AR model

The autoregressive (AR) model of an order p can be written as AR(p) and is defined as  

$$X_t = \alpha_1 X_{t-1} + \cdots + \alpha_{t-p} X_{t-p} + Z_t$$ (8)

where Z_t is a purely random process and $\text{E} (Z_t) = 0, \textrm{Var} (Z_t) = \sigma_Z^2$.The parameters $\alpha_1, \ldots, \alpha_p$ are called the AR coefficients.

The name ``autoregressive'' comes from the fact that X_t is regressed on the past values of itself.

Example: AR(1) or Markov process

The series will converge if $\vert\alpha\vert < 1$.From the properties of the random process Z_t, we can obtain that