# 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.