An ARIMA model is an ARMA model applied after differencing when that step is needed. ARIMA stands for Autoregressive Integrated Moving Average. The orders are written ARIMA(p, d, q).
An ARIMA model is an ARMA model applied after differencing when that step is needed. ARIMA stands for Autoregressive Integrated Moving Average. The orders are written ARIMA(p, d, q).
ARIMA(1, 1, 1) means: difference the series once (d = 1), then use one AR lag (p = 1) and one MA lag (q = 1) on the differenced series.
| Letter | Name | Meaning |
|---|---|---|
| p | Autoregressive order | How many lagged values of the (possibly differenced) series are used |
| d | Integrated / differencing order | How many times the series is differenced |
| q | Moving-average order | How many lagged error terms are used |
Differencing replaces Yt by the change Yt − Yt−1. One difference (d = 1) often removes a slowly wandering level. A series that already looks stable in level may need d = 0, which is just ARMA. Not every time series needs differencing.
Choose d from the plot and from whether the raw ACF dies slowly, not by guessing a large d. Extra differencing can over-correct the series.
A stationary time series has statistical properties that remain reasonably stable over time:
ARMA is aimed at that stable setting. The I in ARIMA is the tool that tries to reach a more stable series by differencing when the original level is not stable enough.
| ARMA | ARIMA | |
|---|---|---|
| Orders | (p, q) with d = 0 | (p, d, q) |
| Differencing | Not used as a built-in step | Allowed through d |
| Typical use | Already roughly stationary series | Series that may need a change-of-level step first |
ARIMA builds on ARMA by allowing differencing through the integrated component. Seasonal ARIMA (SARIMA) is a later topic and adds seasonal orders; it is not required to define ARIMA.