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Keywords:
improvement of prediction; discrete stationary process
improvement of prediction; discrete stationary process
Summary:
Let $\{W_t\}=\{(X'_{t'}, Y'_t)'\}$ be vector ARMA $(m,n)$ processes. Denote by $\hat{X}_t(a)$ the predictor of $X_t$ based on $X_{t-a}, X_{t-a-1}, \ldots$ and by $\hat{X}_t(a,b)$ the predictor of $X_t$ based on $X_{t-a}, X_{t-a-1}, \ldots, Y_{t-b},Y_{t-b-1}, \ldots$. The accuracy of the predictors is measured by $\Delta_X(a)=\text{E}[X_t-\hat{X}_t(a)][X_t-\hat{X}_t(a)]'$ and $\Delta_X(a,b)=\text{E}[X_t-\hat{X}_t(a,b)][X_t-\hat{X}_t(a,b)]'$. A general sufficient condition for the equality $\Delta_X(a)=\Delta_X(a,a)]$ is given in the paper and it is shown that the equality $\Delta_X(1)=\Delta_X(1,1)]$ implies $\Delta_X(a)=\Delta_X(a,a)]$ for all natural numbers $a$.
References:
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