# When we should use $R^2$ instead of adjusted $R^2$?

Following this topic, adjusted $$R^2$$ has been widely used to validate the trustable explanation adding of the additional independent variables. I am wondering when we should use $$R^2$$ instead of adjusted $$R^2$$ . I have the feeling that the OP in that post also asked this question but no one answered it directly.

Generally speaking you should use adjusted $$R^2$$ almost always if you are using it to decide whether to include additional regressors.

The reason for this is that unadjusted $$R^2$$ will never decrease when you add more regressors to the model (see Verbeek A Guide to Modern Econometrics pp 22). Adjusted $$R^2$$ solves for this issue as it penalizes for the additional regressor since adjusted $$R^2$$ is given by*:

$$\text{adj } R^2 = 1 - \frac{1/(N-K) \sum e_i^2}{1/(N-1) \sum (y_i-\bar{y}_i)^2}$$

where, $$N$$ is the number of observations, $$K$$ is the number of independent regressors, $$e$$ are errors and $$y$$ is the dependent variable.

The inclusion of $$K$$ (unadjusted $$R^2$$ has there only 1) will 'punish' you for frivolously adding new regressors because now any time you add new regressor the $$R^2$$ will be lower. Consequently, this is a good safeguard against overfitting because if you add new regressor and you see $$R^2$$ falling that tells you the regressor you added does not improve fit sufficiently for it to be added.

* Although note there are more than one adjusted $$R^2$$, but typically when people talk about adjusted $$R^2$$ they mean the one above that is used quite widely.

• I would add that you shouldn't care much about either of the two in model selection. Jun 10 at 10:10
• Generally speaking, I would never use adjusted $R^2$ for variable selection / model selection, because it is not optimal w.r.t. any known modelling goal. See "Justification for and optimality of $R^2_{adj.}$ as a model selection criterion". Moreover, $R^2_{adj.}$ is not necessarily a more precise estimator of the population $R^2$ than the regular, unadjusted $R^2$ is. Hence, I do not use $R^2_{adj.}$ much at all in practice. Jun 14 at 10:51