An ARIMA model is a generalization of a ARMA model. ARIMA models are applied in cases where data show a clear tendency to non-stationarity. In these cases, to eliminate the non-stationarity, an initial differentiation step is added to the ARMA algorithm (corresponding to the integrated part of the model) that is applied one or more times.
This algorithm is therefore essentially composed of three parts:
- The part AR that determines a regression on its own delayed (that is, previous) values to the evolving variable of interest.
- The MA part. It indicates that the regression error is actually a linear combination of error terms whose values have occurred simultaneously and at various times in the past.
- The integrated part; it indicates that the data values have been replaced with the difference between their current values and the previous values (and this differentiation process may have been performed more than once).
The purpose of each of these features is to make the model suitable for data in the best possible way.
To formulate the representative equation of the ARIMA model we start from the ARMA model equation:
Simply move the AR part to the right side of equation to obtain the following equation (less than the constant c):
By introducing the lag operator (L), we can rewrite this equation as follows:
Remember: The lag operator (L) operates on an element of a time series to produce the previous element, with the meaning that LYt = Yt-1.
Assuming that:
Which expresses precisely the factoring procedure of order d previously carried out to eliminate the non-stationarity. Based on this assumption and setting p = p'-d, we can write the following equation to represent the mathematical formulation of the ARIMA (p,d,q) model using lag polynomials:
The d parameter controls the level of differentiating. Generally d=1 is enough in most cases.