in interpreting the coefficients in this model. Indeed, without a fair amount of extra calculation, it is quite unclear how the coefficients in the ordered probit model should be interpreted.65

Example 21.11 Rating Assignments

Marcus and Greene (1985) estimated an ordered probit model for the job assignments of new Navy recruits. The Navy attempts to direct recruits into job classifications in which they will be most productive. The broad classifications the authors analyzed were technical jobs with three clearly ranked skill ratings: "medium skilled," "highly skilled," and "nuclear qualified/highly skilled." Since the assignment is partly based on the Navy's own assessment and needs and partly on factors specific to the individual, an ordered probit model was used with the following determinants: (1) ENSPE = a dummy variable indicating that the individual entered the Navy with an "A school" (technical training) guarantee, (2) EDMA = educational level of the entrant's mother, (3) AFQT = score on the Air Force Qualifying Test, (4) EDYRS = years of education completed by the trainee, (5) MARR = a dummy variable indicating that the individual was married at the time of enlistment, and (6) AGEAt = trainee's age at the time of enlistment. The sample size was 5,641. The results are reported in Table 21.18. The extremely large t ratio on the AFQT score is to be expected, since it is a primary sorting device used to assign job classifications.

To obtain the marginal effects of the continuous variables, we require the standard normal density evaluated at -x'¡? = -0.8479 and f - = 0.9421. The predicted probabilities are $(-0.8479) = 0.198, $(0.9421) - $(-0.8479) = 0.628, and 1 - $(0.9421) = 0.174. (The actual frequencies were 0.25, 0.52, and 0.23.) The two densities are $(-0.8479) = 0.278 and $(0.9421) = 0.255. Therefore, the derivatives of the three probabilities with respect to AFQT, for example, are d Po 9AFQT

9 Pi 9AFQT

d P2 9AFQT

= (—0.278)0.039 = —0.01084, = (0.278 — 0.255)0.039 = 0.0009, = 0.255(0.039) = 0.00995.

65This point seems uniformly to be overlooked in the received literature. Authors often report coefficients and t ratios, occasionally with some commentary about significant effects, but rarely suggest upon what or in what direction those effects are exerted.

TABLE 21.19 Marginal Effect of a Binary Variable

_-ß'x_j - ß x Prob[y = 0] Prob[y = 1] Prob[y = 2]

MARR = 0 -0.8863 0.9037 MARR = 1 -0.4063 1.3837 Change

0.187 0.629 0.184

0.342 0.574 0.084

Note that the marginal effects sum to zero, which follows from the requirement that the probabilities add to one. This approach is not appropriate for evaluating the effect of a dummy variable. We can analyze a dummy variable by comparing the probabilities that result when the variable takes its two different values with those that occur with the other variables held at their sample means. For example, for the MARR variable, we have the results given in Table 21.19.

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