The autoregressive integrated moving average model, or ARIMA model, is a broadening of the class of ARMA models to include differencing. The word integrated comes from the fact that the stationary model that was fitted is based on the differenced data.

\[y'_t = \phi_1y'_{t-1}+...+\phi_p y'_{t-p}+\theta_iw_{t-1}+...+\theta_qw_{t-q}\] where \(y'_t\) is the differenced series.

The notation for an ARIMA process of order \(p\) for the AR part, order \(q\) for the MA part and differences \(d\) is denoted as ARIMA(\(p, d, q\)), where

\(p=\) order of the autoregressive part,
\(d=\) degree of first differencing involved,
\(q=\) order of the moving average part.

The basic idea is that if differencing the data at some order \(d\) produces an ARMA process, then the original process is said to be ARIMA. Often, a single difference will suffice to yield a stationary series.

Many of the models we have already discussed are special cases of the ARIMA model.

\[ \begin{array}{ll} \hline \text{White noise} & \text{ARIMA(0,0,0)} \\ \text{Random walk} & \text{ARIMA(0,1,0) with no constant}\\ \text{Random walk with drift} & \text{ARIMA(0,1,0) with a constant} \\ \text{Autoregression} & \text{ARIMA(}p\text{,0,0)} \\ \text{Moving average} & \text{ARIMA(0,0,}q\text{)} \end{array} \]