  Regression with $\chi^2$ constructed from the covariance matrix should not be
used for some combinations of covariance matrices and fitting functions. Using
the technique for unsuitable combinations can amplify systematic errors. This
amplification is uncontrolled, and can produce arbitrarily inaccurate results
that might not be ruled out by a $\chi^2$ test. In addition, this technique can
give incorrect (artificially small) errors for fit parameters. I give a test
for this instability and a more robust (but computationally more intensive)
method for fitting correlated data.
