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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.

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