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.
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1$\begingroup$ I would not 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. $\endgroup$– Richard HardyJun 14, 2021 at 10:59
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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.
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1$\begingroup$ I would add that you shouldn't care much about either of the two in model selection. $\endgroup$ Jun 10, 2021 at 10:10
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$\begingroup$ 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. $\endgroup$ Jun 14, 2021 at 10:51