In a study of women's wages, the selection equation for labor force participation includes the number of children under age six, but the wage equation does not. What role does this variable play?
AIt is an omitted variable that biases the wage equation
BIt is the exclusion restriction that helps identify the selection correction
CIt replaces the inverse Mills ratio
DIt must also be included in the wage equation for consistency
Young children plausibly affect whether a woman works but not the wage she would be offered. Including it in the probit and excluding it from the wage equation gives variation in λ that is not a function of x alone, which is what identifies the correction.
Question 2 True / False
In the second step of the Heckman procedure, a coefficient on the inverse Mills ratio that is statistically indistinguishable from zero is evidence that sample selection bias is not a serious problem.
TTrue
FFalse
Answer: True
The coefficient on λ̂ equals ρσ, where ρ is the correlation between the selection and outcome errors. If it is zero, selection is not related to the outcome error, and OLS on the selected sample is consistent. A t-test on it is a standard test for selection bias.
Question 3 Multiple Choice
What is the inverse Mills ratio evaluated at zγ?
AΦ(zγ)/φ(zγ)
Bφ(zγ)/Φ(zγ)
C1 − Φ(zγ)
Dzγ/σ
λ(c) = φ(c)/Φ(c), the standard normal density divided by the standard normal CDF. It is large when the probability of selection is low and falls toward zero as selection becomes nearly certain.
Question 4 Short Answer
Explain why OLS on only working women can understate the return to education when selection is positive on the wage error.
Think about your answer, then reveal below.
Model answer: Women with little education work only if their wage error is high enough to make working worthwhile, while highly educated women work even with modest errors. So among workers, the average error is higher at low education than at high education. This raises observed wages at low education relative to high education and flattens the estimated education slope.
The selection term λ is larger for groups with lower participation probabilities, and including it removes this pattern from the error.