Auto-Regression

A regressive dependence of the values of $X_n$ of a given random sequence $\{X_n : n=0, \pm1, \ldots\}$ on the preceding values of $X_{n-1}, \ldots, X_{n-m}$. A linear auto-regression scheme of order $m$ is defined by a linear regression equation between $X_n$ and $X_{n-k}$, $k=1,\ldots,m$, i.e.

\begin{equation} \tag{*} X_n = \beta_1 X_{n-1} + \dots + \beta_m X_{n-m} + \epsilon_n , \end{equation}

where $\beta_1, \ldots, \beta_m$ are constants and the random variables $\epsilon_n$ are identically distributed with average zero, variance $\sigma^2$ and are uncorrelated (sometimes they are assumed to be independent). An auto-regression scheme is a useful stochastic model for the description of certain time series (the concept of a linear auto-regression scheme was first introduced by G. Yule in 1921) in order to analyze time series describing a system which is oscillating under the effect of internal forces and random external shocks. The auto-regression scheme (*) may be regarded as a stochastic process of a special type: an auto-regressive process.