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
 A: 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.
A: Many statistical methods rely on taking the normalized sample mean of the data and comparing it to a critical value under the null hypothesis. For example you might have $H_0:\mu=a$, and your test statistic is $T=\frac{\bar{X}-a}{\hat{\sigma}/\sqrt{n}}$ $\big(\bar{X}$ = sample mean, $\hat{\sigma}$ = square root of the sample variance $\big)$, which you then compare to some critical value to reject/fail to reject the null hypothesis.
When there is dependence in the data $\hat{\sigma}$ is no longer the correct quantity to use to normalize your test statistic. Instead you would want to use the square root of the long-run variance. Failing to account for this would lead your hypothesis test to have incorrect Type I errors, and more/less power depending on how the long-run variance compares to the variance.
With real data, it is likely not the end of the world if you do not take the dependence into account (and use the square root of the sample variance instead of the square root of the long-run variance). With that said, because the type of hypothesis test I have described is so central to statistics, when possible it is a good idea to incorporate dependence into the errors of time series data.
A: Even in ARIMA models, the noise is still IID; it's just that the model part has an auto-correlation component, a moving-average component, and a differencing component.  Now, if you were to take data generated by one of these models and then model it with some simpler model (e.g., lacking the auto-correlation component) then the "noise" from the simpler approximation is going to be non-IID due to the fact that it does not take account of a part of the original model.
