How to interpret mean squared error matrix in the context of multiple time series?

Question

Consider the time series of 2 variables $x_1$ and $x_2$, put together $y=(x_1,x_2)$: $$ y_t = \begin{bmatrix} v_1\\ v_2 \end{bmatrix} +\begin{bmatrix} c & d\\ e & f \end{bmatrix}y_{t-1} + u_t. $$

The fixed, nonsingular covaraince matrix is: $\Sigma_u = \begin{bmatrix} \sigma_1^{2} & 0\\ 0 & \sigma_2^{2} \end{bmatrix}$.

The mean squared error matrix of the 1-step ahead forecasts is hence also: $\begin{bmatrix} \sigma_1^{2} & 0\\ 0 & \sigma_2^{2} \end{bmatrix}$.

How should I interpret this mean squared error matrix?

Answer 1

Judging by the name it should be \begin{aligned} &\mathbb{E} \begin{bmatrix} \varepsilon_1^2 & \varepsilon_1 \varepsilon_2\\ \varepsilon_2 \varepsilon_1 & \varepsilon_2^2 \end{bmatrix} \\ = \ &\mathbb{E}\begin{bmatrix} (x_{1,t+1}-\hat x_{1,t+1|t})^2 & (x_{1,t+1}-\hat x_{1,t+1|t})(x_{2,t+1}-\hat x_{2,t+1|t})\\ (x_{2,t+1}-\hat x_{2,t+1|t})(x_{1,t+1}-\hat x_{1,t+1|t}) & (x_{2,t+1}-\hat x_{2,t+1|t})^2 \end{bmatrix} \end{aligned} where $\hat x_{i,t+1|t}$ is the forecast of $x_{i,t+1}$ made at time $t$, and $\varepsilon_i$ is the forecast error of $x_i$ (the random variable, not the realization) for $i=\{1,2\}$.