Suppose we have random sample of size T of a real-valued vector $z_t$, which we can assume to have N-dim normal distribution with 0 mean and $I_N$ variance under the null hypothesis. We want to test two hypothesis:
- $H_0: E(z_t z_t^T)=I_N, H_1: E(z_t z_t^T) \neq I_N$,
- $H_0: Cov(z_{i,t}^2, z_{j,t}^2)=0, H_1: Cov(z_{i,t}^2, z_{j,t}^2) \neq 0 \, \forall i \neq j.$
Are there any simple statistical tests to test any of these two hypothesis?
Sure. The (asymptotic) likelihood ratio test for testing $H_0:\Sigma=\Sigma_0$ vs. $H_1:\Sigma\ne \Sigma_0$, where $\Sigma$ is the variance of $z_1$, is given by $$ -2\ln(\Lambda_T)>\chi_{N(N+1)/2}^2, $$ where $$ \Lambda_T=\left(\frac{e}{T}\right)^{-TN/2}\operatorname{det}(\Sigma_0^{-1}S_T)^{T/2}\exp\left(-\frac{1}{2}\operatorname{tr}(\Sigma_0^{-1}S_T)\right), $$ and $S_T=\sum_{t=1}^T(z_t-\bar{z}_T)(z_t-\bar{z}_T)^{\top}$. (See, e.g., Section 8.1 here.)