Consider using data file mroz to estimate a model explaining a married woman's hours of work, \(H O U R S\), as a function of her education, EDUC, her experience, EXPER, and her husband's hours of work, HHOURS.

a. Use all observations to estimate the regression model

\[H O U R S=\beta_{1}+\beta_{2} E D U C+\beta_{3} E X P E R+\beta_{4} H H O U R S+e\]

Is OLS a consistent estimator in this case?

b. Use only the observations for which HOURS \(>0\) to estimate the regression model in (a). Is OLS a consistent estimator in this case?

c. Estimate a probit model for the woman's decision to be in the labor force, \(L F P=1\), \(L F P=\Phi\left(\gamma_{1}+\gamma_{2}\right.\) EXPER \(+\gamma_{3}\) KIDSL6 \(+\gamma_{4}\) KIDS \(618+\gamma_{5}\) MTR \(+\gamma_{6}\) LARGECITY \()\). Which if any of the variables help explain the woman's labor force participation decision?

d. Using the estimates from the probit model, obtain

\[\tilde{w}=\tilde{\gamma}_{1}+\tilde{\gamma}_{2} E X P E R+\tilde{\gamma}_{3} K I D S L 6+\tilde{\gamma}_{4} \text { KIDS618+ } \tilde{\gamma}_{5} M T R+\tilde{\gamma}_{6} L A R G E C I T Y\]

Create the inverse Mills ratio \(\tilde{\lambda}=\phi(\tilde{w}) / \Phi(\tilde{w})\). What are the sample mean and variance of \(\tilde{\lambda}\) ?

e. Estimate the model HOURS \(=\beta_{1}+\beta_{2} E D U C+\beta_{3} E X P E R+\beta_{4} H H O U R S+\beta_{\lambda} \tilde{\lambda}+e\) using the observations for which HOURS \(>0\). Compare these estimates to those in parts (a) and (b). Are the standard errors from this estimation correct?

f. Estimate the model in (e) using heteroskedasticity robust standard errors. Use the option HC3 if it is available. These standard errors are not absolutely correct but an improvement over the ones in (e).

g. Estimate the model in (e) using bootstrap standard errors, with \(B=400\) bootstrap replications. Compare these standard errors to those in (e) and (f).

h. Estimate the model in (e) using proper econometric software for this Heckit model. Compare the results to those in (e)-(g). Be sure to identify whether your software is using a two-step estimator, like part (e), or full information maximum likelihood.