In this exercise, we illustrate some features of instrumental variables estimation, and two-stage least squares, when the potential endogenous variable is binary. Use the data file \(r w m 88\) for this problem, and do not worry too much about the economic reasoning behind the model.

a. Estimate by OLS the regression of DOCVIS on AGE, FEMALE, WORKING, HHNINC2, and \(A D D O N\). Use heteroskedasticity robust standard errors. Does it appear that having add-on insurance is a significant factor affecting the number of doctor visits?

b. \(A D D O N\) might be endogenous. Estimate a first stage equation using OLS with \(A D D O N\) as dependent variable and AGE, FEMALE, WORKING, HHNINC2, WHITEC, and SELF as explanatory variables. Since the dependent variable is binary use heteroskedasticity robust standard errors. Are WHITEC and SELF jointly significant? Why does this matter if our objective is two-stage least squares estimation?

c. Obtain the fitted value from part (b), \(\widehat{A D D O N}\), and reestimate the model in (a) using \(\widehat{A D D O N}\) in place of \(A D D O N\). Use heteroskedasticity robust standard errors. Does it appear that having add-on insurance is a significant factor affecting the number of doctor visits?

d. Use your software command designed for two-stage least squares and estimate the model in (a) using external instruments WHITEC and SELF. Use heteroskedasticity robust standard errors. How do these estimates compare to those in part (c)? Has two-stage least squares performed as expected?

e. Since \(A D D O N\) is binary, estimate the first stage equation in (b) using probit. Compute the estimated probability that \(A D D O N=1, P H A T\). Reestimate the model in (a) using PHAT in place of \(A D D O N\). Use heteroskedasticity robust standard errors. Are the results the same as in part (d)? Why not?

f. Use your software command designed for two-stage least squares and estimate the model in (a) using external instrument PHAT. Use heteroskedasticity robust standard errors. How do these estimates compare to those in part (e)? Has two-stage least squares performed as expected?