This exercise uses the data and model in Exercise 15.18.

a. Estimate the model assuming random effects and with the characteristics of the sex workers included in the model. Carry out a test of the joint significance of the sex worker characteristics at the \(5 \%\) level. Are these coefficients jointly significant? Are they individually significant?

b. What is your \(95 \%\) interval estimate of the risk premium, the coefficient on NOCONDOM, based on these random effects estimates?

c. Test for the presence of random effects using the LM statistic in equation (15.35). Use the 5\% level of significance.

d. Based on the random effects estimates, how much extra does a client have to pay to have unprotected sex with an attractive secondary-educated sex worker?

e. Using the \(t\)-test statistic in equation (15.36) and a 5\% significance level, test whether there are any significant differences between the fixed effects and random effects estimates of the coefficients on NOCONDOM, RICH, REGULAR, ALCOHOL, BAR, and STREET. If there are significant differences between any of the coefficients, should we rely on the fixed effects estimates or the random effects estimates? Explain your choice.

f. Reconsider the random effects model from part (a), but assume NOCONDOM is correlated with the random effects. Reestimate the model using the Hausman-Taylor estimator with NOCONDOM treated as endogenous. Compare the results with those obtained in part (b). How much extra does a client have to pay to have unprotected sex with an attractive secondary-educated sex worker? What is your \(95 \%\) interval estimate of the risk premium, the coefficient on NOCONDOM, based on the Hausman-Taylor estimates?

Data From Exercise 15.18:-

The data file mexican contains data collected in 2001 from the transactions of 754 female Mexican sex workers. There is information on four transactions per worker. \({ }^{17}\) The labels ID and TRANS are used to describe a particular woman and a particular transaction. There are three categories of variables.

1. Sex worker characteristics: (i) \(A G E\), (ii) an indicator variable ATTRACTIVE equal to 1 if the worker is attractive, and (iii) an indicator variable SCHOOL if she has completed secondary school or higher.

2. Client characteristics: (i) an indicator variable REGULAR equal to 1 if the client is a regular, (ii) an indicator variable \(R I C H\) equal to 1 if the client is rich, and (iii) an indicator variable \(A L C O\) \(H O L\) if the client has consumed alcohol before the transaction.

3. Transaction characteristics: (i) the log of the price of the transaction LNPRICE, (ii) an indicator variable NOCONDOM equal to 1 if a condom was not used, and (iii) two indicator variables for location, BAR equal to 1 if the transaction originated in bar and STREET equal to 1 if the transaction originated in the street.

a. Using OLS, estimate a relationship with LNPRICE as the dependent variable, and as explanatory variables the sex worker characteristics, client characteristics, and transaction characteristics. Discuss the signs and significance of the estimated coefficients.

b. Gertler, Shah, and Bertozzi argue that the coefficient of NOCONDOM is a risk premium. Some sex workers are willing to take the risk of having unprotected sex because of the extra price some clients are willing to pay to avoid using a condom. What is your \(95 \%\) interval estimate of the risk premium based on these OLS estimates?

c. What are some factors that might be included in an unobserved heterogeneity error component in this model? A crucial assumption for the consistency of the OLS estimator is that the unobserved heterogeneity term is uncorrelated with the explanatory variables. Without carrying out a formal test, what are your thoughts about this exogeneity assumption for the model in (a)?

d. Estimate the model in part (a) using the fixed effects estimator, omitting sex worker characteristics. (i) Why did we omit the sex worker characteristics? and (ii) Which coefficient estimates are significantly different from zero at a \(5 \%\) level of significance?

e. Using the fixed effects estimation in (d), carry out an \(F\)-test for the presence of individual sex worker differences. Use the \(1 \%\) level of significance.

f. Using the fixed effects estimates, how is the price affected when clients are rich, are regular, and have consumed alcohol? How does the location of the transaction influence the price?

g. What is your \(95 \%\) interval estimate of the risk premium based on these fixed effects estimates? Compare this interval estimate to the one in part (b).