Question:
Suppose we estimate the following regression equation:
\[
\ln(x_t) = \alpha_0 + \alpha_1 \ln(y_t) + \epsilon_t, \quad \alpha_0, \alpha_1>0
\]
where \(x_t\) and \(y_t\) are some variables, \(\alpha_0\) and \(\alpha_1\) are the intercept and the slope, respectively, and \(\epsilon_t\) is the residual term. What is the interpretation of the coefficient \(\alpha_1\)?
Suppose we estimate the following regression equation:
\[
\ln(x_t) = \alpha_0 + \alpha_1 \ln(y_t) + \epsilon_t, \quad \alpha_0, \alpha_1>0
\]
where \(x_t\) and \(y_t\) are some variables, \(\alpha_0\) and \(\alpha_1\) are the intercept and the slope, respectively, and \(\epsilon_t\) is the residual term. What is the interpretation of the coefficient \(\alpha_1\)?
Show Hint
In a log-log regression model, the coefficient represents the elasticity, i.e., the percentage change in the dependent variable for a 1% change in the independent variable.
Updated On: Nov 21, 2025
- A 1% increase in \(y_t\) causes a \(\alpha_1\) % increase in \(x_t\)
- A 1% increase in \(y_t\) causes a \(\alpha_1 \times 0.01\) unit increase in \(x_t\)
- A one unit increase in \(y_t\) causes a 100 \(\times \alpha_1\)% increase in \(x_t\)
- A one unit increase in \(y_t\) causes a \(\alpha_1\) unit increase in \(x_t\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the regression equation.
The given regression equation is a log-log model, where both the dependent variable (\(x_t\)) and the independent variable (\(y_t\)) are in logarithmic form. In this form, the coefficient \(\alpha_1\) represents the elasticity of \(x_t\) with respect to \(y_t\). This means that a 1% change in \(y_t\) leads to a \(\alpha_1\)% change in \(x_t\).
Step 2: Evaluate each option.
Option (A): Correct, the interpretation of the coefficient \(\alpha_1\) is that a 1% increase in \(y_t\) causes a \(\alpha_1\)% increase in \(x_t\).
Option (B): Incorrect, \(\alpha_1 \times 0.01\) is not the correct interpretation in a log-log model.
Option (C): Incorrect, the model does not imply a 100 times \(\alpha_1\)% increase for a 1 unit change in \(y_t\).
Option (D): Incorrect, the equation is in logarithmic terms, not linear terms.
Hence, the correct answer is (A).
The given regression equation is a log-log model, where both the dependent variable (\(x_t\)) and the independent variable (\(y_t\)) are in logarithmic form. In this form, the coefficient \(\alpha_1\) represents the elasticity of \(x_t\) with respect to \(y_t\). This means that a 1% change in \(y_t\) leads to a \(\alpha_1\)% change in \(x_t\).
Step 2: Evaluate each option.
Option (A): Correct, the interpretation of the coefficient \(\alpha_1\) is that a 1% increase in \(y_t\) causes a \(\alpha_1\)% increase in \(x_t\).
Option (B): Incorrect, \(\alpha_1 \times 0.01\) is not the correct interpretation in a log-log model.
Option (C): Incorrect, the model does not imply a 100 times \(\alpha_1\)% increase for a 1 unit change in \(y_t\).
Option (D): Incorrect, the equation is in logarithmic terms, not linear terms.
Hence, the correct answer is (A).
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