assumption of independent errors $\epsilon_i$).
