why fit ARMA model to residuals when doing residual analysis?

I started my Time Series Analysis not long ago and I am currently at the residual analysis.

I found, in the course, the tutor was demonstrating residual analysis by fitting an $$AR$$, then $$ARMA(p,0,q)$$ to the residuals. To be very specific, the residuals are from the prediction and expectation of actual series in which we were interested.

By fitting these models to the residuals process, the tutor went ahead analysing the residuals of the residuals (I'll call it RoR, to save some typing here). i.e. plotting RoR itself, $$ACF$$ of RoR, $$PACF$$ of RoR and finally the $$Q-Q$$ Plot for RoR.

I didn't get why we are analysing the RoR.

Sure, if we are looking to use those two models to predict the residual errors such that we could improve our prediction performance for the time series we're interested in, then I get it.

But, if the perfect fitting should, in theory, generate residuals analogous to/resembling normal white noise, why would we fit the residuals to any model and analyse RoR (given white noise is not predictable)?

if I am lucky enough and my model does provide a perfect fit, then should/would I carry on do the residual analysis by fitting models to it, rather than just analysing the residuals itself?

• Why do you think the original model was fit perfectly? – jbowman Dec 2 '18 at 0:57
• @jbowman if say I did get lucky and my residual analysis showed me an almost perfect white noise process, I would be inclined to say it’s fitted almost perfectly. That’s also why I don’t understand the necessity of any further residual analysis , e.g. fit a model on the residuals. – stucash Dec 2 '18 at 1:02
• Well, in that case there isn't any. The tutor is showing you how it's done when your residuals aren't well-modeled by white noise. – jbowman Dec 2 '18 at 1:06
• @jbowman thanks, I think this is the typical scenario that we could use models to understand the residuals further so I think I get that. If there’s no residuals that’s perfectly white noise then your help here suffices :) thanks again – stucash Dec 2 '18 at 1:09