# Negative Adjusted $R^2$ in twoway effects within model

I am having serious trouble understanding the results of my Fixed Effects panel regression. I am using two fixed effects (on year and regions) and I get a negative Adjusted R2 (i am using the plm package in R). Why? Please find a screenshot of my output below.

I think I am not prone to the usual mistake - all my variables are fairly time variant. Other than I could not find useful explanations for my problem.

Might it have something to do with the unbalancedness of the panel?

Any suggestions would be wonderful!

• Your explanatory variables do not have much explanatory power (as evidenced by the low standard $R^2$), and you have quite a few of them, so that the penalty for complex models as incorporated into the adjusted $R^2$ leads to a negative value. – Christoph Hanck Jan 9 '20 at 17:06
• But the result is the same even if I just use one explanatory variable – BeSeLuFri Jan 9 '20 at 19:11
• You may also want to show these results. – Christoph Hanck Jan 10 '20 at 5:28

It is well-known that the relationship between $$R^2$$ and adjusted $$R^2$$ in a linear regression (and ultimately, a fixed-effects regression can also be seen as a linear regression, see e.g. Difference between fixed effects dummies and fixed effects estimator?) is (see e.g. Is $R^2_{adjusted}$ both unbiased and consistent under the alternative in simple regression?) $$R^2_{adjusted}=1-(1-R^2)\frac{n-1}{n-k}$$ For a simple linear regression ($$k=2$$) as discussed in the comments to the original question we obtain $$R^2_{adjusted}=1-(1-R^2)\frac{n-1}{n-2}$$ Hence, $$R^2_{adjusted}<0$$ in a simple linear regression if $$R^2<1-\frac{n-2}{n-1}$$ or $$R^2<\frac{n-1-(n-2)}{n-1}=\frac{1}{n-1}$$ Hence, adjusted $$R^2$$ is negative when the original $$R^2$$ is very small. In the general case, we obtain $$R^2_{adjusted}<0$$ if $$R^2<\frac{k-1}{n-1}$$ Hence, a somewhat larger $$R^2$$ is possible to still obtain a negative adjusted $$R^2$$.
At the same time, $$R^2_{adjusted}<0$$ can be seen to mostly be a small-sample issue (relative to $$k$$, of course, as the OP's example nicely illustrates) in that the difference between $$R^2$$ and $$R^2_{adjusted}$$ vanishes as $$n$$ increases and that we always (provided there is a constant in the column space of the regressors) have $$R^2\geq0$$.
• Thank you very much! Thus, the negative Adj. R2 is a result of the region and time dummies being included in k. E.g. $$1 - (1 - 0.080791) * ((2669 - 1) / (2669 - (13 + 311 + 8))) = -0.0494$$ with n = 2669, R2=0.080791, k = 12 regressors+intercept+311 region dummies + 8 time dummies (with 312 regions and 9 years). Perhaps I should ask this as a new question: But, does it make any sense to include the time and region dummies in the calculation of Adjusted R2? Without it my Adjusted R2 would be $$1 - (1 - 0.080791) * ((2669 - 1)/(2669 - 13)) = 0.07663795$$ – BeSeLuFri Jan 10 '20 at 9:03