I have problem about the assumptions and model verification of ARIMA models. I know that Gaussian distributed assumption is not necessary for fitting ARIMA models but I wonder why a lot of people standardize their data before fitting an ARIMA and why we check the residuals for being white Gaussian?
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I don't know about standardization, but I can speak to checking residuals for white noise. The traditional Box-Jenkins approach to time series (ARIMA estimation) is about fitting the data until the residuals are white noise. That's how you know that you have built an appropriate model.
Another way to look at this is to consider that an ARIMA model is built up from white noise. It's what happens when you start with white noise and transform the independent noise in various ways. Model fitting works backwards -- you start from the series you have and model terms that "unravel" the data to produce the original white noise.
