# Unit 11 Characteristic Equations for AR(1) Models

Consider the zero-mean form of an AR(1) model $X_t - \phi_1 X_{t-1} = a_t$ We can rewtite in operator form as

$(1-\phi_1 B)X_t = a_t$

Where B is an operator that turns $$X_t$$ into $$X_{t-1}$$. We can then rewrite this akgebraically (for solving purposes) as a characteristic equation: $1-\phi_1 Z = 0$.

Now recall that AR(1) is stationary $$iff$$ $$\mid \phi_1 \mid$$ is less than 1. Lets solve the characteristic equation for the root now:

$r = z = \frac{1}{\phi_1}$

If $$X_t$$ is stationary , $$\mid r \mid$$ is greater than one. This does not feel important now, but when we get to higher order time series this will save us a lot of thinking. Lets look at a numerical example just to make sure we got it:

$X_t = -1.2 X_{t-1} + a_t$ $X_t + 1.2 X_{t-1} = a_t$ $(1+1.2B)X_t = a_t$ $1+1.2Z = 0$ $r = z = -\frac{1}{1.2} = -0.8333333$ Note that in this case, $$\phi_1 = -1.2$$\$. This is a weird thing with this notation, get used to it.