I am just beginning to learn about ARIMA and have been interested in using linear regression with ARIMA errors (I have been reading through these sections in FPP), and have some questions:
Are you building the ARIMA (or SARIMA) model on the residuals of the linear regression model during training, or on the values of your target variable during training? I have seen it done both ways and assumed it would be built on the residuals.
A slightly more a general question, but why would you not use SARIMAX over linear regression when working with time series? Can there be any advantage of not using it (i.e. is it possible the model would perform worse, assuming you picked the optimal orders for the ARIMA model)? If the residuals are normally distributed, had no autocorrelation, stationary etc. from your linear regression model, then would there be no point in using (S)ARIMA? (I understand if the residuals were genuinely white noise, then there would be no point).
Appreciate any insight into this!