ARMA(p,q) is generally denoted as a special case of ARIMA(p,d,q), when d = 0. However, ARIMA(p,d,q) is actually ARMA(p+d,q) so an ARIMA is actually an ARMA model, right? Then, how come ARIMA model is generalized version of ARMA models?

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    $\begingroup$ Why do you think "ARIMA$(p,d,q)$ is actually ARMA$(p+d,q)$"? $\endgroup$ – corey979 Apr 7 '19 at 15:50
  • $\begingroup$ It is here in Definition section: en.wikipedia.org/wiki/Autoregressive_integrated_moving_average $\endgroup$ – cross-entropy Apr 7 '19 at 15:53
  • $\begingroup$ Note also the phrase "having the autoregressive polynomial with $d$ unit roots", which distinguishes from a stationary ARMA process whose AR polynomial has no roots in the unit circle. A fast search gives e.g. these slides as an overview of the unit root topic. $\endgroup$ – corey979 Apr 7 '19 at 17:18
  • $\begingroup$ So, are the following true then? 1. An ARIMA is a non-stationary ARMA; 2. A stationary ARMA is an ARIMA with d = 0. $\endgroup$ – cross-entropy Apr 7 '19 at 17:32
  • $\begingroup$ 2. True. 1. A nonstationary ARMA can be just that, a nonstationary ARMA (e.g. ARMA$(1,0)$ with the AR coefficient >1), not necessarily an ARIMA. $\endgroup$ – corey979 Apr 7 '19 at 17:44

If you do not impose any restrictions on the coefficients then yes, the general ARMA model is the most general, and it subsumes the ARIMA model. The general ARMA model includes both the ARIMA model as well as "explosive" cases. However, it is common to impose the implicit condition that the auto-regressive part of the ARMA model is stationary (autoregressive roots outside the unit circle), and in this latter case, the ARIMA model (with stationary AR part) is the more general, and it subsumes the stationary ARMA model. The relevant subset relations are shown in the diagram below.

This issue can be rather confusing because time-series books and notes ofen fail to explicitly differentiate between the general ARMA model (allowing any coefficient values) and the stationary ARMA model (requiring the roots of the AR part to be outside the unit circle). Many texts regard the "explosive" cases as being of no interest, and they also wish to exclude the ARIMA model from the ARMA form, so they will often implicitly assume they are dealing with the stationary ARMA model without mentioning the constraint explicitly.

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