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

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

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

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

Autoregressive Operator

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:

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

Parameter

The parameters of $M$ consist of:

the constant mean level $\mu$

the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shock $a_t$

the coefficients $\phi_1$ to $\phi_p$.

Also see

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

• Number of Parameters of Autoregressive Model: $M$ has $p + 2$ parameters

• Results about autoregressive models can be found here.

Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

$1$: Introduction:

$1.2$ Stochastic and Deterministic Dynamic Mathematical Models

$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Autoregressive models: $(1.2.2)$