Definition:Autoregressive Model

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 $M$ be a model where the current value of $S$ is expressed as a finite linear aggregate of the past values along with a shock:


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

where $a_t$ is an independent shock at timestamp $t$.

$M$ is known as an autoregressive (AR) process of order $p$.

Autoregressive Operator

Also see

 * Definition:Regression Model, which helps to explain the terminology.