AR(p) Process Model

Definition

A process where the current value is a linear combination of its own past values plus a random shock. $Z_t={\varphi }_1{Z}_{t-1}+{\varphi }_2{Z}_{t-2}+\cdots +{\varphi }_p{Z}_{t-p}+a_t$ where $\{a_t\}$ is white noise independent of past $Z$ values.

Tip

$Z_t=a_t+{\sum }_{i=1}^p{\varphi }_i{a}_{t-i}$

Stationarity Condition

An AR(p) process is weakly 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.

Characteristic Equation

Definition

By expressing an AR(p) process using the backshift operator: (1 - \phi_1 B - \phi_2 B^2 - \dots - \phi_p B^p) Z_t = a_t$$ The **characteristic equation** is: 1 - \phi_1 x - \phi_2 x^2 - \dots - \phi_p x^p = 0$$

Properties

Assuming AR(p) is stationary,

Property	Expression
Variance (${\gamma }_0$)	$\frac{{\sigma }_e^2}{1-{\varphi }_1{\rho }_1-\cdots -{\varphi }_p{\rho }_p}$
Autocovariance	${\varphi }_1{\gamma }_{k-1}+{\varphi }_2{\gamma }_{k-2}+\cdots +{\varphi }_p{\gamma }_{k-p},k\ge 1$
Autocorrelation (ACF)	${\varphi }_1{\rho }_{k-1}+\cdots +{\varphi }_p{\rho }_{k-p},k\ge 1$

Only valid on stationary AR(p)