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I just started learning about times-series modeling and I'm confused by the following scenario:

Let's assume we train a ARMA(p, q) model on a time-series $\{x_1, x_2, ..., x_t\}$.

Later in a test set, which starts at time $t+d$ with $d>max(p,q)$ we want to predict the future value $x_{t+d+1}$, given the history $\{x_{t + d - p}, ..., x_{t+d}\}$. How can we obtain the values $\{\epsilon_{t + d - q}, ..., \epsilon_{t+d}\}$?

Do we set the $\epsilon$-values to zero? If yes, what is the advantage of ARMA(p, q) over AR(p) in this case?

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use your model based on values up to t and predict t+1 and given the actual at t+1 compute e(t+1) . Now introduce the actual value at t+1 and predict t+2 . Use actual at t+2 to compute e(t+2) ......etc ... In this way your new e's will be one-period out forecast errors.

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