linearmodels.system.results.GMMSystemResults.system_rsquared¶
- property GMMSystemResults.system_rsquared : Series¶
Alternative measure of 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.