What effect does having public health insurance have on the number of doctor visits a person has during a year? Using 1988 data, in the data file rwm88_small, from Germany, we will explore this question. The data file \(r w m 88\) contains more observations.

a. Estimate the regression model with the dependent variable DOCVIS and the explanatory variables PUBLIC, FEMALE, HHKIDS, MARRIED, SELF, EDUC2, HHNINC2. Test the null hypothesis that the coefficient on \(P U B L I C\) is less than or equal to zero, versus the alternative that it is greater than zero at the \(1 \%\) level of significance.

b. Test for the presence of heteroskedasticity. Obtain the squared least squares residuals from the regression in (a), regress them on all the explanatory variables, and carry out an \(F\)-test of their joint significance. What do we conclude about the presence of heteroskedasticity at the \(1 \%\) level of significance?

c. Estimate the regression model with the dependent variable DOCVIS and the explanatory variables FEMALE, HHKIDS, MARRIED, SELF, EDUC2, HHNINC2 separately for those with public insurance and those who do not have public insurance. Use equation (7.37) to obtain the estimate of the average treatment effect of public insurance.

d. Estimate the regression model with the dependent variable DOCVIS and the explanatory variables PUBLIC, FEMALE, HHKIDS, MARRIED, SELF, EDUC2, HHNINC2 in "deviation from the mean" form. That is, for each variable \(x\), create the variable \(\tilde{x}=x-\bar{x}\), where \(\bar{x}\) is the sample mean. Using robust standard errors, test the significance of the coefficient on PUBLIC.

e. Estimate the regression model with the dependent variable DOCVIS and the explanatory variables FEMALE, HHKIDS, MARRIED, SELF, EDUC2, HHNINC2, along with PUBLIC and PUB\(L I C\) times each of the variables in deviation about the mean form. What is the estimated average treatment effect? Using a robust standard error, is it statistically significant at the 5\% level?