a. Assume that y,= 1 if and only if wtp,>r. If u, is independent of (x,, r) and is distributed as Normal (0, 2), find P(y, = 1|x,,r). In particular, show that this probability follows a probit model with parameters depending on ẞ and a. b. Lety be the K x 1 vector of estimates on x, and let & be the coefficient on r, from the probit of y, on xi, r. Given these estimates, how would you estimate ß and o? c. How would you estimate ß and directly? d. Now suppose u, is independent of (x,,r) with cdf G(-; 8), where & is an R x 1 vector of parameters. Write down the log likelihood for observation i as a function of ß and 8. e. Does it make sense to compare the estimates of ß for different choices of G(-; 6) in part d? Explain.
a. Assume that y,= 1 if and only if wtp,>r. If u, is independent of (x,, r) and is distributed as Normal (0, 2), find P(y, = 1|x,,r). In particular, show that this probability follows a probit model with parameters depending on ẞ and a. b. Lety be the K x 1 vector of estimates on x, and let & be the coefficient on r, from the probit of y, on xi, r. Given these estimates, how would you estimate ß and o? c. How would you estimate ß and directly? d. Now suppose u, is independent of (x,,r) with cdf G(-; 8), where & is an R x 1 vector of parameters. Write down the log likelihood for observation i as a function of ß and 8. e. Does it make sense to compare the estimates of ß for different choices of G(-; 6) in part d? Explain.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Let wtpi denote the willingness of person i from a population to pay for a new public project, such as a new park or the widening of an existing highway. You are interested in the e¤ects of various socioeconomic variables on wtp, and you specify the population model 
Rather than observe wtpi, each person in the sample is presented with a cost of the project, ri. At this cost the person either favors or does not favor the project. Let yi = 1 if person i favors the project and zero otherwise.
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