In the above Ordinary Least Squares (OLS) regression, it was assumed that the noise term was i.i.d. normal (identically and independently distributed normally). However, this assumption about the noise term is not always the most appropriate as can sometimes be noted in the residual diagnostics.
In cases where the variance of the noise is not identical
at all points, it is better to perform a
General Least Squares regression that
gives less weight to 
values that are more uncertain.
In cases where the noise is more extreme than that expected
from a normal distribution (i.e. fatter tails), it is better
to perform robust regression. This is appropriate if
it is found that the standardized residuals have large
magnitudes. Robust regression is also advisable when dealing
with small samples as often occurs in climate studies.
There are many different ways to do robust
regression including Least Absolute Deviations (
norm),
M-Estimators, Least Median Squares, and Ranked Residuals.
More details can be found in standard texts such as
Draper and Smith (1998).