## Quick Reference

A model for a time series having no trend (the constant mean is taken as 0). Let *X*_{1}, *X*_{2}, …, be successive instances of the random variable *X*, measured at regular intervals of time. Let *ε** _{j}* be the random variable denoting the random error at time

*j*. A

*p*th-order autoregressive model (or autoregressive process) relates the value at time

*j*to the preceding

*p*values by

*X** _{j}*=

*α*

_{1}

*X*

*+*

_{j−1}*α*

_{2}

*X*

*+…+*

_{j−2}*α*

_{p}*X*

*+*

_{j−p}*ε*

*,*

_{j}where *α*_{1}, *α*_{2},…, *α** _{p}* are constants. Such a model is written in brief as AR(

*p*). The AR(1) process is a Markov chain. Autoregressive models can also be expressed as moving average models. Models combining both type of process include ARMA models and ARIMA models.

The Yule–Walker equations, introduced by Yule in 1927 and Walker in 1931, relate *α*_{1}, *α*_{2},…, *α** _{p}* to the population autocorrelation values

*ρ*

_{1},

*ρ*

_{2},…,

*ρ*

*:*

_{p}*ρ*_{1}=*α*_{1}+*α*_{2}*ρ*_{1}+…+*α*_{p}*ρ** _{p−1}*,

*ρ*_{2}=*α*_{1}*ρ*_{1}+*α*_{2}+…+*α*_{p}*ρ** _{p−2}*,

⋮

*ρ** _{p}*=

*α*

_{1}

*ρ*

*+*

_{p−1}*α*

_{2}

*ρ*

*+…+*

_{p−2}*α*

*.*

_{p}*Subjects:*
Probability and Statistics.