AR Stationarity
When is an AR(p) model stationary?
Answer
In order to be (weakly) stationary we need the characteristic equation of the AR model to have all its roots outside of the unit circle in the complex plane. For a general AR(\(p\)) process,
\[
\Phi(z) \;=\; 1 - \phi_1 z - \phi_2 z^2 - \cdots - \phi_p z^p \;=\; 0,
\]
all solutions \(z_i\) must satisfy \(\lvert z_i\rvert > 1\).
Example (AR(1)):
\[
X_t = \phi\,X_{t-1} + \varepsilon_t
\quad\Longrightarrow\quad
\Phi(z) = 1 - \phi z = 0
\;\Longrightarrow\;
z = \frac{1}{\phi}.
\]
Stationarity requires \(\lvert z\rvert > 1\), hence \(\bigl\lvert\tfrac{1}{\phi}\bigr\rvert > 1\), i.e. \(\lvert\phi\rvert < 1\).