<p>Generally speaking you should use adjusted <span class="math-container">$R^2$</span> almost always if you are using it to decide whether to include additional regressors.</p>
<p>The reason for this is that unadjusted <span class="math-container">$R^2$</span> will never decrease when you add more regressors to the model (see Verbeek A Guide to Modern Econometrics pp 22). Adjusted <span class="math-container">$R^2$</span> solves for this issue as it penalizes for the additional regressor since adjusted <span class="math-container">$R^2$</span> is given by*:</p>
<p><span class="math-container">$$\text{adj } R^2 = 1 - \frac{1/(N-K) \sum e_i^2}{1/(N-1) \sum (y_i-\bar{y}_i)^2}$$</span></p>
<p>where, <span class="math-container">$N$</span> is the number of observations, <span class="math-container">$K$</span> is the number of independent regressors, <span class="math-container">$e$</span> are errors and <span class="math-container">$y$</span> is the dependent variable.</p>
<p>The inclusion of <span class="math-container">$K$</span> (unadjusted <span class="math-container">$R^2$</span> has there only 1) will 'punish' you for frivolously adding new regressors because now any time you add new regressor the <span class="math-container">$R^2$</span> will be lower. Consequently, this is a good safeguard against overfitting because if you add new regressor and you see <span class="math-container">$R^2$</span> falling that tells you the regressor you added does not improve fit sufficiently for it to be added.</p>
<hr />
<p><sub>* Although note there are more than one adjusted <span class="math-container">$R^2$</span>, but typically when people talk about adjusted <span class="math-container">$R^2$</span> they mean the one above that is used quite widely. </sub></p>

CC BY-SA 4.0
2021-06-10T10:10:26.300Z
I would add that you shouldn't care much about either of the two in model selection.
Papayapap
CC BY-SA 4.0
2021-06-14T10:51:54.643Z
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"](https://stats.stackexchange.com/questions/408267). 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.
Richard Hardy
CC BY-SA 4.0
2021-06-10T08:17:40.147Z
2021-06-10T08:17:40.147Z
1muflon1
<p>Following <a href="https://stats.stackexchange.com/questions/353807/which-is-better-r-squared-or-adjusted-r-squared/353811#353811">this topic</a>, <strong>adjusted <span class="math-container">$R^2$</span></strong> has been widely used to validate the trustable explanation adding of the additional independent variables. I am wondering when we should use <span class="math-container">$R^2$</span> instead of <strong>adjusted <span class="math-container">$R^2$</span></strong> . I have the feeling that the OP in that post also asked this question but no one answered it directly.</p>

CC BY-SA 4.0
2021-06-14T10:59:29.003Z
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"](https://stats.stackexchange.com/questions/408267). Moreover, $R^2_{adj.}$ is not necessarily a more precise estimator of the population $R^2$ than the regular, unadjusted $R^2$ is.
Richard Hardy
<econometrics>
When we should use $R^2$ instead of adjusted $R^2$?
Phil Nguyen
# When we should use $R^2$ instead of adjusted $R^2$?

(Asked by: username_0 on 2021-06-10)

Following [this topic](https://stats.stackexchange.com/questions/353807/which-is-better-r-squared-or-adjusted-r-squared/353811#353811), **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.

**Tags:** econometrics
> **Comments:**
> **username_1**: 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"](https://stats.stackexchange.com/questions/408267). Moreover, $R^2_{adj.}$ is not necessarily a more precise estimator of the population $R^2$ than the regular, unadjusted $R^2$ is.  

---

## 1 answers

### Answer by username_2 on 2021-06-10. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_3**: I would add that you shouldn't care much about either of the two in model selection.  
> **username_1**: 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"](https://stats.stackexchange.com/questions/408267). 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.  

---

