# AR(2) Characteristic Equation Equivalence

In a recent question I was given the AR(2) process $$Y_t = \phi_1Y_{t-1} + \phi_2Y_{t-2} + \epsilon_t$$

And I determined that the characteristic equation should be $$\phi(z)=1-\phi_1z-\phi_2z^2$$

However, when finding the roots of the equations, I have been told that $$\phi(z)=1-\phi_1z-\phi_2z^2 = 0$$ Is equivalent to $$C(z) =z^2 -\phi_1z-\phi_2=0$$ Does this make sense? How do you come about this conclusion?

The roots of $$\phi(z)$$ and $$C(z)$$ are reciprocals of each other. To see it, just substitute $$z^{-1}$$ instead of $$z$$ in $$\phi(z)$$ and multiply each side of the equation with $$z^2$$. Thus, you'll obtain $$C(z)$$. Here, equivalence does not mean that $$C(z)$$ and $$\phi(z)$$ have the same $$z$$ roots; but they're closely related.