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?