## 8.3 Autoregressive models

In a multiple regression model, we forecast the variable of interest using a linear combination of predictors. In an autoregression model, we forecast the variable of interest using a linear combination of *past values of the variable*. The term *auto*regression indicates that it is a regression of the variable against itself.

Thus, an autoregressive model of order \(p\) can be written as
\[
y_{t} = c + \phi_{1}y_{t-1} + \phi_{2}y_{t-2} + \dots + \phi_{p}y_{t-p} + \varepsilon_{t},
\]
where \(\varepsilon_t\) is white noise. This is like a multiple regression but with *lagged values* of \(y_t\) as predictors. We refer to this as an **AR(\(p\)) model**, an autoregressive model of order \(p\).

Autoregressive models are remarkably flexible at handling a wide range of different time series patterns. The two series in Figure 8.5 show series from an AR(1) model and an AR(2) model. Changing the parameters \(\phi_1,\dots,\phi_p\) results in different time series patterns. The variance of the error term \(\varepsilon_t\) will only change the scale of the series, not the patterns.

For an AR(1) model:

- when \(\phi_1=0\), \(y_t\) is equivalent to white noise;
- when \(\phi_1=1\) and \(c=0\), \(y_t\) is equivalent to a random walk;
- when \(\phi_1=1\) and \(c\ne0\), \(y_t\) is equivalent to a random walk with drift;
- when \(\phi_1<0\), \(y_t\) tends to oscillate around the mean.

We normally restrict autoregressive models to stationary data, in which case some constraints on the values of the parameters are required.

- For an AR(1) model: \(-1 < \phi_1 < 1\).
- For an AR(2) model: \(-1 < \phi_2 < 1\), \(\phi_1+\phi_2 < 1\), \(\phi_2-\phi_1 < 1\).

When \(p\ge3\), the restrictions are much more complicated. R takes care of these restrictions when estimating a model.