The College Premium in the Eighties: Returns to College or Returns to Ability

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1 The College Premium in the Eighties: Returns to College or Returns to Ability Christopher Taber March 4, 2008

2 The problem of Ability Bias were A i is ability Y i = X i β + αs i + A i + u i If A i is correlated with S i then α is biased upward Willis and Rosen find evidence of this Instrumental Variables can also deliver a consistent estimate of α

3 Dynamics Lets think about how this evolves over time Y it = X it β + α ts i + γ t A i + u it Then abstracting from X it and u it E (Y it S i ) = α t S i + γ t E (A i S i ) We know that the measured return to schooling rose rapidly during the 80s

4 In this context, this can be for three reasons: α t E(A i S i ) > 0 and γ t E (A i S i ) increases over time Third is a cohort effect so is probably least likely

5 First way to check this is using observable measure of ability Can also check using IV I try Willis and Rosen s instrument, parental background

6 TABLE II Preliminary Regression Results NLSY White Males Dependent Variable: Log Weekly Wage (Huber Standard Errors in Parentheses) Variable (1) (2) (3) Ages : College 0.370(0.050) 0.276(0.053) 0.307(0.062) 0.288(0.094) AFQT Score (0.135) - - General Science Score (0.008) (0.015) Gen. Sci. College (0.014) 0.028(0.024) Mathematics Knowledge Score (0.005) 0.008(0.009) Math College (0.009) 0.010(0.014) Auto and Shop Score (0.006) 0.018(0.012) Auto College (0.011) 0.001(0.017) Word Knowledge Score (0.005) 0.007(0.010) Word College (0.012) (0.020) Highest Grade College 0.104(0.021) 0.084(0.021) 0.091(0.024) 0.079(0.032) : College 0.447(0.037) 0.350(0.040) 0.347(0.046) 0.318(0.072) AFQT Score (0.119) - - General Science Score (0.009) (0.013) Gen. Sci. College (0.013) (0.019) Mathematics Knowledge Score (0.005) 0.020(0.007) Math College (0.008) 0.004(0.010) Auto and Shop Score (0.006) 0.021(0.009) Auto College (0.009) (0.013) Word Knowledge Score (0.006) (0.009) Word College (0.009) (0.013) Highest Grade College 0.113(0.013) 0.094(0.014) 0.086(0.015) 0.083(0.026) : College 0.485(0.035) 0.402(0.038) 0.416(0.041) 0.412(0.064) AFQT Score (0.099) - - General Science Score (0.008) 0.003(0.011) Gen. Sci. College (0.011) 0.000(0.014) Mathematics Knowledge Score (0.005) 0.015(0.007) Math College (0.007) (0.009) Auto and Shop Score (0.005) 0.000(0.007) Auto College (0.008) 0.005(0.009) Word Knowledge Score (0.005) (0.007) Word College (0.007) 0.005(0.010) Highest Grade College 0.123(0.013) 0.107(0.013) 0.106(0.014) 0.093(0.023) (continued on following page)

7 TABLE II(cont) Variable (1) (2) (3) Experience 0.103(0.010) 0.106(0.010) 0.101(0.010) 0.076(0.028) Experience Squared (0.070) (0.069) (0.069) (0.248) Local Unemployment Rate (0.010) (0.010) (0.010) (0.014) Age Jan (0.007) 0.008(0.007) 0.009(0.007) 0.009(0.010) Period Intercepts Yes Yes Yes Yes R Number of Observations Number of Individuals Standard errors are calculated for OLS results allowing for heteroskedasticity in individual variance covariance matrices.

8 TABLE III Instrumental Variable Results Using Family Background as Instrument Dependent Variable: Log Weekly Wages (Huber Standard Errors in Parentheses) Variable OLS IV OLS IV : Schooling 0.079(0.015) 0.159(0.069) 0.054(0.019) 0.100(0.090) General Science Score (0.009) (0.011) (0.010) (0.013) Mathematics Knowledge Score 0.008(0.006) (0.012) 0.012(0.006) 0.006(0.015) Auto and Shop Score 0.023(0.007) 0.031(0.010) 0.020(0.007) 0.024(0.013) Word Knowledge Score 0.009(0.007) 0.004(0.008) 0.000(0.007) 0.000(0.007) : Schooling 0.083(0.010) 0.125(0.034) 0.078(0.012) 0.121(0.040) General Science Score (0.007) (0.008) (0.008) (0.009) Mathematics Knowledge Score 0.020(0.004) 0.013(0.007) 0.019(0.004) 0.012(0.008) Auto and Shop Score 0.012(0.004) 0.016(0.006) 0.012(0.005) 0.016(0.007) Word Knowledge Score 0.001(0.004) (0.004) (0.005) (0.005) : Schooling 0.107(0.009) 0.133(0.025) 0.102(0.010) 0.115(0.028) General Science Score (0.005) (0.006) (0.006) (0.006) Mathematics Knowledge Score 0.017(0.003) 0.012(0.005) 0.018(0.003) 0.015(0.006) Auto and Shop Score 0.011(0.003) 0.014(0.004) 0.010(0.004) 0.012(0.005) Word Knowledge Score 0.001(0.003) (0.003) (0.003) (0.004) Experience 0.085(0.012) 0.093(0.021) 0.069(0.015) 0.085(0.022) Experience Squared (0.070) (0.133) (0.097) (0.144) Local Unemployment Rate (0.010) (0.011) (0.011) (0.012) Age Jan (0.007) (0.007) 0.001(0.007) 0.001(0.007) Period Intercepts Yes Yes Yes Yes Droupouts Included Yes Yes No No Number of Observations Number of Individuals I treat schooling, experience and experience squared as endogenous and instrument excluding parent s education, age, and age squared interacted with time periods. Standard errors are calculated for OLS results allowing for heteroskedasticity in individual variance covariance matrices. This table also uses individuals over 22 so that the number of panel observations per individual will not strongly depend on schooling (this does subtantially affect the results).

9 Interpretation of IV results: β ols t β IV t p E(A i S i ) β t + γ t E(Si 2 ) p E(A i PE i ) β t + γ t E(S i PE i ) Hard to see why problems with instrument would bias you towards this result

10 Next step is to control form Mills Ratio as in Willis and Rosen

11 TABLE IV Selection Equation for Attend College Av erage Derivatives from Probit High School Graduates Only NLSY White Males (Standard Errors in Parentheses) Variable (1) (2) Highest Grade Mother 0.022(0.008) 0.021(0.008) Highest Grade Father 0.033(0.006) 0.034(0.006) Number of Siblings (0.008) (0.008) General Science Score 0.020(0.006) 0.021(0.006) Word Knowledge Score 0.010(0.004) 0.010(0.004) Mathematics Knowledge Score 0.036(0.003) 0.036(0.003) Auto and Shop Score (0.004) (0.004) Age Jan. 1, (0.007) - Age 14 on Jan. 1, (0.060) Age 15 on Jan. 1, (0.056) Age 16 on Jan. 1, (0.058) Age 17 on Jan. 1, (0.062) Age 18 on Jan. 1, (0.062) Age 19 on Jan. 1, (0.056) Age 20 on Jan. 1, (0.060) Number of Individuals 1527 These results give the derivatives of the predicted probability at the means of the covariates.

12 -- -- TABLE V Results from Two Step Sample Selection Model NLSY White Males Dependent Variable: Log Weekly Wage (Huber Standard Errors in Parentheses) (1) (2) Variable (3) (4) : College 0.307(0.062) 0.428(0.167) 0.432(0.159) 0.412(0.151) Intercept 5.241(0.108) 5.175(0.135) 5.177(0.126) 5.262(0.101) Highest Grade College 0.091(0.024) 0.089(0.024) 0.089(0.024) 0.089(0.024) General Science Score (0.008) (0.009) (0.009) (0.009) Gen. Sci. College 0.014(0.014) (0.005) (0.005) (0.005) Mathematics Knowledge Score 0.013(0.005) 0.009(0.008) 0.009(0.007) 0.008(0.008) Math College 0.003(0.009) 0.017(0.008) 0.017(0.008) 0.018(0.008) Auto and Shop Score 0.014(0.006) 0.014(0.016) 0.014(0.014) 0.016(0.015) Auto College 0.006(0.011) (0.013) (0.012) (0.013) Word Knowledge Score (0.005) 0.004(0.013) 0.004(0.009) 0.007(0.011) Word College (0.012) 0.005(0.013) 0.005(0.011) 0.002(0.012) λ c (0.163) (0.093) (0.093) λ h (0.113) (0.093) (0.093) : College 0.347(0.046) 0.366(0.142) 0.379(0.145) 0.283(0.136) Intercept 5.236(0.101) 5.124(0.134) 5.220(0.118) 5.266(0.108) Highest Grade College 0.086(0.015) 0.087(0.015) 0.085(0.015) 0.087(0.015) General Science Score (0.009) (0.011) (0.010) (0.010) Gen. Sci. College 0.008(0.013) (0.006) (0.006) (0.006) Mathematics Knowledge Score 0.016(0.005) 0.009(0.009) 0.015(0.007) 0.012(0.008) Math College 0.003(0.008) 0.020(0.009) 0.015(0.007) 0.018(0.008) Auto and Shop Score 0.015(0.006) 0.015(0.015) 0.008(0.013) 0.015(0.015) Auto College (0.009) 0.002(0.009) (0.009) 0.002(0.009) Word Knowledge Score (0.006) 0.015(0.011) 0.003(0.007) 0.016(0.011) Word College (0.009) (0.011) (0.009) (0.011) λ c (0.115) (0.087) 0.169(0.087) λ h (0.127) (0.087) (0.087) : College 0.416(0.041) 0.265(0.123) 0.260(0.122) 0.287(0.121) Intercept 5.224(0.094) 5.278(0.121) 5.300(0.105) 5.361(0.099) Highest Grade College 0.106(0.014) 0.107(0.014) 0.107(0.014) 0.107(0.014) General Science Score (0.008) (0.008) (0.008) (0.008) Gen. Sci. College 0.016(0.011) 0.003(0.005) 0.004(0.005) 0.003(0.005) Mathematics Knowledge Score 0.018(0.005) 0.022(0.007) 0.023(0.006) 0.021(0.007) Math College (0.007) 0.009(0.007) 0.008(0.006) 0.010(0.007) Auto and Shop Score 0.012(0.005) 0.016(0.012) 0.015(0.012) 0.018(0.012) Auto College (0.008) (0.008) (0.007) (0.008) Word Knowledge Score 0.002(0.005) 0.001(0.010) (0.007) 0.002(0.010) Word College (0.007) (0.009) (0.008) (0.009) λ c (0.095) 0.098(0.073) 0.117(0.073) λ h (0.114) 0.098(0.073) 0.044(0.073) (continued on following page)

13 -- -- TABLE V(cont.) Variable (1) (2) (3) (4) Experience 0.102(0.010) 0.101(0.010) 0.101(0.010) 0.103(0.010) Experience Squared (0.069) (0.070) (0.069) (0.070) Local Unemployment Rate (0.010) (0.010) (0.010) (0.010) Age Jan (0.007) 0.009(0.007) 0.009(0.007) - Cohort Dummies No No No Yes Number of Observations Number of Individuals Standard errors are calculated for OLS results allowing for heteroskedasticity in individual variance covariance matrices and correcting for estimation of Mills ratio in first step. In this case I use cohort dummies in both the first and second stage, in the others only the cohort trend is included in both stages.

14 Notice that λ ( X i β) = E ( u i X i β + u i > 0, X i ) so lim λ ( X X i β i β) = E (u i X i ) = 0 lim λ ( X i β i β) = lim E (u i u i > κ, X i ) = κ X Thus λ is high for individuals who look like they shouldn t have gone to college, but do We think of these as people with high levels of unobserved ability They did well during the 1980s

15 Dynamic Programming Model How do we want to think of high school graduation? Don t want to just condition on it, or ignore it. Let S=1, high school dropout S=2, exactly high school degree S=3 Attend College

16 The set up can be seen as S = 2 S > 1 S = 3 S = 1

17 Define the relavant utility functions as E 1 [V 1 ] = X 1 β 1 + α 1 E 1[θ] + ω 1 E 2 [V 2 ] = 0 E 2 [V 3 ] = X 3 β 3 + α 3 E 2[θ] + ω 3, where V 2 = 0 is a normalization

18 The key unknown is the value function at the complete high school node dated at time 1 E 1 [Ṽ2] E 1 [max{0, V 3 }] = E 1 [max{x 3 β 3 + α 3 E 2[θ] + ω 3, 0}] ( X = σ 3 [Φ 3 β 3 + α3 s ) ( θs X 3 β 3 + α s σ 3 Where θ S = E 1 (θ) = E 2 (θ) σ 3 3 θs ) ( X3 β 3 + α s + φ σ 3 3 θs )]

19 That is the model To estimate it we need the probability of each choice ( ) Pr(d 1 = 1 X, θ s ) = Φ X 1 β 1 + α1 s θs E 1 [Ṽ2] Pr(d 3 = 1 X, θ s, d 1 = 0) = Φ ( X 3 β 31 + α s 3 θs).

20 Define Wages as log(w tj ) = Z tj γ tj + δ s tj θs + δ w tj θw + ε tj. We can then estimate by maximum likelihood L(W ts, d 1, d 2, d 3 ) = d 1 {Φ d } T ( log (Wt2 ) Z t2 +d 2 {(1 Φ d ) (1 Φ c ) φ γ t2 δt2 s θs δ w ) } t2 θw σ ε2t t=1 { T ( log(wt3 ) Z t3 +d 3 (1 Φ d ) Φ c φ γ t3 δt3 s θs δt3 w ) } θw σ ε3t dg s (θ s )dg w (θ w ). t=1 where ( ) Φ d = Φ X 1 β 1+α1 s θs E 1 [Ṽ2] Φ c = Φ ( X 3 β 3+α s 3 θs)

21 Base Schooling Model Without Selection Tw o Independent Dimensions of Heterogeneity Each Dimension Distributed N(0.5,0.02) NLSY White Males (Standard Errors in Parentheses) Dynamic Programming Selection Equation Complete Attend Variable High School College α w - - α s - - Intercept (0.015) (0.384) Highest Grade Mother 0.078(0.022) 0.055(0.020) Highest Grade Father 0.034(0.018) 0.084(0.015) Number of Siblings (0.021) (0.022) Age Jan. 1, (0.020) 0.029(0.019) General Science Score 0.017(0.017) 0.051(0.015) Word Knowledge Score 0.019(0.010) 0.026(0.010) Mathematics Knowledge Score 0.071(0.017) 0.092(0.008) Auto and Shop Score (0.015) (0.010) σ ε 0.571(0.307) - Regression Equation Dependent Variable: Log Weekly Wage Variable College High School : δ w 3.404(0.139) 3.342(0.119) δ s 0.428(0.248) 0.134(0.197) General Science Score 0.020(0.011) 0.002(0.007) Word Knowledge Score (0.008) 0.004(0.005) Mathematics Knowledge Score 0.017(0.007) 0.008(0.005) Auto and Shop Score 0.010(0.007) 0.011(0.005) Highest Grade College 0.091(0.024) : δ w 2.366(0.148) 2.617(0.123) δ s 2.059(0.155) 1.747(0.160) General Science Score 0.010(0.007) (0.006) Word Knowledge Score (0.006) 0.005(0.004) Mathematics Knowledge Score 0.020(0.004) 0.010(0.004) Auto and Shop Score 0.001(0.004) 0.012(0.005) Highest Grade College 0.091(0.013) -

22 TABLE VI(cont.) Regression Equation(cont.) Variable College High School : δ w 1.317(0.170) 2.034(0.160) δ s 3.433(0.098) 2.809(0.128) General Science Score 0.004(0.007) (0.006) Word Knowledge Score (0.006) 0.009(0.004) Mathematics Knowledge Score 0.017(0.004) 0.013(0.004) Auto and Shop Score 0.005(0.004) 0.012(0.005) Highest Grade College 0.113(0.011) - Experience 0.120(0.008) 0.120(0.008) Experience Squared (0.047) (0.047) Local Unemployment Rate (0.005) (0.005) Age Jan. 1, (0.007) 0.004(0.007) Av erage Marginal Treatment Treatment Effect Effect (0.060) 0.317(0.060) (0.041) 0.383(0.041) (0.038) 0.457(0.038) Log Likelihood Number of Individuals 1836 Likelihood is calculated using 7 Gauss NewtonQuadrature Points to approximate Normal Distribution.

23 Results for Full Schooling Model Tw o Independent Dimensions of Heterogeneity Each Dimension Distributed N(0.5,0.02) NLSY White Males (Standard Errors in Parentheses) Dynamic Programming Selection Equation Complete Attend Variable High School College α w - - α s 1.481(1.229) 1.468(0.637) Intercept (0.015) (0.543) Highest Grade Mother 0.080(0.023) 0.054(0.020) Highest Grade Father 0.032(0.019) 0.084(0.015) Number of Siblings (0.022) (0.022) Age Jan. 1, (0.021) 0.033(0.019) General Science Score 0.016(0.017) 0.053(0.015) Word Knowledge Score 0.020(0.010) 0.028(0.011) Mathematics Knowledge Score 0.071(0.018) 0.096(0.009) Auto and Shop Score (0.015) (0.011) σ ε 0.681(0.350) - Regression Equation Dependent Variable: Log Weekly Wage Variable College High School : δ w 3.381(0.155) 3.333(0.119) δ s 0.171(0.259) 0.161(0.198) General Science Score 0.007(0.011) 0.001(0.008) Word Knowledge Score (0.008) 0.003(0.005) Mathematics Knowledge Score 0.016(0.007) 0.007(0.005) Auto and Shop Score 0.018(0.006) 0.012(0.005) Highest Grade College 0.115(0.024) : δ w 2.452(0.133) 2.592(0.122) δ s 1.855(0.155) 1.769(0.160) General Science Score 0.001(0.007) (0.006) Word Knowledge Score (0.005) 0.005(0.004) Mathematics Knowledge Score 0.023(0.004) 0.013(0.004) Auto and Shop Score 0.004(0.004) 0.012(0.005) Highest Grade College 0.111(0.013) - (continued on following page)

24 TABLE VII(cont.) Regression Equation(cont.) Variable College High School : δ w 1.508(0.166) 1.993(0.158) δ s 3.330(0.104) 2.878(0.131) General Science Score (0.007) (0.006) Word Knowledge Score (0.006) 0.010(0.004) Mathematics Knowledge Score 0.023(0.004) 0.018(0.005) Auto and Shop Score 0.006(0.004) 0.011(0.005) Highest Grade College 0.132(0.011) - Experience 0.120(0.008) 0.120(0.008) Experience Squared (0.048) (0.048) Local Unemployment Rate (0.005) (0.005) Age Jan. 1, (0.007) 0.001(0.007) Av erage Marginal Treatment Treatment Effect Effect (0.060) 0.337(0.060) (0.043) 0.331(0.042) (0.052) 0.349(0.051) Log Likelihood Number of Individuals 1836 Likelihood is calculated using 7 Gauss NewtonQuadrature Points to approximate Normal Distribution.

25 Variable Results for Selection Model Tw o Independent Dimensions of Heterogeneity Each Dimension Distributed N(0.5,0.02) NLSY White Males (Standard Errors in Parentheses) Selection Equation Attend College α w - α s 1.648(0.632) Intercept (0.532) Highest Grade Mother 0.054(0.020) Highest Grade Father 0.085(0.015) Number of Siblings (0.023) Age Jan. 1, (0.019) General Science Score 0.054(0.015) Word Knowledge Score 0.028(0.011) Mathematics Knowledge Score 0.095(0.009) Auto and Shop Score (0.011) Regression Equation Dependent Variable: Log Weekly Wage Variable College High School : δ w 3.389(0.156) 3.337(0.118) δ s 0.153(0.258) 0.139(0.190) General Science Score 0.006(0.011) 0.002(0.007) Word Knowledge Score (0.008) 0.003(0.005) Mathematics Knowledge Score 0.017(0.007) 0.007(0.005) Auto and Shop Score 0.018(0.006) 0.012(0.005) Highest Grade College 0.117(0.024) : δ w 2.467(0.130) 2.595(0.116) δ s 1.834(0.155) 1.749(0.152) General Science Score 0.001(0.007) (0.005) Word Knowledge Score (0.005) 0.004(0.004) Mathematics Knowledge Score 0.022(0.004) 0.012(0.004) Auto and Shop Score 0.004(0.004) 0.012(0.005) Highest Grade College 0.112(0.013) - (continued on following page)

26 TABLE VIII(cont.) Regression Equation(cont.) Variable College High School : δ w 1.531(0.164) 1.997(0.153) δ s 3.315(0.104) 2.865(0.122) General Science Score (0.007) (0.006) Word Knowledge Score (0.006) 0.009(0.004) Mathematics Knowledge Score 0.022(0.004) 0.016(0.004) Auto and Shop Score 0.005(0.004) 0.011(0.005) Highest Grade College 0.133(0.011) - Experience 0.121(0.008) 0.121(0.008) Experience Squared (0.048) (0.048) Local Unemployment Rate (0.005) (0.005) Age Jan. 1, (0.007) (0.007) Av erage Marginal Treatment Treatment Effect Effect (0.058) 0.341(0.059) (0.042) 0.326(0.042) (0.051) 0.334(0.050) Log Likelihood Number of Individuals 1527 Likelihood is calculated using 7 Gauss NewtonQuadrature Points to approximate Normal Distribution. The likelihood also includes a probit for high school selection so the likelihood can be directly compared with the other specifications.

27 Results for Full Schooling Model Tw o Independent Dimensions of Heterogeneity Alternative Heterogeneity Distributions NLSY White Males (Standard Errors in Parentheses) Mixture of Mixture of Normals 1 Normals 2 Uniform Dynamic Programming Selection Equation Complete Attend Complete Attend Complete Attend Variable High School College High School College High School College α w α s (1.424) 2.122(0.665) 0.100(1.655) 7.393(1.427) 2.948(0.654) (0.396) Regression Equation Variable College High School College High School College High School : δ w 4.012(0.200) 3.999(0.212) 3.574(0.209) 3.667(0.154) 1.195(0.059) 1.191(0.041) δ s 1.586(0.346) 1.730(0.302) (0.342) 1.031(0.353) (0.094) 0.692(0.078) : δ w 2.303(0.234) 2.494(0.254) 3.326(0.140) 2.546(0.212) 1.161(0.052) 0.909(0.053) δ s 3.228(0.187) 3.187(0.197) 1.071(0.288) 2.782(0.260) 0.566(0.075) 1.414(0.070) : δ w 0.514(0.248) 1.153(0.316) 2.936(0.229) 1.612(0.305) 1.128(0.069) 0.662(0.051) δ s 4.476(0.101) 4.298(0.137) 3.140(0.259) 4.019(0.194) 1.137(0.077) 1.813(0.077) Av erage Marginal Average Marginal Average Marginal Treatment Treatment Treatment Treatment Treatment Treatment Effect Effect Effect Effect Effect Effect (0.067) 0.264(0.066) 0.331(0.073) 0.297(0.079) 0.501(0.065) 0.519(0.066) (0.048) 0.261(0.046) 0.126(0.055) 0.107(0.063) 0.887(0.051) 0.909(0.054) (0.050) 0.287(0.050) (0.071) (0.079) 1.175(0.059) 1.190(0.063) Likelihood Individ Both Mixtures use 8 Gauss Newton Quadrature Points to approximate Normal Distribution. The first has expected vaue zero for both distributions, but the second has double the variance. The second mixture has the same variance for each, but the expected values are one standard deviation apart. The uniform is approximated with 5 equally spaced points with equal probability.

28 TABLE X Results for Alternative Restiction: Skill Price Falling over Time Coefficients on Skills (Standard Errors in Parentheses) Dynamic Programming Selection Equation Complete Attend Variable High School College α w - - α s (1.127) (0.579) Regression Equation Variable College High School : δ s 3.489(0.148) 3.458(0.123) δ w 0.658(0.211) (0.173) : δ s 2.439(0.136) 2.831(0.111) δ w 2.195(0.156) 1.610(0.142) : δ s 1.320(0.152) 2.264(0.148) δ w 3.524(0.090) 2.682(0.115) Av erage Marginal Treatment Treatment (0.069) 0.484(0.069) (0.049) 0.515(0.048) (0.042) 0.548(0.042) Log Likelihood

29 Results for Alternative Restriction: Allowing Differential Selection for High School and College Skills Three Independent Dimensions of Heterogeneity Each Dimension Distributed N(0.5,0.02) NLSY White Males (Standard Errors in Parentheses) Local Local Opt. 1 Opt. 2 Dynamic Programming Selection Equation Complete Attend Complete Attend Variable High School College High School College α w α s H 1.246(1.523) 1.662(1.040) 1.210(1.415) 1.913(1.110) α s C 3.391(3.508) 1.151(1.088) 1.726(3.600) (1.235) Regression Equation Dependent Variable: Log Weekly Wage Variable College High School College High School : δ w 3.381(0.155) 3.334(0.119) 3.512(0.151) 3.336(0.119) δ s H (0.197) (0.195) δ s C 0.166(0.259) (0.233) : δ w 2.453(0.135) 2.595(0.121) 2.388(0.131) 2.597(0.120) δ s H (0.158) (0.156) δ s C 1.861(0.157) (0.162) : δ w 1.510(0.170) 1.995(0.157) 1.302(0.143) 2.000(0.155) δ s H (0.129) (0.128) δ s C 3.343(0.113) (0.102) - Av erage Marginal Average Marginal Treatment Treatment Treatment Treatment Effect Effect Effect Effect (0.060) 0.337(0.060) 0.387(0.063) 0.367(0.062) (0.043) 0.327(0.043) 0.459(0.044) 0.432(0.044) (0.052) 0.340(0.051) 0.557(0.050) 0.516(0.051) Log Likelihood Number of Individuals

30 TABLE XII Results for Alternative Restriction: Both Factors Influence Schooling Tw o Independent Dimensions of Heterogeneity Each Dimension Distributed N(0.5,0.02) NLSY White Males (Standard Errors in Parentheses) Local Local Opt. 1 Opt. 2 Dynamic Programming Selection Equation Complete Attend Complete Attend Variable High School College High School College α s (1.549) (0.618) (1.596) (0.758) α s (1.549) (0.618) 2.787(1.596) 1.962(0.758) Regression Equation Dependent Variable: Log Weekly Wage Variable College High School College High School : δ s (0.148) 3.489(0.123) 3.386(0.154) 3.382(0.122) δ s (0.203) (0.176) 0.184(0.258) 0.130(0.193) : δ s (0.132) 2.929(0.103) 2.452(0.131) 2.606(0.120) δ s (0.162) 1.607(0.139) 1.844(0.151) 1.746(0.159) : δ s (0.148) 2.385(0.145) 1.495(0.165) 1.955(0.159) δ s (0.092) 2.799(0.121) 3.322(0.105) 2.898(0.132) Av erage Marginal Average Marginal Treatment Treatment Treatment Treatment Effect Effect Effect Effect (0.071) 0.501(0.070) 0.347(0.060) 0.337(0.060) (0.047) 0.617(0.047) 0.333(0.043) 0.327(0.043) (0.049) 0.713(0.048) 0.351(0.052) 0.340(0.051) Log Likelihood Number of Individuals Likelihood is calculated using 7 Gauss NewtonQuadrature Points to approximate Normal Distribution.

Tables and Figures. This draft, July 2, 2007

Tables and Figures. This draft, July 2, 2007 and Figures This draft, July 2, 2007 1 / 16 Figures 2 / 16 Figure 1: Density of Estimated Propensity Score Pr(D=1) % 50 40 Treated Group Untreated Group 30 f (P) 20 10 0.01~.10.11~.20.21~.30.31~.40.41~.50.51~.60.61~.70.71~.80.81~.90.91~.99