I am performing a generalized least squares regression based on a design matrix $X$, a response vector $Y$ and a (non-diagonal) covariance matrix $C$, assuming Gaussian errors. I'm not sure what goodness-of-fit tests are applicable. As a first step I could go for a simple chi-squared approach, using the usual formula $\chi^2 = r^T C^{-1} r$, where $r$ is the vector of residuals, but:

  1. Is this formula applicable to the case of a non-diagonal matrix $C$?
  2. Can the generated $\chi^2$ statistic be used in the same way as if $C$ were diagonal (e.g., the weighted least squares case)
  3. Are there better goodness-of-fit statistics that are more frequently used for a problem like this one?

If the residuals, $r$ have a known population covariance matrix $C$ then $r^TC^{-1}r\sim \chi_n^2$ as long as $r\sim \mathcal{N}(0, C)$, regardless of the structure of $C$. For a proof, Google "quadratic forms of random variables". So it has the exact same meaning as goodness-of-fit for weighted least squares and I don't see anything reason why you couldn't use it in a similar way. However, in my experience statistical packages will give you coefficient of determination as a goodness-of-fit statistic.

  • $\begingroup$ Won't the coefficient of determination will be the same if I multiply $C$ by an arbitrary factor? My initial choice of $\chi^2$ was motivated by the need to estimate whether my assigned uncertainties ($C$) are sufficient to explain the fit residuals $r$. It seems to me that I'm losing that piece of information if I use $R^2$. $\endgroup$ – Mathieu Mar 23 '18 at 11:05
  • $\begingroup$ Yes, but note that the GLS estimator is unchanged by scaling of C. The relative structure of C is what provides improved efficiency to GLS, not the absolute scale, just as the magnitude of the variance in LLS is unimportant as long as it is approximately constant. $\endgroup$ – deasmhumnha Mar 23 '18 at 13:03

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