linearmodels.system.results.SystemResults.system_rsquared

property SystemResults.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