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
I didn't get why we are analysing the
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?