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

Question

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!

Answer 1

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 $R^2\geq0$.