Consider the wage model in Example 10.5 and the 428 observations on married women who participate in the labor force. Use only MOTHEREDUC as an instrument in this exercise.

a. Estimate the first-stage equation by OLS and obtain the fitted values

Save the least squares residuals. Call them REDUCHAT. Calculate the sum of squared residuals, \(\sum\) REDUCHAT \({ }_{i}^{2}\).

b. Estimate the regression \(\widehat{E D U C}=a_{1}+a_{2} E X P E R+a_{3} E X P E R^{2}+\) error and save the OLS residuals. Call them \(R E D U C\). Calculate the sum of squared residuals, \(\sum R E D U C_{i}^{2}\).

c. Estimate the regression MOTHEREDUC \(=c_{1}+c_{2} E X P E R+c_{3} E X P E R^{2}+\) error and save the OLS residuals. Call them \(R M O M\). Calculate the sum of squared residuals, \(\Sigma R M O M_{i}^{2}\).

d. Estimate the regression \(R E D U C=\theta_{1} R M O M+\) error. Compare the estimated value of \(\theta_{1}\) from this regression to the estimated \(\theta_{1}\) from the first-stage equation. What \(R^{2}\) value did you obtain from this regression? What is the sum of squared residuals?

e. Show that \(\sum R M O M_{i}^{2}=\hat{\theta}_{1}^{2} \sum R E D U C_{i}^{2}\).

f. Refer to equation (10.25) and discuss the importance of the quantities in (e) for the precision of the IV/2SLS estimator.

Data From Example 10.5:-

In addition to education a worker's experience is also important in determining their wage. Because additional years of experience have a declining marginal effect on wage use the quadratic model 

where EXPER is years of experience. This is the same specification. We assume that EXPER is an exogenous variable that is uncorrelated with the worker's ability and therefore uncorrelated with the random error \(e\). Two instrumental variables for years of education, \(E D U C\), are mother's and father's years of education, MOTHEREDUC and FATHEREDUC, introduced in the previous examples. The first-stage equation is  

Using the 428 observations in the data file mroz the estimated first-stage equation is reported in Table 10.1. The IV/2SLS estimates, with correctly computed standard errors, are

The estimated return to education is approximately \(6.1 \%\), and the estimated coefficient is statistically significant with a \(t=1.96\).

Data From Equation 10.25:-

Transcribed Image Text:

EDUC=+EXPER+13EXPER + MOTHEREDUC