PANEL Procedure

R-Square

The R-square statistic is the proportion of variability in the dependent variable that is attributed to the independent variables. Because of the transformations that are used prior to fitting the final regression model, the conventional R-square measure is not appropriate for most models that the PANEL procedure supports. In random-effects models that use a GLS transform, PROC PANEL calculates the modified R-square statistic proposed by Buse (1973),

upper R squared equals 1 minus StartFraction normal upper S normal upper S normal upper E Over bold y Superscript prime Baseline bold upper D Superscript prime Baseline ModifyingAbove bold upper Omega With caret Superscript negative 1 Baseline bold upper D bold y EndFraction

where normal upper S normal upper S normal upper E is the error sum of squares from the final model fit, ModifyingAbove bold upper Omega With caret Superscript negative 1 slash 2 represents the GLS transform, and bold upper D equals bold upper I Subscript upper M Baseline minus a Superscript negative 1 Baseline bold upper J Subscript upper M Baseline ModifyingAbove bold upper Omega With caret Superscript negative 1 Baseline comma for a equals bold j prime Subscript upper M Baseline ModifyingAbove bold upper Omega With caret Superscript negative 1 Baseline bold j Subscript upper M.

In GLS models that do not have an intercept, the alternate R-square measure, which is attributed to Theil (1961), is calculated as follows:

upper R squared equals 1 minus StartFraction normal upper S normal upper S normal upper E Over bold y Superscript prime Baseline ModifyingAbove bold upper Omega With caret Superscript negative 1 Baseline bold y EndFraction

In fixed-effects models, the R-square measure is

upper R squared equals 1 minus StartFraction normal upper S normal upper S normal upper E Over bold y Subscript w Superscript prime Baseline bold y Subscript w Baseline EndFraction

where bold y Subscript w is the within-transformed dependent variable.

In the case of pooled OLS estimation, all three of the R-square formulas reduce to the usual R-square statistic for linear models.

Last updated: June 19, 2025