19.9 ARMA(p,q)
A stationary stochastic process $ ( Y_{t} ) $ follows an autoregressive-moving-average model of order \((p, q)\), or $ (p, q), $ if and only if
\[\begin{equation} Y_{t}=\mu+\phi_{1} Y_{t-1}+\phi_{2} Y_{t-2}+\ldots+\phi_{p} Y_{t-p}+\omega_{t}-\theta_{1} \omega_{t-1}-\theta_{2} \omega_{t-2}-\ldots-\theta_{q} \omega_{t-q} \label{eq:arma_general} \end{equation}\]
for all \(t=0, \pm 1, \pm 2, \ldots\), where \(\left(\omega_{t}\right) \sim \operatorname{IID}\left(0, \sigma_{\omega}^{2}\right)\) y \(\mu, \phi_{1}, \phi_{2}, \ldots, \phi_{p}, \theta_{1}, \theta_{2}, \ldots, \theta_{q}\) are parameters such that all the roots of the polynomial equation
\[ 1-\phi_{1} x-\phi_{2} x^{2}-\ldots-\phi_{p} x^{p}=0 \] are outside the unit circle (condition of stationarity).
A model $ (p, q) $ described by the equation is invertible if all the roots of the polynomial equation
\[ 1-\theta_{1} x-\theta_{2} x^{2}-\ldots-\theta_{q} x^{q}=0 \] are outside the unit circle (condition of invertibility).
The equation can be alternatively written as
\[ \phi(B) Y_{t}=\mu+\theta(B) \omega_{t}, \] where
\[ \phi(B) \equiv 1-\phi_{1} B-\phi_{2} B^{2}-\ldots-\phi_{p} B^{p} \]
is the autoregressive operator or polynomial (AR) of the model, and
\[ \theta(B) \equiv 1-\theta_{1} B-\theta_{2} B^{2}-\ldots-\theta_{q} B^{q} \]
is the moving average operator or polynomial (MA).
19.9.1 ARMA(1,1)
When a stationary process $ ( Y_{t} )$ follows a model $ (1,1), $
\[ Y_{t}=\mu+\phi_{1} Y_{t-1}+\omega_{t}-\theta_{1} \omega_{t-1} \]
with \(\left|\phi_{1}\right|<1,\left|\theta_{1}\right|<1\) and \(\left(\omega_{t}\right) \sim \operatorname{IID}\left(0, \sigma_{A}^{2}\right)\), it can be seen that
- Mean:
\[ \mu_{Y}=\frac{\mu}{1-\phi_{1}} \]
- ACF:
\[ \rho_{k}=\left\{\begin{array}{lr} {\left[\left(\phi_{1}-\theta_{1}\right)\left(1-\phi_{1} \theta_{1}\right)\right] /\left(1-2 \phi_{1} \theta_{1}+\theta_{1}^{2}\right)} & \text { si } k=1 \\ \rho_{1} \phi_{1}^{k-1} & \text { para todo } k>1 . \end{array}\right. \]
- Variance: \[ \sigma_{Y}^{2}=\frac{1-2 \phi_{1} \theta_{1}+\theta_{1}^{2}}{1-\phi_{1}^{2}} \sigma_{A}^{2}=\left[1+\frac{\left(\phi_{1}-\theta_{1}\right)^{2}}{1-\phi_{1}^{2}}\right] \sigma_{A}^{2} \]