Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $(Z_t)_{t \in \mathbb{Z}}$ be an independent and identically distributed white-noise sequence in $L^2(\Omega, \mathcal{F}, \mathbb{P})$ with

\begin{align*} \mathbb{E}[Z_t] = 0, \qquad \operatorname{Var}(Z_t) = \sigma^2 \in (0, \infty), \qquad t \in \mathbb{Z}. \end{align*}

Fix integers $p, q \ge 0$ and coefficients $\phi_1, \dots, \phi_p, \theta_1, \dots, \theta_q \in \mathbb{R}$, and let $(Y_t)_{t \in \mathbb{Z}}$ be the **causal** and **invertible** $\operatorname{ARMA}(p,q)$ process satisfying

\begin{align*} Y_t = \sum_{i=1}^p \phi_i Y_{t-i} + Z_t + \sum_{j=1}^q \theta_j Z_{t-j}, \qquad t \in \mathbb{Z}. \end{align*}

Here causality means there exist real coefficients $(\psi_j)_{j \ge 0}$ with $\sum_{j=0}^\infty |\psi_j| < \infty$ such that $Y_t = \sum_{j=0}^\infty \psi_j Z_{t-j}$ with convergence in $L^2(\Omega, \mathcal{F}, \mathbb{P})$, and invertibility means there exist real coefficients $(\pi_j)_{j \ge 0}$ with $\sum_{j=0}^\infty |\pi_j| < \infty$ such that $Z_t = \sum_{j=0}^\infty \pi_j Y_{t-j}$ with convergence in $L^2(\Omega, \mathcal{F}, \mathbb{P})$.

Fix $t \in \mathbb{Z}$ and let $\mathcal{F}_t = \sigma(Y_s : s \le t)$. For each integer $h \ge 1$ define the $h$-step-ahead forecast

\begin{align*} \hat{Y}_t(h) := \mathbb{E}[Y_{t+h} \mid \mathcal{F}_t]. \end{align*}

Adopt the conventions

\begin{align*} \hat{Y}_t(r) := Y_{t+r} \ \ (r \le 0), \qquad \hat{Z}_t(s) := Z_{t+s} \ \ (s \le 0), \qquad \hat{Z}_t(s) := 0 \ \ (s \ge 1), \end{align*}

for integers $r, s$. Then for every integer $h \ge 1$,

\begin{align*} \hat{Y}_t(h) = \sum_{i=1}^p \phi_i\, \hat{Y}_t(h - i) + \hat{Z}_t(h) + \sum_{j=1}^q \theta_j\, \hat{Z}_t(h - j). \end{align*}

In particular, the one-step forecast is

\begin{align*} \hat{Y}_t(1) = \sum_{i=1}^p \phi_i\, Y_{t+1-i} + \sum_{j=1}^q \theta_j\, Z_{t+1-j}, \end{align*}

and for every $h \ge 1$ the forecast $\hat{Y}_t(h)$ is determined by the observed values $\{Y_s : s \le t\}$, the past innovations $\{Z_s : s \le t\}$, and the earlier forecasts $\hat{Y}_t(r)$ with $1 \le r < h$. Equivalently, $\hat{Y}_t(h)$ is obtained from the ARMA recursion at index $t+h$ by replacing each future innovation $Z_{t+1}, \dots, Z_{t+h}$ by $0$ and each future value $Y_{t+1}, \dots, Y_{t+h-1}$ by its forecast.