When adjusted R² stops falling

Idea. Adjusted $R^2$ charges a penalty for each extra regressor. The new regressor must reduce the unexplained share $1-R^2$ by enough to pay for the lost degree of freedom.

Step 1: equal adjusted $R^2$. $$0.6\cdot \frac{49}{46}=(1-R^2)\cdot \frac{49}{45},$$ so $$1-R^2=0.6\cdot \frac{45}{46}=\frac{27}{46}.$$

Step 2. $$R^2=1-\frac{27}{46}=\frac{19}{46}\approx \mathrm{0.413.}$$ So the new regressor must add at least about 0.013 to $R^2$.

The t-statistic rule. Adjusted $R^2$ rises exactly when the new regressor's t-statistic exceeds 1 in absolute value. Here, at the threshold, $$\begin{array}{rl}{t}^2& =\frac{R_{\text{new}}^2-R_{\text{old}}^2}{1-R_{\text{new}}^2}\cdot (n-k-1)\\ & =\frac{0.6/46}{27/46}\cdot 45=1.\end{array}$$ A t-statistic of 1 is far from significant, so adjusted $R^2$ is a lenient criterion for adding variables.