Is it a problem to get a negative adjusted r-squared?

Question

Background: I have the cross-sectional model:

$Y_{i} = a + b X_{1,i} + c X_{2,i} + d X_{3,i} + e X_{4,i} + \nu_i$.

The application is corporate finance. So each $Y_i$ is something like the change in return on assets over a 1 year period for firm $i$, and the regressors are typical corporate finance variables.

In corporate finance, very small values of $R^2$ are common, even sometimes $1\%$. My $R^2$ is around $1\%$ but my Adjusted $R^2$ is $-0.2\%$.

I have never seen papers report negative Adjusted $R^2$ but this could just be because they omit the publication of their adjusted $R^2$ when they see that it is negative.

Question

Is there some problem when the adjusted $R^2$ is negative?

Answer 1

The formula for adjusted R square allows it to be negative. It is intended to approximate the actual percentage variance explained. So if the actual R square is close to zero the adjusted R square can be slightly negative. Just think of it as an estimate of zero.

Answer 2

If $R^2_{adj} \leq 0$, then: \begin{align} 1 &\leq \dfrac{n-1} {n-p} .\dfrac{SSE} {SST} \\[8pt] \dfrac{SST}{SSE} &\leq \dfrac{n-1}{n-p} \\[8pt] \dfrac{SST-SSE}{SSE} &\leq \dfrac{n-1-n-p}{n-p} \\[8pt] R^2 &\leq \dfrac{p-1}{n-p} \end{align} and we know that if $R^2\leq1$, then: $$ 1 \leq \dfrac{p-1}{n-p} $$ $R^2_{adj}$ must be negative. Therefore, \begin{align} n-p &\leq p-1 \\[8pt] \dfrac{n+1}{2} &\leq p \end{align}

This means the number of variables must be more than $\dfrac{n+1}{2}$ to get a negative adjusted R-squared.

Answer 3

The above derivation is incorrect. \begin{align*} R_{adj}^2 &< 0\\ 1-\frac{n-1}{n-p-1}(1-R^2) &<0\\ \frac{n-p-1}{n-1} &< 1-R^2\\ R^2 < 1-\frac{n-p-1}{n-1}\\ R^2 < \frac{p}{n-1}. \end{align*}