Some comments:
1) I agree with you, there are many ways to check whether the residuals are white noise,
but the Breusch-Godfrey test (the one our students usually employ in an introductory
course of econometrics) cannot be run this way. Again, they should make this "by
hand".
2) When FGLS estimator is applied to deal with the autocorrelation problem (for example
using Cochrane-Orcutt), statistics based on the rho-differenced data give the mean (and
the standard deviation) of the original dependent variable, whereas all the other
statistics are based on the weighted data. Why this difference?
3) With Cochrane-Orcutt $uhat gives the GLS residuals, but with WLS the residuals are the
originals (not the weighted ones). Again, why this difference? In essence the procedure is
the same (GLS/FGLS).
Thanks a lot in advance.
Best
Javi