In Example 16.14, we described an ordinal probit model for postsecondary education choice, and estimated a simple model in which the choice depended simply on the student's GRADES. Expand the ordered probit model to include family income (FAMINC, in \(\$ 1000\) ), family size (FAMSIZ), the dummy variables \(B L A C K\) and \(P A R C O L L=1\) if a parent has at least a college degree, and 0 otherwise. The estimates of this model are Model 2 in Table 16.13.

a. Using the estimates in Table 16.13, Model 1, calculate the probability that a student will choose no college, a 2-year college, and a 4-year college if the student's grades are GRADES \(=7(\mathrm{~B}-)\). Recompute these probabilities assuming that \(G R A D E S=3(\mathrm{~A}-)\). Discuss the probability changes. Are they what you anticipated? Explain.

b. Discuss the Model 2 estimates, their signs and significance.

c. Test the joint significance of the variables added in (b) using a likelihood ratio test at the \(1 \%\) level of significance.

d. Compute the probability that a black student from a household of four members with \(\$ 100,000\) income, and with at least one parent having at least a college degree, so that PARCOLL \(=1\), will attend a 4-year college if (i) GRADES \(=7\) and (ii) GRADES \(=3\).

e. Repeat (d) for a "nonblack" student and discuss the differences in your findings.

Data From Example 16.14:-

To illustrate, we use the college choice data introduced in Section 16.3 and contained in the data file nels_small. We treat PSECHOICE as an ordered variable with 1 representing the least favored alternative (no college) and 3 denoting the most favored alternative (4-year college). The estimation results are in Table 16.5 .

The estimated coefficient of GRADES is negative, indicating that the probability of attending a 4-year college goes down when GRADES increase (indicating a worse performance), and the probability of the lowest ranked choice, attending no college, increases. Let us examine the marginal effects of an increase in GRADES on attending a 4-year college. For a student with median grades (6.64), the marginal effect is -0.1221 , and for a student in the 5th percentile (2.635), the marginal effect is -0.0538 . These are similar in magnitude to the marginal effects shown in Table 16.3.

Data From Table 16.3:-

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TABLE 16.13 Estimates for Exercise 16.14 Model 1 Model 2 PSECHOICE Coefficient Standard Error Coefficient Standard Error GRADES -0.3066 0.0192 -0.2953 0.0202 FAMINC 0.0053 0.0013 FAMSIZ -0.0241 0.0302 BLACK 0.7131 0.1768 PARCOLL 0.4236 0.1016 -2.9456 0.1468 -2.5958 0.2046 -2.0900 0.1358 -1.6946 0.1971 InZ -875.8217 -839.8647