# Testing cross-covariance in the residuals of a VAR(p) model

Suppose I have a vector autoregressive model of order $$p$$: $$y_t = c + A_1 y_{t-1} + ... + A_p y_{t-p} + u_t$$, where $$y_t$$ is a $$K\times 1$$ vector and $$A_i$$ are $$K\times K$$ matrices. We assume the errors follow a white noise process: $$u_t\sim WN(0, \Sigma_u)$$. I would like to test whether $$u_t$$ is indeed uncorrelated with $$u_{t-h}$$ for any $$h\in \mathbb{N}$$. I have found that this means that the cross-covariance matrix $$E(u_t u_{t-h}^T)=0$$ for all $$h>0$$. I would like to test this (for a given $$h$$) by estimating $$E(u_t u_{t-h}')$$ with $$\hat{K} = \frac{1}{T} \sum_{t=h+1}^T \hat{u}_t \hat{u}_t'$$, where the residuals come from my (least squares) estimation of the VAR($$p$$) model.

My problem now is that I do not know a test for this. I would appreciate any help for a test statistic.