You have estimated the following regression model: [math: ln(Y)^=3.4+0.5ln(X)+0.4ln(X)×Female]\widehat {ln(Y)} = 3.4 + 0.5 ln(X)+ 0.4 ln(X) imes Female, where [math: Female] is a dummy variable taking value 1 for a female and 0 for a male individual. Which of the following statements is correct about the estimated elasticity of [math: Y] with respect to [math: X]?[Fill in the blank]

Question

You have estimated the following regression model: [math: ln(Y)^=3.4+0.5ln(X)+0.4ln(X)×Female]\widehat {ln(Y)} = 3.4 + 0.5 ln(X)+ 0.4 ln(X) 	imes Female, where [math: Female] is a dummy variable taking value 1 for a female and 0 for a male individual. Which of the following statements is correct about the estimated elasticity of [math: Y] with respect to [math: X]?[Fill in the blank]

BlackTom AI · Accepted Answer

To evaluate the estimated elasticity of Y with respect to X, note that the regression is in logs: ln(Y) = 3.4 + 0.5 ln(X) + 0.4 ln(X) × Female. The elasticity d ln(Y) / d ln(X) equals the derivative of ln(Y) with respect to ln(X), which is the sum of the coefficients on ln(X) and on the interaction term ln(X) × Female.

Option a: 'A 10% increase in X is predicted to increase Y by 5% for men and 4% for women.' For men (Female = 0), elasticity is 0.5, so a 10% increase in X yields about a 5% increase in Y. This part would be correct for men, but for women (Female = 1) the elasticity would be 0.5 + 0.4 = 0.9, i.e., about 9% for a 10% increase, not 4%. So this option misstates the effect for women.

Option b: 'A 10% increase in X is predicted to increase Y by 50% for men and 90% for women.' A 10% increase in X does not produce such large percentage changes when the elasticity is around 0.5 to 0.9; 0.5 or 0.9 correspond to 5% or 9% changes, not 50% or 90%. This vastly overstates the impact and is inconsistent with the log-linear specification.

Option c: 'A 10% increase in X is predicted to increase Y by 9% for men and 4% for women.' This reverses the correct pattern. For men (Female = 0) the elasticity is 0.5, giving about 5% for a 10% X change, not 9%. For women (Female = 1) the elasticity is 0.9, giving about 9% for a 10% X change, not 4%.

Option d: 'A 10% increase in X is predicted to increase Y by 5% for men and 9% for women.' This matches the computed elasticities: men have elasticity 0.5 → 5% change per 10% X change; women have elasticity 0.5 + 0.4 = 0.9 → 9% change per 10% X change. The logic aligns with the model specification and the interpretation of elasticities in a log-log specification.

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