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: $(1.2.2)$
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1$: Introduction: