Autoregressive–moving-average model

Builds on Autoregressive model Moving-average model Stationary process Root of a polynomial

Definition

Let $(\varepsilon_t)_{t \in \Z}$ be a white noise with variance $\sigma^2 > 0$ . Let $p, q \ge 0$ , and let $\phi(z) = 1 - \phi_1 z - \cdots - \phi_p z^p$ and $\theta(z) = 1 + \theta_1 z + \cdots + \theta_q z^q$ be real polynomials with no common root. An ARMA( $p$ , $q$ ) process is a weakly stationary process $(X_t)_{t \in \Z}$ with mean $\mu$ such that $X_t - \mu - \sum_{i=1}^{p} \phi_i (X_{t-i} - \mu) = \varepsilon_t + \sum_{j=1}^{q} \theta_j \varepsilon_{t-j} \qquad \text{for all } t \in \Z.$ With the backshift operator $B X_t = X_{t-1}$ , this reads $\phi(B)(X_t - \mu) = \theta(B) \varepsilon_t$ .

Remarks

The case $q = 0$ is the autoregressive model and the case $p = 0$ is the moving-average model. A stationary solution exists, and is unique, exactly when $\phi(z) \ne 0$ for $\abs{z} = 1$ . It is causal, a function of present and past shocks only, when $\phi$ has no root with $\abs{z} \le 1$ , and invertible when $\theta$ has no root with $\abs{z} \le 1$ . The condition of no common roots excludes redundant parameters.

Example: $X_t = 0.5 X_{t-1} + \varepsilon_t + 0.3 \varepsilon_{t-1}$ has $\phi(z) = 1 - 0.5 z$ with root $2$ and $\theta(z) = 1 + 0.3 z$ with root $-10/3$ . Both roots lie outside the unit disk and they differ, so this is a causal and invertible ARMA( $1$ , $1$ ) process.

Non-example: $X_t - 0.5 X_{t-1} = \varepsilon_t - 0.5 \varepsilon_{t-1}$ fails the condition on common roots, since both polynomials equal $1 - 0.5 z$ . Its stationary solution is just $X_t = \varepsilon_t$ , a white noise, so the two parameters are not identifiable.

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Autoregressive model concept
Moving-average model concept
Root of a polynomial concept
Stationary process concept
Autoregressive–moving-average model definition

Tag	`F8T3MJX3VJ`
URI	`m2://F8T3MJX3VJ`
Kind	definition
Writing	W0 stub
MSC	62M10, 91B84
Defines	Autoregressive–moving-average model
Uses	Autoregressive model, Moving-average model, Stationary process, Root of a polynomial

Autoregressive–moving-average model

Definition

Remarks

Reading path

Further reading

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