Explain Training R-Squared After Adding a Random Feature

# Explain Training R-Squared After Adding a Random Feature For an ordinary least-squares linear regression fitted on the same training observations, what happens to training R-squared after adding a completely random feature: can it increase, remain the same, or decrease? State the fitting assumptions and distinguish the result from adjusted R-squared and test-set performance. ### What a Strong Answer Covers - Nested model spaces and the ability to set the added coefficient to zero. - The relationship between residual sum of squares and training R-squared. - Why chance correlation can improve training fit without genuine predictive value. - The scope limits for regularized fitting, changed samples, adjusted R-squared, and test performance. ```hint Reuse the old solution The expanded model can reproduce every old fitted value by assigning zero to the new coefficient. ``` ### Follow-up Questions - When does the added feature leave training R-squared exactly unchanged? - Why can adjusted R-squared decrease even when ordinary training R-squared increases?