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1 Sample Selection Example Bill Evans

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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

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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.

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4 ε i and v i are assumed to be bivariate normal E(ε i ) = E(v i ) =0 Var(ε i ) = σ 2 Var(v i ) = 1 Corr(ε i,v i ) = ρ Cov(ε i,v i ) = ρ σ In this case, ρ=0.25 and σ=0.46

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5 Y i = β 0 + β 1 educ i + β 2 age i + ε i E[Y i | SSR] = β 0 + β 1 educ i + β 2 age i + E[ε i | SSR] E[ε i | SSR] = E[ε i | v i >-w i γ] = ρ σ φ(w i γ)/Φ(w i γ)

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6 λ i = φ(w i γ)/Φ(w i γ) w i γ = γ 0 +educ γ 1 +age γ 2 +z γ 3 γ 2 and γ 3 are both constructed to be positive cov(educ, λ i ) < 0 and cov(age, λ i ) < 0

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7 The omitted variable λ i 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

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8 Numbe rof non-missing observations

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

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

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

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

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

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14 Comparison of Estimates Covariate OLS w/ All data OLS w/ Selected sample MLE of Heckman SS model Educ0.0803 (0.0010) 0.0703 (0.0015) 0.0817 (0.0064) Age0.0122 (0.0035) 0.0119 (0.0046) 0.0125 (0.0006) Constant4.484 (0.169) 4.670 (0.258) 4.445 (0.127)

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15 Comparison of Estimates Covariate OLS w/ All data OLS w/ Selected sample MLE of Heckman SS model Educ0.08030.0703 [-12.5%] 0.0817 [1.7%] Age0.01220.0119 [-2.5%] 0.0125 [2.5%] [% difference from OLS w/ all data]

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

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17 No where close on rho

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18 Comparison of Estimates Covariate OLS w/ All data OLS w/ Selected sample MLE of Heckman SS model Function form Ident. Educ0.08030.0703 [-12.5%] 0.065 [-19.2%] Age0.01220.0119 [-2.5%] 0.0115 [-5.7%] [% difference from OLS w/ all data]

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