Consider a probit model explaining the choice to attend a 4-year college rather than a 2 -year

Question:

Consider a probit model explaining the choice to attend a 4-year college rather than a 2 -year college by high-school graduates who chose to attend a postsecondary school. Define $F O U R Y R=1$ if a high-school graduate chooses 4-year college and FOURYR $=0$ if the high school graduate chooses a 2-year college. We use explanatory variables GRADES, 13 point scale with 1 indicating highest grade (A+) and 13 the lowest $(\mathrm{F}) ;$ FAMINC, gross family income in $\$ 1000$ units; and HSCATH $=1$ if the student attended a Catholic high school and $H S C A T H=0$ otherwise. Table 16.9 contains some probit model estimates.

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a. Using Model (2), how large an effect on the probability of attending a 4-year college does attending a catholic high school have for a student with $G R A D E S=5$ (B) and family income of $\$ 100,000$.

b. Comparing Models (2) and (3), are the interaction variables HSCATH $\times$ GRADES and $H S C A T H \times$ FAMINC jointly significant at $5 \%$ using a likelihood ratio test?

c. Can we interpret the Model (3) results as saying an increase in family income reduces the probability of attending a 4-year college for someone graduating from a Catholic high school? What is the marginal effect of an additional $\$ 1000$ in family income for a Catholic high school student with GRADES $=5$ (B) and family income of $\$ 50,000$ ?

d. Using Model (3), compute the probability of attending a 4-year college for someone graduating from a Catholic high school with GRADES $=5$ (B) and family income of $\$ 100,000$. Compare this probability to a student who did not attend a Catholic high school but has GRADES $=5$ (B) and family income of $\$ 100,000$.

e. Using Models (1) and (3), test the null hypothesis that the probit model parameters are the same for students who attend and do not attend a Catholic high school. Use a likelihood ratio test at the $5 \%$ level of significance.

f. Using Models (4) and (5), estimate the probit model separately for $H S C A T H=0$ and $H S C A T H=1$. Compute the sum of the log-likelihood functions values. Compare the sum to the log-likelihood for Model (3). Algebraically show that this is not an accident.

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