Definition:Autoregressive Model/Autoregressive Operator

Definition
Let $S$ be a stochastic process based on an equispaced time series.

Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$

Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
 * $\tilde z_t = z_t - \mu$

Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$

Let $M$ be an autoregessive model on $S$ of order $p$:


 * $\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$

Let $\map \phi B$ be defined as:
 * $\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$

where $B$ denotes the backward shift operator.

Then $\map \phi B$ is referred to as the autoregressive operator.

Hence the autoregessive model can be written in the following compact manner:


 * $\map \phi B \tilde z_t = a_t$