Multivariate context:

In this case we define a multivariate object, the covariance matrix $\Sigma_t$,determined from the moving past 250 day history of the 10 series k. The elements of $\Sigma_t$ are determined by:  

$$s_t^{_{(jk)}} = s_t^{_{(kj)}} = {1\over250}\sum_{i=0}^{249}x_{t-i}^{_{(j)}}x_{t-i}^{_{(k)}}.$$ (12)

Note that the diagonal elements (j=k) are exactly the variances ${\sigma_t^{_{(k)}}}^2$ obtained in the univariate context.

Then, using standard multivariate probability theory the forecast distribution p_t^(P) for x_t^(P) is the Gaussian distribution with variance:  

$${\sigma_t^{_{(P)}}}^2 = {1\over100}\left(\sum_{k=1}^{10}s_t^{_{(kk)}} + 2\sum_{k=1}^{10}\sum_{j<k}^{10}s_t^{_{(jk)}}\right).$$ (13)

The determination of $\sigma_t^{_{(P)}}$ as above along with x_t^(P) constitutes the full specification of the prediction-realization pairs that form the starting point of the multivariate performance analysis of the model.