I'm sorry if this is a duplicate, but I can't seem to find the answer to this.
If $Z_t$ is a white noise process and $X_t$ satisfies
$$ \phi(B) X_t = \theta(B) Z_t $$
(where $B$ is the lag operator), then when is $X_t$ stationary?
If $\theta(z)$ and $\phi(z)$ have no common roots and $\phi(z)$ has a root with $|z| = 1 $ then $X_t$ is necessarily non-stationary [Brockwell and Davis, Remark 3 in Chapter 3].
I also know that a moving average of a stationary process with absolutely summable coefficients is also stationary [Brockwell an Davis, Proposition 3.1.2]. So if $\phi(z)$ has no roots in with $|z| \leq 1$ then the Maclaurin series for $1/\phi(z)$ has radius of convergence greater than 1. Therefore $X_t$ is a moving average of $\theta(B) Z_t$ with absolutely summable coefficients, and hence stationary.
By stationarity, I mean weak (=second order) stationarity.
The lecturer has a habit of putting this on the final, so I'd like to know if there's a general theorem governing it.
From a past paper: