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.
 A: Estimate your model equation-by-equation using OLS. This is the efficient estimator for a VAR where the right-hand-side variables are the same in all equations. In that case, GLS collapses to equation-by-equation OLS. MLE collapses to equation-by-equation OLS in case the errors are normal (and you say they are). A more detailed explanation could be found e.g. in Chapter 3 of Lütkepohl "New Introduction to Multiple Time Series Analysis" (2005).
A: 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 $\epsilon_i$).
