A reader left a comment under the MSARIMA post on LinkedIn saying that in order to compare ARIMAs via information criteria, we need to make sure that the candidate models have the same order of differencing. They are right — for the conventional ARIMA. BUT! In the state space formulation, the problem disappears. Here is why.
In the conventional ARIMA, taking differences is treated as a pre-processing step, where we switch from, for example, the sales of a product to the sales increase/decrease. If you had a sample of 100 observations, after first differences you are left with 99 – the first observation has nothing to be subtracted from. So yes, with the conventional approach, ARIMA(1,0,1) and ARIMA(1,1,1) are estimated on different sample sizes, and their information criteria are not directly comparable.
But! MSARIMA (and ADAM in general) is a state space model, and this pre-processing step is not needed. The differences are embedded in the model as additional components: ARIMA(0,1,1) has one, ARIMA(0,2,1) has two. The sample stays the same; what changes is the number of estimated parameters – each new component needs its initial value, and the information criterion penalises that. So the comparison stays fair: same sample, same likelihood basis, different number of parameters. I have several examples of how ARIMA is formulated in the ADAM SSOE framework.
What this implies: any ARIMA model of any orders (non-seasonal, seasonal, or multiseasonal) can be compared with other ARIMA models directly via information criteria, on the same sample of data.
This is not a new idea, by the way. I used this approach in the ARIMA algorithm I developed for the DemandWorks company (now part of Netstock) more than ten years ago, and John Boylan and I later built the State Space ARIMA on the same principle (the paper).
So, yes: if you use the conventional ARIMA (auto_arima from pmdarima, ARIMA from aeon or statsforecast in Python; stats or forecast implementations in R), you NEED to make sure that you compare models with the same order of differencing. But if you use the smooth implementation, you don’t need to worry – everything is already taken care of for you. This comes as a bonus from the specific formulation of the model.