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.