While studying univariate time series analysis, I got curious about how I can apply the AIC metric to determine the appropriate order for an ARIMA model.
As far as I've investigated, in many texts ARIMA models assume the error terms to be white noise, not Gaussian white noise, which means that there is no distributional assumption on the error terms.
Then, how can we define AIC properly for such ARIMA models? Or is it implicitly assumed to be normal?
That is a very good question.
Especially in the time series forecasting subdiscipline, people are frequently rather sloppy about their terminology, using the term "white noise" to refer to Gaussian white noise implicitly. See here for an otherwise excellent textbook that does so.
Conversely, people who look at ARIMA from a more "statistical" point of view tend to be a little more precise. For instance, Shumway & Stoffer (2016) in section 3.1 on p. 78 explicitly refer to Gaussian white noise in their introduction to ARIMA processes. Brockwell & Davis (2016) introduce ARIMA with general white noise innovations, but when they come to the AIC(c) criterion in section 5.2.2 on p. 151, they are careful to note the condition on the white noise to be Gaussian when they give the explicit formula.
Also, non-Gaussian ARIMA processes are very rarely studied. I would honestly not be able to recall anyone doing so (but then, I'm a forecaster, not a statistician, so my view on the field is biased). There are people looking at integer-valued INARMA processes, but these are different than "ARMA with non-Gaussian white noise innovations".
Bottom line: when the Gaussianity is not explicitly mentioned, but an AIC (or any other information criterion) is calculated, you can probably safely assume it is Gaussian. (If you are a reviewer of a paper, it would be good to ask the authors to be more precise in such a situation.)
