Notes: Bold coefficients are significant at the .05 level; other coefficients are not. The omitted categories in the model are White, male, SAT < 1000, bottom 90% of high school class, middle SES, SEL-3, coed institution. Graduation rates are 6-year, first-school graduation rates, as defined in the notes to Appendix Table D.3.1. Institutional selectivity categories are as defined in the notes to Appendix Table D.3.1. See Appendix B for definition of socioeconomic status (SES).

SEL-1 = institutions with mean combined SAT scores of 1300 and above. SEL-2 = institutions with mean combined SAT scores between 1150 and 1299. SEL-3 = institutions with mean combined SAT scores below 1150. Source: Bowen and Bok, op. cit., p. 381.

William G. Bowen and Derek Bok, The Shape of the River: Long Term Consequences of Considering Race in College and University Admissions, Princeton University Press, Princeton, N.J., 1998, p. 381.

CHAPTER FIFTEEN: QUALITATIVE RESPONSE REGRESSION MODELS 629

a. What general conclusion do you draw about graduation rates of all matriculants and black-only matriculants?

b. The odds ratio is the ratio of two odds. Compare two groups of all matriculants, one with a SAT score of greater than 1299 and the other with a SAT score of less than 1000 (the base category). The odds ratio of 1.393 means the odds of matriculants in the first category graduating from college are 39 percent higher than those in the latter category. Do the various odds ratios shown in the table accord with a priori expectations?

c. What can you say about the statistical significance of the estimated parameters? What about the overall significance of the estimated model?

15.12. In the probit model given in Table 15.11 the disturbance ui has this variance:

where fi is the standard normal density function evaluated at F-1(Pi).

a. Given the preceding variance of ui, how would you transform the model in Table 15.10 to make the resulting error term homoscedastic?

b. Use the data in Table 15.10 to show the transformed data.

c. Estimate the probit model based on the transformed data and compare the results with those based on the original data.

15.13. Since R2 as a measure of goodness of fit is not particularly well suited for the dichotomous dependent variable models, one suggested alternative is the x2 test described below:

where Ni = number of observations in the ith cell

Pi = actual probability of the event occurring ( = ni/Ni) PI* = estimated probability

G = number of cells (i.e., the number of levels at which Xi is measured, e.g., 10 in Table 15.4)

It can be shown that, for large samples, x 2 is distributed according to the X2 distribution with (G - k) df, where k is the number of parameters in the estimating model (k < G).

Apply the preceding x2 test to regression (15.7.1) and comment on the resulting goodness of fit and compare it with the reported R2 value.

15.14. Table 15.20 gives data on the results of spraying rotenone of different concentrations on the chrysanthemum aphis in batches of approximately fifty. Develop a suitable model to express the probability of death as a function of the log of X, the log of dosage, and comment on the results. Also compute the x2 test of fit discussed in exercise 15.13.

15.15. Fourteen applicants to a graduate program had quantitative and verbal scores on the GRE as listed in Table 15.21. Six students were admitted to the program.

630 PART THREE: TOPICS IN ECONOMETRICS

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