Why is non-iid noise so important to traditional time-series approaches? I've been reading the whitepaper that accompanied Facebook's release of Prophet, it's time-series modeling library. One topic the authors drew attention to was that noise was assumed to be iid; they note, that this assumption goes against the grain for traditional time-series solutions, such as ARIMA. Likewise, their solution doesn't account for autocorrelation or moving averages whatsoever.
In general, the Prophet model accounts for piecewise linear (or logistic growth) trends, seasonality, and holiday effects (where seasonality is captured via a fourier series.)
I'm curious, why are autocorrelation, moving average, and non-iid noise emphasized in traditional time-series approaches, such as ARIMA? Wouldn't it be easier to just use a GLM where seasonal controls (whether that be month, week, etc) could be used to augment the overall linear (or logistic trend)?
https://www.youtube.com/watch?v=OaTAe4W9IfA
https://www.youtube.com/watch?v=fIbgWVMRnis

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.
