How do I find an $h$-step prediction interval (forecast interval) for a zero-mean ARMA(p,q) process
$$
x_t = \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p} + \varepsilon_t + \theta_1\varepsilon_{t-1} + \dots + \theta_q\varepsilon_{t-q} \ ?
$$
Solved – Multistep prediction interval for ARMA(p,q) process
arimaforecastingprediction interval
Best Answer
The $h$-step-ahead point forecast, or the predicted conditional mean, is obtained by predicting one step ahead, $$ \hat x_{t+1} = \varphi_1 x_{t} + \dots + \varphi_p x_{t-p+1} + 0 + \theta_1 e_{t} + \theta_2 e_{t-1} + \dots + \theta_q e_{t-q+1}, $$ taking that forecast and plugging it in place of the true value $x_{t+1}$ and iterating such as $$ \hat x_{t+2} = \varphi_1 \hat x_{t+1} + \dots + \varphi_p x_{t-p+2} + 0 + \theta_1\times 0 + \theta_2 e_{t} + \theta_3 e_{t-1} + \dots + \theta_q e_{t-q+2}, $$ until $\hat x_{t+h}$ is reached. Here $e_t$ stands for the estimate of $\varepsilon_t$.
The $h$-step-ahead $(1-\alpha)$-level prediction interval (large sample approximation) is constructed as $$ [ \ \hat x_{t+h} - q_{\alpha/2}(\hat\sigma^2(h)); \ \hat x_{t+h} + q_{1-\alpha/2}(\hat\sigma^2(h)) \ ] $$ where $q_\alpha$ is the $\alpha$-level quantile of the error distribution with variance $\hat\sigma^2(h)$, $$ \hat\sigma^2(h) = \hat\sigma^2\sum_{j=0}^{h-1} \hat\psi_j^2 $$ where $\hat\sigma^2$ is the estimated error variance and $\hat\psi_j$ are the estimated coefficients of a moving-average representation of the ARMA(p,q) process.
Under normally distributed errors, the interval is $$ \hat x_{t+h} \pm z_{\alpha/2}\hat\sigma(h) $$ where $z_\alpha$ is the $\alpha$-level quantile of the N(0,1) distribution.
This derivation ignores parameter estimation uncertainty, so the actual intervals should be wider (recall the qualifier large sample approximation above). The problem diminishes with the sample size, though, and vanishes asymptotically.
(Based on Brockwell and Davis "Introduction to time series and forecasting" (3rd ed., 2016), p. 93-94.)
Keywords: multi-step, multi-period, multistep, multiperiod, multiple step, multiple period, steps ahead, periods ahead, forecast, predict, forecasting, prediction, point, interval, ARMA, ARIMA.