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?

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

There are three fundamental representations of time series: autocovariance function, Wold's MA(∞) and spectral density. If one of the aforementioned representations for two series is the same, then they are the same series.

Try representing your series as MA(∞) and from there you should be able to figure out the rest.

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  • $\begingroup$ An autocovariance function is sufficient to determine the time-series?! That sounds like a statement that is too strong to be true. Covariance is a very weak invariant -- dependent variables can have 0 covariance. $\endgroup$ – Andrew NC Sep 26 '17 at 14:42
  • $\begingroup$ 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. $\endgroup$ – Andrew NC Sep 26 '17 at 14:44
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    $\begingroup$ Please refer to Cochrane's Time Series for Macroeconomics and Finance, page 26. $\endgroup$ – PhD In Procrastination Sep 26 '17 at 14:46
  • $\begingroup$ Oh, okay, so that further assumes that the $Z_t$'s are iid Gaussian. $\endgroup$ – Andrew NC Sep 26 '17 at 14:54
  • $\begingroup$ And again, in the situation of my question, there may not be an MA($\infty$) presentation. $\endgroup$ – Andrew NC Sep 26 '17 at 14:55

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