I'm struggling to fully understand this topic. I've read the blog post that is highly cited on previous Cross Validated topics: https://robjhyndman.com/hyndsight/arimax/, but I still have some questions. I would appreciate any wisdom.
1) Am I understanding "Regression With ARIMA Errors" correctly? This procedure fits the coefficients for the exogenous variables in a simple linear regression for the time series y, and then constructs an ARIMA model for the residuals of this model.
2) It seems to me that the coefficients for the exogenous variables would be poorly estimated in this approach because the variance in y that is explained by the ARIMA components is unaccounted for. Am I right about this? Why is this considered an acceptable approach?
3) If studying the effect of the exogenous variables is the primary goal of my analysis, does that mean I need a real ARIMAX model instead of Regression with ARIMA Errors?