I only have a very basic understanding of time series analysis. As I am learning ARIMA and then ARCH/GARCH models, I have some subtle (at least for me) questions on the common procedure to build such models and what statistical tests to use.
Let's say I have followed the Box-Jenkins methodology to find the best $p$ and $q$ for ARIMA. In most literature, the Ljung-Box test is used in the end to test whether autocorrelation at different lags is zero.
But the assumptions of zero mean and constant variance (homoscedasticity) are never mentioned or checked in all the literature I see. Aren't these two along with the autocorrelation the white noise assumption for residuals? Are those two assumptions not that important or that the usual Ljung-Box tells more than just autocorrelation (I feel like zero autocorrelation at different lags somehow implies constant variance, correct me if I am wrong)
This question I think is somehow related to the first one. From the expression or equation of the ARCH/GARCH models, my understanding is that they are trying to model the inconstant variance (heteroskedasticity) of the residuals. Therefore, in my opinion, the natural way to employ ARCH/GARCH is that after building an ARIMA model, if we found residuals with inconstant variance, we fit an ARCH/GARCH process to the variance of residuals (However, like I asked above, inconstant variance is barely, if not never, checked as an assumption).
Somehow follow-up question of 2. If we have already found that the residuals have inconstant variance (heteroskedasticity), do we fit the ARCH/GARCH process to the variance of residuals or directly to the variance of original time series $y$? Is there a common practice to this? Or we could just fit both to see which one is better based on prediction and other metrics?
Thank you in advance. I know this might be too much to ask. But any input would be greatly appreciated!