R-squared

The TSCSREG Procedure

R-squared

The conventional R-squared measure is inappropriate for all models that the TSCSREG procedure estimates using GLS since a number outside the 0-to-1 range may be produced. Hence, a generalization of the R-squared measure is reported. The following goodness-of-fit measure (Buse 1973) is reported:

$R^2=1- \frac{\hat{u}^{'}\hat{V}^{-1}\hat{u}}{y^{'}D^{'}\hat{V}^{-1}{Dy} }$

where ${\hat{u} }$ are the residuals of the transformed model, ${\hat{u}=y-X ( X^{'}\hat{V}^{-1}X)^{-1} X^{'} \hat{V}^{-1}y }$ ,

and ${D=I_{M} - j_{M} j^{'}_{M} (\frac{\hat{V}^{-1}}{j^{'}_{M}\hat{V}^{-1}j_{M} }) }$ .

This is a measure of the proportion of the transformed sum of squares of the dependent variable that is attributable to the influence of the independent variables.

If there is no intercept in the model, the corresponding measure (Theil 1961) is

$R^2=1- \frac{\hat{u}^{'}\hat{V}^{-1}\hat{u}}{y^{'}\hat{V}^{-1}y }$

Clearly, in the case of OLS estimation, both the R-squared formulas given here reduce to the usual R-squared formula.

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