Financial Risk Manager Part 1

Explanation

When two regression models have identical R-squared values, it indicates that the additional variable in the second model ($x_3$) does not contribute to the explanatory power of the model for the dependent variable $y$.

Key Points:

• R-squared represents the proportion of variance in the dependent variable explained by the independent variables
• Identical R-squared values mean $x_3$ provides no additional explanatory power
• Adjusted R-squared penalizes models for including unnecessary predictors
• Since $x_3$ doesn't improve model fit, the adjusted R-squared for Model 2 will be lower than Model 1

Why Other Options Are Incorrect:

• B: Adjusted R-squared only increases if new predictors improve model fit more than expected by chance
• C: Adjusted R-squared accounts for number of predictors, so identical R-squared doesn't imply identical adjusted R-squared
• D: If $x_3$ were statistically significant, it would improve R-squared, which contradicts the given condition

Mathematical Insight:

The adjusted R-squared formula: $R^2_{adj} = 1 - \frac{(1-R^2)(n-1)}{n-k-1}$ where $n$ is sample size and $k$ is number of predictors. Since Model 2 has more predictors ($k=2$) than Model 1 ($k=1$) with the same $R^2$, the denominator increases, making $R^2_{adj}$ smaller for Model 2.