Financial Risk Manager Part 1

Explanation

The correct answer is C: The adjusted R² is always less than or equal to the R².

Key Concepts:

R² (Coefficient of Determination):

• Measures how well the regression predictions approximate the real data points
• An R² of 100% indicates that all changes in the dependent variable are completely explained by changes in the independent variable(s)
• Limitation: Always increases as more independent variables are added to the model, even if those variables are only weakly associated with the response
• This can lead to overfitting, where the model describes random error or noise instead of the underlying relationship

Adjusted R²:

• Accounts for the number of predictors in the model
• Increases only if the new variable improves the model more than would be expected by chance
• Can decrease if a variable improves the model less than expected by chance
• Always less than or equal to R² because it penalizes for adding unnecessary variables

Why Other Options Are Incorrect:

A: The adjusted R² is not always greater than R² - in fact, it's usually less than or equal to R²

B: While both metrics are non-negative and can take values between 0 and 1, they do not always have positive values - both can be zero if there's no linear relationship

D: The adjusted R² does not always increase with more independent variables - unlike R², it only increases when new predictors genuinely enhance the model beyond what would occur by chance