ARMA(1,1) Process Model

Definition

Let $\{a_t\}$ be white noise with variance ${\sigma }_a^2$. ARMA(1,1) is defined as:

$Z_t=\varphi Z_{t-1}+a_t-\theta a_{t-1}$

Stationarity

Stationary if $|\varphi |<1$

Invertible if $|\theta |<1$.

Properties

Assuming the model is stationary,

Property	Expression
Autocovariance	$\{\begin{array}{ll}\frac{1-2\theta \varphi +{\theta }^2}{1-{\varphi }^2}{\sigma }_e^2,& k=0\\ \varphi {\gamma }_0-\theta {\sigma }_e^2,& k=1\\ \varphi {\gamma }_{k-1},& k\ge 2\end{array}$
Autocorrelation	$\frac{(1-\varphi \theta )(\varphi -\theta )}{1-2\theta \varphi +{\theta }^2}{\varphi }^{k-1},k\ge 1$

The ACF decays exponentially like an AR(1) but starting from ${\rho }_1$ instead of ${\rho }_0$ like MA(1).

General Linear Process Representation

ARMA(1,1) process can be represented as general linear process with ${\psi }_j=(\varphi -\theta ){\varphi }^{j-1}$ :

$$Z_t=a_t+(\varphi -\theta )\sum _{j=1}^{\infty }{\varphi }^{j-1}{a}_{t-j}$$