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Can someone explain how confidence intervals for ARIMA forecasts are derived? I can't seem to find any good explanation of it. From what I've read it seems like because an ARIMA process can be expressed as an infinite valued MA process then the forecast values are normally distributed. If this is true then how do you derive the standard deviation of the forecast?

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  • $\begingroup$ It's from a prediction interval. $\endgroup$ Commented Oct 15, 2019 at 6:50

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For reference, let your model be:

$$X_t=\phi X_{t-1} + \epsilon_t$$

Say you have data from 1 to $T$. So your point forecast for $X_{T+1}$ would be $$E(X_{T+1}|\{X_t\}_{t=1}^T) = \phi X_T$$ and the confidence interval of $X_{T+1}$ would depend on the distribution you assume for $\epsilon_T$. So forecast follows normal if you assume so for the error term.

There is another important point to be made here. If you are modelling your data, then you actually don't know the model parameter, $\phi$. So you estimate $\phi$ by some method (OLS, MLE, etc.). Based on this estimate you have a point forecast which is $\hat{E}(X_{T+1}|\{X_t\}_{t=1}^T)$, i.e., you have an estimate of $E(X_{T+1}|\{X_t\}_{t=1}^T)$ and so another layer of uncertainty in your forecast. In general the parameter estimates are asymptotically normal. So this is another place from where normality comes into picture. Note however that here we define the confidence interval of forecast as that coming from uncertainty in parameter estimate. In contrast, prediction interval of forecast is the one that also includes the uncertainty from $\epsilon$.

For excellent discussion on this see here.

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  • $\begingroup$ Thank you that cleared up a few things. So I guess what I'm really asking about is the prediction interval. After I make a point forecast how do I calculate the upper and lower bounds for the forecast? $\endgroup$ Commented Oct 16, 2019 at 22:27
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    $\begingroup$ Taking example from my answer, assume you get $\hat{\phi}$. Whichever, method you use you will also get (estimate of) $Var(\hat{\phi})$. From the estimated residuals that you will get estimate of $\hat{\sigma} \equiv Var(\epsilon_{T+1})$. Assuming independence of $\hat{\phi}$ and $\hat{\sigma}$ (as far as I know, this assumption is very reasonable for OLS. For other methods I am not sure), estimate of $Var(X_{T+1})=Var(\hat{\phi}) + \hat{\sigma}$. So assuming normality of error term, you can use this estimate of variance to get bounds of confidence interval. $\endgroup$
    – Dayne
    Commented Oct 17, 2019 at 6:26

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