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I know that McElroy R^2 is a measure of goodness of fit for Seemingly Unrelated Regressions (SUR models), but how can one judge that the estimated equations are well fit by using the McElroy R^2?

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This kind of $R^2$ is a monotonic transformation of the F statistic from the the test that all the coefficients in the system are zero. It is bounded on $[0,1]$, so it has the same kind of interpretation as the ordinary $R^2$:

$$ \text{McElroy }R^2 = 1-\frac{U'W*U}{Y'W*Y},$$ where

  • U is the stacked vector of residuals
  • Y is the stacked vector of dependent variables
  • W is the variance-covariance matrix of residuals

If your model is the cat's pajamas, the vector of residuals will be close to zero, so $R^2 \approx 1$ since the second term drops out. If the model has no explanatory power, then $U\approx Y$, so $R^2 \approx 1-1=0$.

The full citation is McElroy, Marjorie B. (1977) "Goodness of Fit for Seemingly Unrelated Regressions", Journal of Econometrics, 6(3), November; 381-387. I can't seem to find an ungated link.

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