AR(p) Process Model

Definition

The p-th order autoregressive process: $Z_t={\varphi }_1{Z}_{t-1}+{\varphi }_2{Z}_{t-2}+\cdots +{\varphi }_p{Z}_{t-p}+a_t$ where $\{a_t\}$ is white noise.

Stationarity Condition

An AR(p) process is stationary if all roots of its characteristic polynomial $1-{\varphi }_{1}x-\cdots -{\varphi }_p{x}^p=0$ lie outside the unit circle ($|x|>1$). Necessary (but not sufficient) conditions:

• ${\varphi }_1+\cdots +{\varphi }_p<1$
• $|{\varphi }_p|<1$

Yule-Walker Equations

Linear equations that relate the parameters ${\varphi }_i$ of an AR(p) process to its autocorrelations ${\rho }_k$: ${\rho }_k={\varphi }_1{\rho }_{k-1}+{\varphi }_2{\rho }_{k-2}+\cdots +{\varphi }_p{\rho }_{k-p}$ for $k=1,2,\dots ,p$. Solving these equations allows for estimating the parameters from the sample autocorrelations.

Variance

The variance ${\gamma }_0$ of a stationary AR(p) process is: ${\gamma }_0=\frac{{\sigma }_a^2}{1-{\varphi }_1{\rho }_1-{\varphi }_2{\rho }_2-\cdots -{\varphi }_p{\rho }_p}$ where ${\sigma }_a^2$ is the variance of the white noise shocks.