# Is the AR-characteristic polynomial independent of the ARMA equations?

Say that I begin with a time series $X_t$, and say that it satisfies two different ARMA equations:

$$\Phi_1(B)X_t=\Theta_1(B)Z_t$$ and also $$\Phi_2(B)X_t=\Theta_2(B)Z_t.$$ Then must $\Phi_1=\Phi_2$? How about if we further require that the degree of $\Phi_1$ be the same as the degree of $\Phi_2$?

The reason that I am asking is that the augmented Dickey Fuller test checks "whether the AR-characteristic polynomial has a unit root". But does that even mean, if the AR-characteristic polynomial is not invariant of the ARMA equations?

• Related post here. – Richard Hardy Sep 26 '17 at 15:03

• Also, since I am not assume the ARMAs I am beginning with are causal, there isn't an MA($\infty$) presentation, only one that goes both backward and forward. – Andrew NC Sep 26 '17 at 14:44
• Oh, okay, so that further assumes that the $Z_t$'s are iid Gaussian. – Andrew NC Sep 26 '17 at 14:54
• And again, in the situation of my question, there may not be an MA($\infty$) presentation. – Andrew NC Sep 26 '17 at 14:55