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The autoregressive (AR) model is a stochastic process modelling time series, which specifies the value of the series linearly in terms of the previous values.

Overview

The autoregressive (AR) model is a stochastic process modelling time series, which specifies the value of the series linearly in terms of the previous values. Mathematically, the AR model is specified by its order $p$, i.e. how many previous values in the series affect the present value. Explicitly, an order $p$ model, AR(p) for a time series $Y_t$ is written as:

$$Y_t = c + \sum_{i=1}^p \phi_i Y_{t-i} + \epsilon_t$$

Here, $c$ represents a constant, $\phi_i$ ($i=1,2,\ldots p$) represent the parameters of the model, and $\epsilon_t$ is white noise. Different order models will have different conditions for stationarity. For example, an AR(1) process is stationary if $|\phi_i| < 1$ ($i=1,2,\ldots p$).

The AR model is a special case of the ARMA model, which has additional moving average terms.