Estimating the return to education for married women mroz.csv: 753 observations and 22 variables
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1 Return to education Estimating the return to education for married women mroz.csv: 753 observations and 22 variables 1. inlf =1 if in labor force, hours hours worked, kidslt6 # kids < 6 years 4. kidsge6 # kids age woman s age in yrs 6. educ years of schooling 7. wage estimated wage from earns., hours 8. repwage reported wage at interview in hushrs hours worked by husband, husage husband s age 11. huseduc husband s years of schooling 12. huswage husband s hourly wage, faminc family income, mtr fed. marginal tax rate facing woman 15. motheduc mother s years of schooling 16. fatheduc father s years of schooling 17. unem unem. rate in county of resid. 18. city =1 if live in SMSA 19. exper actual labor mkt exper 20. nwifeinc (faminc - wage*hours)/ lwage log(wage) 22. expersq exper^2
2 Return to education We use the data on married working women to estimate the return to education in the simple regression model log(wage) = β 0 + β 1 educ + u. OLS estimates (for comparison): log(wage) = (0.185) where n = 428 and R 2 = (0.014) educ The estimate for β 1 implies an almost 11% return for another year of education.
3 Father s education as IV We have to maintain that: fatheduc is uncorrelated with u, and educ and fatheduc are correlated. Simple regression of educ on fatheduc: êduc = (0.28) (0.029) fatheduc where n = 428 and R 2 =
4 IV regression Using fatheduc as an IV for educ gives log(wage) = (0.446) (0.035) educ where n = 428 and R 2 = The IV estimate of the return to education is 5.9%, which is barely more than one-half of the OLS estimate. This suggests that the OLS estimate is too high and is consistent with omitted ability bias.
5 R code install.packages("ivpack") library(ivpack) data = read.csv("mroz.csv",header=true) attach(data) n = nrow(data) reg1 = lm(lwage~educ) reg2 = lm(educ~fatheduc) reg3 = ivreg(lwage ~ educ fatheduc)
6 IV regression 1 Call: ivreg(formula = lwage ~ educ fatheduc) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) educ Signif. codes: 0 *** ** 0.01 * Residual standard error: on 426 degrees of freedom Multiple R-Squared: ,Adjusted R-squared: Wald test: on 1 and 426 DF, p-value: This is done by two-stage least squares.
7 More instruments Suppose now we entertain the following structural model log(wage) = β 0 + β 1 educ + β 2 exper + β 3 expersq + u, with motheduc, fatheduc, and huseduc as instrumental variables for educ. The reduced form is educ = δ 0 + δ 1 motheduc + δ 2 fatheduc + δ 3 huseduc + δ 4 exper + δ 5 expersq + ɛ.
8 R code install.packages("ivpack") library(ivpack) data = read.csv("mroz.csv",header=true) attach(data) n = nrow(data) reg1 = lm(lwage ~ educ + exper + expersq) reg2 = lm(educ ~ motheduc + fatheduc + huseduc + exper + expersq) reg3 = ivreg(lwage ~ educ + exper + expersq motheduc + fatheduc + huseduc + exper + expersq)
9 > summary(reg1) Call: lm(formula = lwage ~ educ + exper + expersq) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) ** educ e-13 *** exper ** expersq * --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 424 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 3 and 424 DF, p-value: 1.302e-15
10 > summary(reg2) Call: lm(formula = educ ~ motheduc + fatheduc + huseduc + exper + expersq) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** motheduc *** fatheduc *** huseduc < 2e-16 *** exper expersq Signif. codes: 0 *** ** 0.01 * Residual standard error: on 422 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: 63.3 on 5 and 422 DF, p-value: < 2.2e-16
11 > summary(reg3) Call: ivreg(formula = lwage ~ educ + exper + expersq motheduc + fatheduc + huseduc + exper + expersq) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) educ *** exper ** expersq * --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 424 degrees of freedom Multiple R-Squared: ,Adjusted R-squared: Wald test: on 3 and 424 DF, p-value: 2.817e-07
12 Hausman s test reg2 = lm(educ ~ motheduc + fatheduc + huseduc + exper + expersq) error = reg2$res reg4 = lm(lwage ~ educ + exper + expersq + error) summary(reg4) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) educ *** exper ** expersq * error Signif. codes: 0 *** ** 0.01 * Since the estimated coeficient of error, , is statistically significant at the 10% level, then educ is endogenous. Not at the 5% levels though!
13 Sargan s test reg3 = ivreg(lwage ~ educ + exper + expersq motheduc + fatheduc + huseduc + exper + expersq) error = reg3$res reg5 = lm(error ~ motheduc + fatheduc + huseduc + exper + expersq) p = 3 q = 1 sarg = (n-6)*summary(reg5)$r.sq p.val = 1-pchisq(sarg,p-q) p.val [1] Conclusion: Fail to reject H 0 : instruments are valid.
14 With or without huseduc? reg6 = ivreg(lwage ~ educ + exper + expersq motheduc + fatheduc + exper + expersq) summary(reg6) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) educ exper ** expersq * --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 424 degrees of freedom Multiple R-Squared: ,Adjusted R-squared: Wald test: on 3 and 424 DF, p-value: 2.787e-05 Notice that educ becomes statistically irrelevant!
15 With or without huseduc? educ is endogenous at the 10% significance level. Fail to reject H 0 : instruments are valid. # Hausman s test reg2 = lm(educ ~ motheduc + fatheduc + exper + expersq) error = reg2$res reg4 = lm(lwage ~ educ + exper + expersq + error) summary(reg4) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) educ * exper *** expersq * error Signif. codes: 0 *** ** 0.01 * # Sargan s test reg3 = ivreg(lwage ~ educ + exper + expersq motheduc + fatheduc + exper + expersq) error = reg3$res reg5 = lm(error ~ motheduc + fatheduc + exper + expersq) p = 2 q = 1 sarg = (n-5)*summary(reg5)$r.sq p.val = 1-pchisq(sarg,p-q) p.val [1]
16 Summary Model 1 log(wage) = β 0 + β 1 educ + u IV for educ: fatheduc ˆβ 1 = , s.e.( ˆβ 1 ) = , p-value = Model 2 log(wage) = β 0 + β 1 educ + β 2 exper + β 3 expersq + u IV for educ: motheduc, fatheduc, and huseduc ˆβ 1 = , s.e.( ˆβ 1 ) = , p-value = Model 3 log(wage) = β 0 + β 1 educ + β 2 exper + β 3 expersq + u IV for educ: motheduc and fatheduc ˆβ 1 = , s.e.( ˆβ 1 ) = , p-value =
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