Financial Risk Manager Part 1

Explanation

Correct Answer: B - Adjusted R-squared ($\bar{R}^2$) will never be greater than the regression R-squared ($R^2$).

Key Concepts:

1. R-squared ($R^2$): Measures the proportion of variance in the dependent variable explained by the independent variables. It always increases when adding more variables, even if they're irrelevant.

2. Adjusted R-squared ($\bar{R}^2$): Adjusts for the number of predictors in the model. It penalizes the addition of irrelevant variables.

Why Option B is Correct:

• $\bar{R}^2 = 1 - \frac{(1-R^2)(n-1)}{n-p-1}$ where n = sample size, p = number of predictors
• Since $\frac{n-1}{n-p-1} \geq 1$ when $p \geq 0$, $\bar{R}^2 \leq R^2$
• $\bar{R}^2$ can be equal to $R^2$ only when $p=0$ (no predictors)

Why Other Options are Incorrect:

A: Is always positive - False. $\bar{R}^2$ can be negative when the model fits worse than a horizontal line (when $R^2$ is very small relative to the number of predictors).

C: Cannot increase when an additional independent variable is incorporated - False. $\bar{R}^2$ can increase if the new variable significantly improves the model beyond what would be expected by chance.

D: Is always negative - False. $\bar{R}^2$ is typically positive when the model has explanatory power.

Practical Implication:

Researchers use $\bar{R}^2$ for model selection because it accounts for model complexity, preventing overfitting by penalizing unnecessary variables.