How to estimate the variance of correlated observations? Assume we have n observations $x_i$ (i from 1 to n), each from the a normal distribution with mean 0 and some variance component: $X_i \sim N(0, \sigma^2)$. The random variables $X_i$s have some (let's assume known) correlation structure. I.e.: $Corr(X_i, X_j) = \rho_{ij}$.
How should we estimate $\sigma^2$?
How would the correlated structure impact an estimator such as $\hat \sigma^2 = \frac{\sum(x_i^2)}{n}$? Could we devise a better estimator (given the correlation structure is known)?

You can apply GLS equations, e.g. top of p.5 in these lectures:  http://halweb.uc3m.es/esp/Personal/personas/durban/esp/web/notes/gls.pdf
The equation for MSE is $\sigma^2=1/n \sum_{ij} x_i r^{-1}_{ij} x_j$ if you set X to zero and correspond your r to their V
This equation doesn’t work for extreme cases such as correlation 1 between all variables. This case your usual equation will still produce a number
