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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?

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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.

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