Expected value of kronecker product

Question

$W$ is a vector valued random variable distributed as $W \sim N(\mu, \Sigma)$. I need to calculate $E[W \otimes W]$. How can I calculate this expectation?

Answer 1

If $W\sim \mathcal{N}(\mu,\Sigma)$, $$ \mathsf{E}[W\otimes W]=\operatorname{vec}\!\left(\mathsf{E}\!\left[WW^{\top}\right]\right)=\operatorname{vec}\!\left(\Sigma+\mu\mu^{\top}\right). $$

Answer 2

A "tedious" way is to calculate element by element in $E[W_{t+1}\otimes W_{t+1}]$. $E[W_iW_j]=E[W_i]E[W_j]+Cov(W_i,W_j)$.