The Ar(1) Process

1. Core Concept Overview & Intuition

An autoregressive process of order one, AR(1), models each observation as a multiple of the previous observation plus a new random shock. The coefficient controls how strongly the process carries its past forward. Under the stationarity condition, the effect of a shock declines geometrically over time, which gives simple formulas for the variance and autocorrelation.

2. Mathematical Formalization & Core Definitions

The source defines the AR(1) process as

$$
X_t=\alpha X_{t-1}+e_t,\qquad |\alpha|<1,
$$

where the innovations have mean zero and are uncorrelated across time:

$$
E(e_t)=0,\qquad \operatorname{Cov}(e_t,e_{t'})=
\begin{cases}
\sigma^2,&t=t',\
0,&t\ne t'.
\end{cases}
$$

The condition $|\alpha|<1$ is the stationarity condition used for this AR(1) model.