# How to use an ARMA model to make predictions?

After we fit an ARMA model and do all the goodness of fit tests to convince ourselves everything is as it should be, how can we use the model to make predictions? The point of my question is the moving averages part of ARMA, which says predictions depend on the errors in the past (the $\epsilon$ below)

$$y_t = \beta_0 + \sum_{i=1}^p \beta_i y_{t-i} + \sum_{j=1}^q \delta_j \epsilon_{t-j} + \epsilon_t$$

So suppose I'm at $T=t$ and want to make a prediction for $T=t+1$. At this point in time I know $\epsilon_{T<=t}$ so I just sustitute those values and evaluate the model. However, how can I make a prediction for two steps into the future, i.e. at $T=t+2$? To make this prediction I would need $\epsilon_{T=t+1}$ which I don't have yet.

Am I missing the point here?

• You substitute $\epsilon_{t+1}$ with its forecast, which is 0. – Richard Hardy Dec 17 '17 at 19:13
• @RichardHardy Alright, I buy it. Care to put that in a response so I can accept it? – oneloop Dec 17 '17 at 20:52

At time $t$ you do not know $\epsilon_{t+1}$, but you need it to construct the forecast of $y_{t+2}$. What do you do?
Simple: you substitute $\epsilon_{t+1}$ with its forecast. The best forecast for $\epsilon_{t+1}$ under square loss is its conditional expectation $\mathbb{E}(\epsilon_{t+1}|I_t)$ (where $I_t$ is the information available at time $t$). This expectation is $0$: $\mathbb{E}(\epsilon_{t+1}|I_t)=0$. Hence, $\epsilon_{t+1}$ effectively disappears from the forecast of $y_{t+2}$.