Sample Selection Example

1. Sample Selection Example Bill Evans

2. Draw 10,000 obs at random • educ uniform over [0,16] • age uniform over [18,64] • wearnl=4.49 + 0.08*educ + 0.012*age + ε • Generate missing data for wearnl

3. drawn from standard normal [0,1] • d*=-1.5+0.15*educ+0.01*age+0.15*z+v • wearnl missing if d*≤0 • wearn reported if d*>0 • wearnl_all=wearnl with non-missing obs.

4. εi and vi are assumed to be bivariate normal • E(εi) = E(vi) =0 • Var(εi) = σ2 • Var(vi) = 1 • Corr(εi,vi) = ρ • Cov(εi,vi) = ρ σ • In this case, ρ=0.25 and σ=0.46

5. Yi = β0 + β1educi + β2agei + εi • E[Yi | SSR] = β0 + β1educi + β2agei + E[εi | SSR] • E[εi | SSR] = E[εi | vi>-wiγ] = ρ σ φ(wiγ)/Φ(wiγ)

6. λi = φ(wiγ)/Φ(wiγ) • wiγ = γ0+educ γ1+age γ2+z γ3 • γ2 and γ3 are both constructed to be positive • cov(educ, λi) < 0 and • cov(age, λi) < 0

7. The omitted variable λi is negatively correlated with what is observed in the model • Therefore, the coefficients on educ and age in the selected sample will be too low

8. Numbe rof non-missing observations

9. OLS on all data (no missing obs) Generated by the equation wearnl=4.49 + 0.08*educ + 0.012*age + ε

10. OLS on reported data Smaller MSE Notice that the estimates for educ and age are now smaller

11. Probit, why is data non-missing Generated by the equation d*=-1.5+0.15*educ+0.01*age+0.15*z+v

12. Syntax for Heckman model in STATA . heckman wearnl educ age, select(educ age z); Equation of interest Variables in selection equation

13. Notice β’s have increased over OLS w/ missing data Cannot reject null Rho=0 Sigma right on Rho is a little off

14. Comparison of Estimates

15. Comparison of Estimates [% difference from OLS w/ all data]

16. * run heckman sample selection correction; • . * but use functional form to identify the model; • . heckman wearnl educ age, select(educ age);

17. No where close on rho

18. Comparison of Estimates [% difference from OLS w/ all data]