I have questions regarding ARMA Filters.

Is the output of a ARMA Filter stationary or just wide sense stationary?

I do know that you can obtain an ARMA filter by connecting an MA filter with an AR filter (MA,AR). Does that work the other way around? Can i obtain an ARMA filter by switching the order and connecting an AR filter with an MA filter? (AR,MA)

  • $\begingroup$ hi! Can you restrict yourself to one question per question post here? You don't show any own thoughts on the first problem at all, so we wouldn't really know how to help you understand things there; and as stated, we'd close your question for the first part simply being a homework-typical question without any own attempt. The second part: Hmmm. Hmmmmmmmm. You know things about systems if you're asking about ARMA systems; what about their commutativity is unclear to you? $\endgroup$ Jun 19, 2020 at 16:33
  • $\begingroup$ The first question i have stems from the fact, that i have seen examples of both "Stationary" and " wide sense Stationary" in ARMA Filters, which just confuses me. Regarding the second part. I can feel the light bulb in my brain lighting up. Since connections in series are commutative, that should mean that the same applies in this case. So the order won't make a difference. $\endgroup$
    – Noobcoder
    Jun 20, 2020 at 9:45

1 Answer 1


Hi: The use of the term "stationary" is vague because there are a few different kinds. But usually, the term "stationary" is meant to mean wide-sense stationary. But, since you've also used the term wide sense stationary in your question, I'm thinking that you're use of the term stationary might just be referring to the output ( by output, I'm assuming you mean the estimated residual noise term ) having a constant (zero) mean and constant variance.

Wide sense stationary refers to constant mean, constant variance and $corr(X_{t1}, X_{t2})$ being only a function of, the time difference, $t2 - t1$.

But, the general process in ARMA modelling ( in statistical time series that is. dsp could be different), is to take the estimated residuals and check that they have a constant mean of zero, constant variance and that they are uncorrelated. This is because the ARMA framework assumes that the error term is white noise so the estimated residuals should be uncorrelated and have zero mean and constant variance.

So, the test that is done after the ARMA model is identified and estimated is testing that the estimated residuals are white noise. But, white noise is a specific case of stationarity so one could argue that you're kind of testing for wide-sense stationarity ( except that corr should be zero so not even a function of the time difference ).

Note that there is another term called "strict sense stationarity" which is a stronger form of stationarity where one requires that the joint distribution of $X_{1}, \ldots, X_{n}$ is the same as the joint distribution of $X_{t+1}, \ldots, X_{t+n}$.

  • $\begingroup$ I guess I have just found the culprit. I might have been confused about the way they use the terms of stationarity. I thought Stationarity= Strict stationarity. Just to clear things up, white noise is wide sense stationary, and is not Strict sense Stationary? $\endgroup$
    – Noobcoder
    Jun 20, 2020 at 9:50
  • $\begingroup$ @Noobcoder it's both. It's the classical example of a strict sense stationary process, and every strict sense is also a weak sense stationary process, inherently. $\endgroup$ Jun 20, 2020 at 9:52
  • $\begingroup$ strict is much stronger and much more difficult to test for. in practice or textbooks, I have never seen it tested for. $\endgroup$
    – mark leeds
    Jun 20, 2020 at 11:56

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