Solutions to Review Question

by Qiang Gao, updated at Mar 21, 2017

Review Question 1.2.6 (Change in units and $R^2$)

Does a change in the unit of measurement for the dependent variable change $R^2$? A change in the unit of measurement for the regressors? Hint: Check whether the change affects the denominator and the numerator in the definition for $R^2$.

Solution

By definition,

$$R^2 = \frac{ \sum_{i=1}^n (\hat{y}_i - \bar{y})^2 } { \sum_{i=1}^n (y_i - \bar{y})^2 }.$$

(1) If we change the unit of measurement of the dependent variable $y$, then $y_i$, $\hat{y}_i$, and $\bar{y}$ simultaneously scales by a factor $\alpha$. Both the denominator and the numerator are scaled by $\alpha^2$, leaving $R^2$ unchanged.

(2) If we change the unit of measurement of the regressors, then $y_i$, $\hat{y}_i$, and $\bar{y}$ donnot change at all. $R^2$ remains unchanged.

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