For the MANOVA model, we assume $X_{lj}=\mu+\tau_{l}+\epsilon_{lj}$, where $\epsilon_{lj}\sim^{iid} N(\mathbf{0},\Sigma)$.
How can we test/assess whether $\epsilon_{lj}\sim^{iid} N(\mathbf{0},\Sigma)$ or not? We may assume that they have the same covariance matrix.
I'm thinking of using the residual of the usual decomposition used in these types of models and use something like $\hat\epsilon_{lj}'\hat \Sigma^{-1} \hat \epsilon_{lj} \sim^a \chi^2(p) $, where $p$ is the number of independent variables. However, I'm not sure if I can do this. Also, in my example of a possible statistic, I'm not using all the residuals, whereas I should use all of them.
Any help would be appreciated.
