- How do I select the best ARIMA model (by trying all different orders and checking the best MASE/MAPE/MSE? where the selection of performance measurement can be a discussion in it's own..)
Out of sample risk estimates are the gold standard for performance evaluation, and therefore for model selection. Ideally, you cross-validate so that your risk estimates are averaged over more data. FPP explains one cross-validation method for time series. See Tashman for a review of other methods:
Tashman, L. J. (2000). Out-of-sample tests of forecasting accuracy: an analysis and review. International Journal of Forecasting, 16(4), 437–450. doi:10.1016/S0169-2070(00)00065-0
Of course, cross-validation is time consuming and so people often resort to using in-sample criteria to select a model, such as AIC, which is how auto.arima selects the best model. This approach is perfectly valid, if perhaps not as optimal.
- If I generate a new model and forecast for every new day forecast (as in online forecasting), do I need to take the yearly trend into account and how? (as in such a small subset my guess would be that the trend is neglible)
I'm not sure what you mean by yearly trend. Assuming you mean yearly seasonality, there's not really any way to take it into account with less than a year's worth of data.
- Would you expect that the model order stays the same throughout the dataset, i.e. when taking another subset will that give me the same model?
I would expect that barring some change to how the data are generated, the most correct underlying model will be the same throughout the dataset. However, that's not the same as saying that the model selected by any procedure (such as the procedure used by auto.arima) will be the same if that procedure is applied to different subsets of the data. This is because the variability due to sampling will result in variability in the results of the model selection procedure.
- What is a good way, within this method to cope with holidays? Or is ARIMAX with external holiday dummies needed for this?
External holiday dummies is the best approach.
- Do I need to use Fourier series approach to try models with
seasonality=672 as discussed in Long seasonal periods?
You need to do something, because as mentioned in that article, the arima function in R does not support seasonal periods greater than 350. I've had reasonable success with the Fourier approach. Other options include forecasting after seasonal decomposition (also covered in FPP), and exponential smoothing models such as bats and tbats.
- If so would this be like
fit<-Arima(timeseries,order=c(0,1,4), xreg=fourier(1:n,4,672) (where the function fourier is as defined in Hyndman's blog post)
That looks correct. You should experiment with different numbers of terms. Note that there is now a fourier function in the forecast package with a slightly different specification that I assume supersedes the one on Hyndman's blog. See the help file for syntax.
- Are initial P and Q components included with the fourier series?
I'm not sure what you're asking here. P and Q usually refer to the degrees of the AR and MA seasonal components. Using the fourier approach, there are no seasonal components and instead there are covariates for fourier terms related to season. It's no longer seasonal ARIMA, it's ARIMAX where the covariates approximate the season.
Best Answer
This is a very good question. I believe it is very closely related to the question why ARIMA is still one time series analysis and forecasting methods that everyone learns - even though its performance in forecasting is mediocre at best.
My nagging suspicion is that this is not because these methods do a better job at describing reality, and yielding better forecasts. (The proof of the pudding is in the eating, and the proof of the modeling is in the predicting. That, at least, is my opinion.) Rather, it's because time series analysis has historically been the domain of theoretical statisticians and mathematicians. And you can prove theorems about ARIMA and related models. Unit roots! Complex numbers! Characteristic polynomials! And their zeros! Much nicer than methods like exponential smoothing, where the forecasting methods predated a rigorous stochastic model (via state space models) by decades.
Rob Hyndman's "Brief history of forecasting competitions" (2020, IJF) is very enlightening to read in this context. It shows how the earlier forecasting competitions were received by statisticians, who had major difficulties in accepting that simple empirical methods could beat their cherished ARIMA models.