1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Related post here. $\endgroup$ – Richard Hardy Sep 26 '17 at 15:03
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
  • 2
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.