This question is regarding the SOLS estimator: $Y_i = X_i\beta_{SOLS} + u_i$ with $N$ observations. There are two assumptions:
SOLS.A1: $E(X_i^Tu_i) = 0$.
SOLS.A2: $E(X_i^TX_i)$ is nonsingular.
So how do you derive the variance of $\hat{\beta}_{SOLS} = (N^{-1}\sum_{i=1}^NX_i^TX_i)(N^{-1}\sum_{i=1}^NX_i^TY_i)$? The hardest part for me is how to reason that
$$ E\left(N^{-1} \left(\sum_{i=1}^N X_i^Tu_i\right) \left( \sum_{i=1}^N u_i^TX_i \right)\right) = E\left(\sum_{i=1}^N X_i^T u_i u_i^T X_i\right). $$