问一道题：NO.PZ2020010801000017

问题如下：

Both $R^2$ and $\overline R^2$ only increase when adding a new variable. True or false?

选项：

解释：

False. $R^2$ will always increase. $\overline R^2$ can decrease if the degree-of-freedom adjusted residual variance estimator, $\frac{\sum \widehat\epsilon_i^2}{n-k}$, does not decrease when adding a new regressor. This can happen because k is the number of regressors in the model and so if the numerator does not change, adding an additional regressor will increase k and so decrease the denominator.