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?

  • $\begingroup$ I think I need to read up a bit. $\endgroup$
    – Aqqqq
    Mar 15, 2018 at 20:42

1 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 $i=\{1,2\}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.