How can I solve this quadratic problem?

Question

For φ2> 0, show that the inequalities for $φ_1$ and $φ_2$ which ensure the roots of $φ(z) = 1 − φ_1(z) − φ_2(z^2)$ are greater than $1$ are given by: $φ_1 + φ_2 < 1$, $φ_2 − φ_1 < 1$, and $|φ_2| < 1$

This is actually a problem relating to AR models in an econometrics class but clearly I have not done enough with quadratics

Answer 1

Try conditions $\varphi(1) > 0$ and $\varphi(-1) > 0 $

This is related to the unit disc, which is the main idea behind causality and invertibility of AR processes if I remember right

Try plotting the function to see why the equations above are fitting.

For the absolute value of $\varphi_{2}$, try the formula for sum and product of roots. $z_{1}z_{2} = \frac{-1}{\varphi_{2}}$

Combine the results carefully, taking into account the facts about the roots for the causality of the AR process

Also, don't forget the case, where roots are complex (rather trivial due to how roots are related to each other in this case)