Finding stationary values for an AR(2) process I have been given the following AR(2) process:
$X_t + \phi_1 X_{t-1} + \phi_2 X_{t-2} = \epsilon_t$
and I need to figure out for which values of $\phi_2$ the process is stationary, when I have been given that $\phi_1=-1/3$. 
I hope that someone can maybe help me with this, since I have no idea about where to start.
 A: As described in this answer, determining if your AR(2) process is stationary breaks down to the question if all (complex) roots of the polynomial 
$$p(z) = z^2 - \left(-\frac{1}{3}\right)z - \phi_2 $$
lie inside the unit disk, i.e. have an absolute smaller than $1$. Can you take it from here?
It's worth noting that you get the polynomial by transformation from the characteristic polynomial of the process, but I stuck to the formulation of the linked answer for consistency's sake.
Here's what I'd do (but not a complete solution): The (possibly complex) solutions of $p(z) = 0$ are $$z_{\pm}=-\frac{1}{6}\pm \sqrt{\frac{1}{36}+\phi_2}.$$
If $\phi_2 > -\frac{1}{36}$, then both solutions are real. Then $|z_+|<1$ iff $-1<z_+$ and $z_+<1$. For $z_+ < 1$ consider
$$1 > -\frac{1}{6} + \sqrt{\frac{1}{36}+\phi_2} \Leftrightarrow \frac{7}{6}> \sqrt{\frac{1}{36}+\phi_2} \Leftrightarrow \frac{4}{3} >\phi_2.$$
Furthermore, if real, $z_+$ is always positive. Hence, $z_+$ lies inside the disc if $\phi_2 > \frac{4}{3}$. Similarly, $z_-$ needs to be considered. And we arrive at an admissable set for $\phi_2$.
If $\phi_2 < -\frac{1}{36}$, then the solitions are complex and the way I wrote them is not precise, but rather 
$$z_{\pm}=-\frac{1}{6}\pm i \sqrt{-\frac{1}{36}-\phi_2}.$$
Here, $z_+$ and $z_-$ are complex conjugate and we have $|z_+|^2=|z_-|^2=z_+z_-$. So they lie in the unit disc iff
$$1> |z_+|^2 = \frac{1}{36} +(-\frac{1}{36} -\phi_2)=-\phi_2,$$
i.e. if $\phi_2 > -1$. We get another admissable set for $\phi_2$: the open interval $(-1,-1/36)$.
Considering that for $\phi_2 = -1/36$ we have that $z_+=z_-=-1/6$ which is inside the unit disc, we can amend that set to $(-1,-1/36]$.
