AR Characteristic Equation

Definition

By expressing an AR(p) process using the backshift operator: $(1-{\varphi }_{1}B-{\varphi }_2{B}^2-\cdots -{\varphi }_p{B}^p)Z_t=a_t$ The characteristic equation is: $1-{\varphi }_{1}x-{\varphi }_2{x}^2-\cdots -{\varphi }_p{x}^p=0$

Stationarity Condition

An AR(p) process is stationary, if all roots of the characteristic equation lie outside the unit circle, i.e., $|x|>1$

Unit Root

An AR(p) process is called unit root, if one of the solution for its characteristic equation is $|x|=1$

Example: Example Unit Root AR(2)

Example: Unit Root AR(2)

Let $Y_t$ be an AR(2) process defined as: $Y_t=1.5Y_{t-1}-0.5Y_{t-2}+a_t$

Transforming the equation into its characteristic equation:

$$\begin{array}{rl}{a}_t& =Y_t-1.5Y_{t-1}+0.5Y_{t-2}\\ & =Y_t-1.5B{Y}_t+0.5B^2{Y}_t\\ & =Y_t(1-1.5B+0.5B^2)\\ ⟺0& =1-1.5x+0.5x^2\end{array}$$

Solving the polynomial:

$$\begin{array}{rl}1-1.5x+0.5x^2& =0\\ 2-3x+x^2& =0\\ (x-2)(x-1)& =0\end{array}$$

Because one of its solution is $x=1$, then we say that $Y_t$ is an unit root process.