What does an ARIMA model with 6 values mean?

Question

I am trying to get my head around ARIMA modelling.

I get the idea of it at least on a superficial level.

Looking around you see a lot about ARIMA(p,d,q) where p is the autoregressive term, d is the differencing and q is the moving average term.

However when I use the auto.arima function in R and plot the result it gives me an output with "ARIMA(2,1,0)(1,0,0)[12] with Drift"

Can anyone explain how this relates to ARIMA(p,d,q) I believe the [12] might be due to the fact that it is monthly data I am using to build the model so it has frequency 12. But I don't see why there is 6 values (2,1,0)(1,0,0) rather than just 3 in the form (p,d,q).

Can anyone shed any light on this?

Answer 1

This is not an ARIMA model (strictly speaking) but a SARIMA - seasonal ARIMA - model. The first set of coefficients (i.e. (2,1,0) in your example) pertains to usual differencing, AR- and MA-terms, the second set (i.e. (1,0,0) in your example) means seasonal differencing, order of the seasonal AR components and order of the seasonal MA component.

Seasonal differencing means subtracting not the previous observation, but rather the same observation from the previous season (i.e. the $y_{t-12}$ if you have monthly data), likewise seasonal AR means using the observation from the same period of the previous season and the season before that and so on, seasonal MA means the error from the same period of the previous season and the season before that and so on.

This latter of course requires the knowledge of the frequency (number of periods per season), that is given in square brackets.