ARMA vs ARIMA Models

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?

• Why do you think "ARIMA$(p,d,q)$ is actually ARMA$(p+d,q)$"? – corey979 Apr 7 at 15:50
• It is here in Definition section: en.wikipedia.org/wiki/Autoregressive_integrated_moving_average – cross-entropy Apr 7 at 15:53
• 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. – corey979 Apr 7 at 17:18
• So, are the following true then? 1. An ARIMA is a non-stationary ARMA; 2. A stationary ARMA is an ARIMA with d = 0. – cross-entropy Apr 7 at 17:32
• 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. – corey979 Apr 7 at 17:44