# How to predict more than one future values in ARMA model

I want some help with predicting more than one future values in ARMA. I saw that the similar question has been asked here. But it is only helpful for predicting one future value.

For estimation of the error terms for the time series $$y_1\ldots y_n$$, I am using the following equation...

$$\epsilon_t = y_t-(\sum_{i=1}^p\phi_iy_{t-i}+\sum_{i=1}^q\theta_i\epsilon_{t-i})$$

If I wanted to predict $$y_{n+1}$$, I would use the following equation...

$$y_{t+1} = \sum_{i=1}^p\phi_iy_{t-i+1}+\sum_{i=1}^q\theta_i\epsilon_{t-i+1}$$

Since $$E(\epsilon_{t+1})=0$$ (reference)

But then if I want to predict the second value I will use the previous equation with subscript $$t+2$$. But in that equation, I'll need $$\epsilon+1$$ since it will have the term $$\theta_{0}\epsilon_{t+1}$$. And the process will go on. I can't use the value zero everywhere. What should I do in this case. Please help.

P.S.: Currently, my algorithm is to estimate the future value $$y_{t+1}$$ and then calculate the error term $$\epsilon_{t+1}$$ using the same estimated value. And then I continue predicting the future values this way, but is this appropriate?

• The expected value of the error is zero, by assumption for an ARIMA model. So if you want an expectation forecast, you should indeed set future $\epsilon_t=0$. Commented Sep 15, 2023 at 15:22
• yes, but in this way all $\epsilon_{t+}$ values will be 0. I can consider $\epsilon_{t+1}=0$ for predicting $y_{t+1}$. But what should I do when I have to predict $y_{t+2}$? Commented Sep 15, 2023 at 15:28
• You again use $\epsilon_{t+2}=0$, because by assumption, the innovations are IID normally distributed with mean zero. This actually is the reason why ARIMA forecasts usually turn into flat lines very quickly, which is apparently surprising to many people: we have lots of questions about flat or constant ARIMA forecasts, e.g., this. But that is just what ARIMA does: it separates the forecastable variations from the unforecastable residuals. Commented Sep 15, 2023 at 18:45
• yup, I'm doing that but when I predict $y_{t+2}$, I have the term $\theta_{0}\epsilon_{t+1}$. What value should I put there? (sorry, I had a typo in the main issue, I mentioned $\theta_{t+1}\epsilon_{t+1}$) but I fixed it now.) Commented Sep 15, 2023 at 19:42
• You still feed in the expected value of the residual and set $\epsilon_{t+1}=0$. This may be helpful; the entire textbook is very very good. Commented Sep 15, 2023 at 19:48