# linearmodels.system.results.GMMSystemResults.system_rsquared¶

property GMMSystemResults.system_rsquared: Series

Alternative measure of system fit

Returns
Series

The measures of overall system fit.

Notes

McElroy’s R2 is defined as

$1 - \frac{SSR_{\Omega}}{TSS_{\Omega}}$

where

$SSR_{\Omega} = \hat{\epsilon}^\prime\hat{\Omega}^{-1}\hat{\epsilon}$

and

$TSS_{\Omega} = \hat{\eta}^\prime\hat{\Omega}^{-1}\hat{\eta}$

where $$\eta$$ is the residual from a regression on only a constant.

Judge’s system R2 is defined as

$1 - \frac{\sum_i \sum_j \hat{\epsilon}_ij^2}{\sum_i \sum_j \hat{\eta}_ij^2}$

where $$\eta$$ is the residual from a regression on only a constant.

Berndt’s system R2 is defined as

$1 - \frac{|\hat{\Sigma}_\epsilon|}{|\hat{\Sigma}_\eta|}$

where $$\hat{\Sigma}_\epsilon$$ and $$\hat{\Sigma}_\eta$$ are the estimated covariances $$\epsilon$$ and $$\eta$$, respectively.

Dhrymes’s system R2 is defined as a weighted average of the R2 of each equation

$\sum__i w_i R^2_i$

where the weight is

$w_i = \frac{\hat{\Sigma}_{\eta}^{[ii]}}{\tr{\hat{\Sigma}_{\eta}}}$

the ratio of the variance the dependent in an equation to the total variance of all dependent variables.

Return type

Series