multiple linear regression model: scale the dependent variable y by a factor $c ∈ \mathbb{R}, c \neq 0$

Question

In the multiple linear regression model $y = Xβ + u$, if you scale the dependent variable $y$ by a factor $c ∈ \mathbb{R}$, $c \neq 0$, how does the LS estimator $\hat{β}$ change? Does such a change of units change the coefficient of determination $R^2$?

Answer 1

1) Let $y^*=y/c$, $c\neq 0$. The OLS estimator is given by $\hat{\beta} = (X'X)^{-1}X'y$. Given that the design matrix $X$ remains the same, the "new" coefficients vector is $$ \hat{\beta}^*= (X'X)^{-1}X'y^*=\frac{1}{c}(X'X)^{-1}X'y=\frac{\hat{\beta}}{c}. $$

2) Recall that $R^2 = \frac{\sum(\hat{y_i} - \bar{y})^2}{\sum(y_i - \bar{y})^2}$. Note that $\hat{y}=X'\hat{\beta}^*=\frac{1}{c}X'\beta=\frac{\hat{y}}{c}$. In addition, clearly $\bar{y}^* =\frac{1}{c}\bar{y} $, hence, by plugging the results in the $R^2$ equation, you will get $R^{2*}=R^2$.