In Chapter 2, we found that the point estimates produced by
lm() are reasonable despite the violation of the usual
model ordinary least squares (OLS) model assumptions, but the OLS
standard errors and associated tests are unreliable. The justification
was that the OLS estimator is unbiased regardless of the correlation
structure, but its variance depends on a particular assumption about the
correlation (i.e., independence) that is inconsistent with the sample
ACF. However, this is not quite enough to dismiss the OLS standard error
estimates. It could be that in this particular situation the standard
errors are not sensitive to the model violation, in which case the OLS
standard errors might be a useful approximation. Alternatively, the OLS
standard errors might be substantially wrong. How could you investigate
that, given your current level of understanding of statistical methods?
Imagine the issue arose in a statistical consulting situation where you
are supposed to give an immediate suggestion of how to proceed.