I was reviewing time series textbooks recently and have been left confused since.

In particular I have looked into the book of Brockwell and Davis (Introduction to Time Series and Forecasting, Second Edition).

In section 2.3 it is said (just after eq 2.3.4) that there is a unique stationary solution to the ARMA(1,1) equation, if the coefficient on the AR part is not 1 in modulus. To quote the text, they look at the equation

$$ X_t − \phi X_{t−1} 􏰀= Z_t + \theta Z_{t−1}$$ 􏰊􏰔 where $\{Z_t\}\sim\text{WN}(0,\sigma^2)$ and $\phi+\theta\neq􏰀0$.

And for $\left| \phi \right| > 1$ they get the representation

$$ X_t = -\theta\phi^{-1}Z_t - (\theta+\phi)\sum_{j=1}^\infty\phi^{-j-1}Z_{t+j}.$$

The proof seems OK to me. But I immediately had the question: How does this relate to explosive processes? I read that a coefficient larger than 1 (in modulus) means that a process is explosive.

In particular the accepted answer of Non-Stationary: Larger-than-unit root performed a simulation that shows the explosive behaviour.

I have the suspicion that this conclusion only holds if we look at adapted processes, as the solution from the textbook looks into the future (it is non-causal). Straight-forward simulations will be adapted since they only incorporate one random variable in each step.

Is my explanation correct, and if not, what am I missing?

  • 1
    $\begingroup$ I'm having trouble understanding your question. Explosive ARMA(1,1) processes are not stationary. Is the question in your title your real question, or is your question whether your explanation of the answer to that question is correct? If the latter, what is your explanation exactly? What is the solution from the textbook that you refer to? $\endgroup$ – The Laconic Mar 30 at 16:57
  • $\begingroup$ My title might be a bit tongue-in-cheek but either the statement “root outside the unit circle implies explosiveness” or the statement “there is a unique stationary solution for an ARMA(1,1) equation with a root outside the unit circle” is wrong. I can quote the whole passage later, but the solution can be represented as a series of the future innovations, which is stationary as the series is absolutely convergent. $\endgroup$ – dlrlc Mar 30 at 18:07
  • $\begingroup$ I think the explosive ARMA process is stationary but non-ergodic, see my question here stats.stackexchange.com/questions/400781/… $\endgroup$ – Joogs Apr 2 at 15:40
  • $\begingroup$ Might this be a duplicate of this? Or the other way around? $\endgroup$ – Richard Hardy Apr 2 at 16:13
  • $\begingroup$ Yes indeed. You both pointed me to the same question. Aksakal’s answer and comments pointed me to the right direction. Apparently the stationary solution to the equation is not the same as the explosive one. The explosive one is causal but the stationary one is not (obviously it uses future innovations). So the solution is just a time-flipped causal AR. It makes sense when you think about it. By flipping time an AR coefficient of 1/2 becomes 2. I guess why this questions doesn’t come up too often is because most people will only consider causal processes. Makes sense for real data. Thank you! $\endgroup$ – dlrlc Apr 2 at 19:02

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