Interpretation of $R^2$ in fixed-effects panel regression For cross-section analysis, $R^2$ represents the fraction of the variation that is explained by the model.
I understand that a fixed-effects panel regression is designed to optimize for the "between $R^2$". However, what is it's interpretation? What is the interpretation of the "within $R^2$" and the "overall $R^2$" (even though they are not optimized for, I assume they still carry some meaning)?
Thanks!
 A: In the fixed effects regression you should actually look at the within $R^2$ rather than the between. Let's consider the three cases:


*

*overall $R^2$: that's the usual $R^2$ which you would get from regressing your dependent variable $Y_{i,t}$ on the explanatory variables $X_{i,t}$.

*between $R^2$: if you collapse your data and remove the time component by taking the means of your variables for each panel unit individually, the $R^2$ from regressing these time de-meaned data gives the between $R^2$. That's the regression $Y_{i,.}$ on $X_{i,.}$ (where $.$ replaces the time-subscript to show that time has been averaged out for each panel unit $i$). So this disregards all the within information in the data.

*within $R^2$: this comes from the prediction equation $(\widehat{Y}_{i,t}-\widehat{\overline{Y}}_{i,.}) = (X_{i,t} - \overline{X}_{i,.})\widehat{\beta})$, where $\widehat{\overline{Y}}_{i,.}$ and $\widehat{\overline{X}}_{i,.}$ are the grand means of your variables. So the within $R^2$ gives you the goodness of fit measure for the individual mean de-trended data which disregards all the between information in the data.


What you want in this case is a good amount of within information that can be exploited by the FE estimator. So if you are interested in $R^2$ you should be looking at the within version in this case.
