Consider a probit model explaining the choice to attend college by high-school graduates. Define \(C O L L E G E=1\) if a high-school graduate chooses either a 2-year or 4-year college, and zero otherwise. We use explanatory variables GRADES, 13 point scale with 1 indicating highest grade \((\mathrm{A}+)\) and 13 the lowest (F); FAMINC, gross family income in \(\$ 1000\) units; and \(B L A C K=1\) if black. Using a sample of \(N=1000\) graduates the estimated model is

a. What information is provided by the signs of the estimated coefficients? Which coefficients are statistically significant at the \(5 \%\) level?

b. Estimate the probability of attending college for a white student with \(G R A D E S=2(\mathrm{~A})\) and \(F A M I N C=50(\$ 50,000)\). Repeat this probability calculation if GRADES \(=5(\mathrm{~B})\)

c. Estimate the probability of attending college for a black student with \(G R A D E S=5(\mathrm{~B})\) and \(F A M I N C=50(\$ 50,000)\). Compare this probability to the comparable probability for a white student calculated in part (b).

d. Calculate the marginal effect of an increase in family income of \(\$ 1000\) on the probability of attending college for a white student with GRADES \(=5(\mathrm{~B})\).

e. The log-likelihood for the model estimated above is -423.36 . Omitting FAMINC and BLACK the \(\log\)-likelihood of the estimated probit model is -438.26 . Test the joint significance of FAMINC and BLACK at the \(1 \%\) level of significance using a likelihood ratio test.

P(COLLEGE = 1) = (2.5757 0.3068GRADES + 0.0074FAMINC + 0.6416BLACK) (0.0265) (0.0017) (se) (0.2177)