ARMA(1,1) Process Model

Definition

The simplest mixed model: $Z_t=\varphi Z_{t-1}+a_t-\theta a_{t-1}$ It is stationary if $|\varphi |<1$ and invertible if $|\theta |<1$.

Properties

Variance

${\gamma }_0=\frac{1-2\varphi \theta +{\theta }^2}{1-{\varphi }^2}{\sigma }_a^2$

Autocorrelation Function (ACF)

For $k=1$: ${\rho }_1=\frac{(1-\varphi \theta )(\varphi -\theta )}{1-2\varphi \theta +{\theta }^2}$ For $k\ge 2$: ${\rho }_k=\varphi {\rho }_{k-1}$ The ACF decays exponentially like an AR(1) but starting from ${\rho }_1$ instead of ${\rho }_0$.