Answer: The log-log model

## Explanation The correct answer is **C. The log-log model**. ### Understanding the Functional Forms: 1. **Lin-log model**: - Form: \(Y = \beta_0 + \beta_1 \ln(X) + \epsilon\) - This model tests the relationship between absolute changes in Y and relative changes in X (percentage changes in X). 2. **Log-lin model**: - Form: \(\ln(Y) = \beta_0 + \beta_1 X + \epsilon\) - This model tests the relationship between relative changes in Y (percentage changes in Y) and absolute changes in X. 3. **Log-log model**: - Form: \(\ln(Y) = \beta_0 + \beta_1 \ln(X) + \epsilon\) - This model tests the relationship between relative changes in Y and relative changes in X. ### Why Log-Log Model is Correct: - In the log-log model, the coefficient \(\beta_1\) represents the **elasticity** of Y with respect to X. - Elasticity measures the percentage change in Y for a 1% change in X. - When both variables are in logarithmic form, the regression coefficient directly measures the relationship between relative changes (percentage changes) in both variables. - This is exactly what the question asks for: "the linear relationship between relative changes in the dependent variable and relative changes in the independent variable." ### Mathematical Derivation: For the log-log model: \[\ln(Y) = \beta_0 + \beta_1 \ln(X) + \epsilon\] Taking the derivative with respect to X: \[\frac{d\ln(Y)}{d\ln(X)} = \beta_1\] Since \(d\ln(Y) \approx \frac{dY}{Y}\) (percentage change in Y) and \(d\ln(X) \approx \frac{dX}{X}\) (percentage change in X), we get: \[\frac{\% \Delta Y}{\% \Delta X} \approx \beta_1\] Thus, \(\beta_1\) represents the elasticity of Y with respect to X, which is the relationship between relative changes in both variables. ### CFA Context: This concept is important in quantitative methods for understanding different functional forms in regression analysis and their interpretations, particularly in financial modeling where elasticities are commonly used.

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