# Multinomial Choice Models

In many economic situations, the choice may be among m alternatives where m > 2. These may be unordered alternatives like the selection of a mode of transportation, bus, car or train, or an occupational choice like lawyer, carpenter, teacher, etc., or they may be ordered alternatives like bond ratings, or the response to an opinion survey, which could vary from strongly agree to strongly disagree. Ordered response multinomial models utilize the extra information implicit in the ordinal nature of the dependent variable. Therefore, these models have a different likelihood than unordered response multinomial models and have to be treated separately.

13.10.1 Ordered Response Models

Suppose there are three bond ratings, A, AA and AAA. We sample n bonds and the i-th bond is rated A (which we record as yi = 0) if its performance index I* < 0, where 0 is again not restrictive. I* = xi/3 + Ui, so the probability of an A rating or the Pr[yi = 0] is

. margeff

Average partial effects after probit

 y = Pr(emp) variable Coef. Std. Err. z P> z [95% Conf. Interval] hvdrnk90 -.0164971 .009264 -1.78 0.075 -.0346543 .00166 ue88 -.0078854 .0019748 -3.99 0.000 -.011756 -.0040149 age .0147633 .0024012 6.15 0.000 .010057 .0194697 agesq -.000193 .0000287 -6.73 0.000 -.0002493 -.0001368 educ .0069852 .0009316 7.50 0.000 .0051593 .0088112 married .048454 .0070149 6.91 0.000 .0347051 .0622028 famsize .002796 .0019603 1.43 0.154 -.0010461 .0066382 white .0685255 .0062822 10.91 0.000 .0562127 .0808383 hlstat1 .2849987 .0059359 48.01 0.000 .2733645 .2966328 hlstat2 .2318828 .0049776 46.59 0.000 .2221269 .2416386 hlstat3 .1725703 .0049899 34.58 0.000 .1627903 .1823502 hlstat4 .0914458 .0048387 18.90 0.000 .0819621 .1009295 region1 .0050178 .0083778 0.60 0.549 -.0114025 .021438 region2 .0088116 .0071262 1.24 0.216 -.0051556 .0227787 region3 .0259534 .0064999 3.99 0.000 .0132139 .0386929 msa1 -.0109515 .007632 -1.43 0.151 -.02591 .0040071 msa2 .0111628 .0067952 1.64 0.100 -.0021556 .0244811 q1 -.0160925 .0080458 -2.00 0.045 -.0318619 -.0003231 q2 -.0077086 .0076973 -1.00 0.317 -.0227951 .0073779 q3 -.0043814 .0077835 -0.56 0.573 -.0196368 .010874

. estat class Probit model for emp

True

 Classified D -D Total + 8743 826 9569 – 79 174 253 Total 8822 1000 9822 Classified + if predicted Pr(D) >= .5 True D defined as emp!= 0 Sensitivity Pr( + D) 99.10% Specificity Pr( – D) 17.40% Positive predictive value Pr( D +) 91.37% Negative predictive value Pr- D -) 68.77% False + rate for true —D Pr( ++ D) 82.60% False – rate for true D Pr( -| D) 0.90% False + rate for classified Pr- D| +) 8.63% False – rate for classified Pr( D| -) 31.23% 90.79%  (*) dF/dx is for discrete change of dummy variable from 0 to 1

z and P > z correspond to the test of the underlying coefficient being 0

The г-th bond is rated AA (which we record as yi = 1) if its performance index I* is between 0 and c where c is a positive number, with probability

п2i = Pr[y = 1] = P[0 < I* < c] (13.35)

= P[0 < x’fl + Ui < c] = F(c — хів) — F(—хів)

The г-th bond is rated AAA (which we record as yi = 2) if I* > c, with probability

пзі = Pr[yi = 2] = P[I* > c] = P[хів + Ui > c] = 1 — F(c — хів) (13.36)

F can be the logit or probit function. The log-likelihood function for the ordered probit is given by

tog^Ac) = E№=o log(\$(—хів)) + y.=1 log[T(c — хів) — ф(—х’ф)] (13.37)

+ E y.=2 log[1 — ®(c — хШ-

For the probabilities given in (13.34), (13.35) and (13.36), the marginal effects of changes in the regressors are:

дпц/дхі = —f (—хів) в (13.38)

дп2і/дхі = [f (—хів) — f (c — х’і в)]в (13.39)

дпзі/дхі = f (c — хів) в (13.40)

Generalizing this model to m bond ratings is straight forward. The likelihood, the score and the Hessian for the m-ordered probit model are given in Maddala (1983, pp. 47-49).

Illustrative Example: Corporate Bond Rating. This data set is obtained from Baum(2006) by issuing the following command in Stata:

This data set contains ratings of 98 corporate bonds coded as 2 to 5 (rating83c). The rating 2 corresponds to the lowest rating BA B C and 5 to the highest rating AAA. These are given in Table 13.11.

Table 13.11 Corporate Bond Rating

. tab rating83c

 Bond rating, 1982 Freq. Percent Cum. BA B C 26 26.53 26.53 BAA 28 28.57 55.10 AA A 15 15.31 70.41 AAA 29 29.59 100.00 Total 98 100.00

This is modeled as an ordered logit with two explanatory variables: ia83, the income to asset ratio in 1983, and the change in that ratio from 1982 to 1983 (dia). The summary statistics are given in Table 13.12.

Table 13.12 Ordered Logit

 Variable Obs Mean Std. Dev. Min Max rating83c 98 3.479592 1.17736 2 5 ia83 98 10.11473 7.441946 -13.08016 30.74564 dia 98 .7075242 4.711211 -10.79014 20.05367

 rating83c Coef. Std. Err. z P> |z| [95% Conf. Interval] ia83 .0939166 .0296196 3.17 0.002 .0358633 .1519699 dia -.0866925 .0449789 -1.93 0.054 -.1748496 .0014646 /cut1 -.1853053 .3571432 -.8852932 .5146826 /cut2 1.185726 .3882099 .4248488 1.946604 /cut3 1.908412 .4164896 1.092108 2.724717 Income/assets has a positive effect on the rating, while the change in that ratio has a negative effect! Both are significant.

Table 13.13 Predicted Bond Rating

. predict pbabc pbaa paa paaa, pr. sum pbabc pbaa paa paaa, separator(0)

 Variable Obs Mean Std. Dev. Min Max pbabc 98 .2729981 .1224448 .0388714 .7158453 pbaa 98 .2950074 .0456984 .0985567 .3299373 paa 98 .1496219 .0274841 .0449056 .1787291 paaa 98 .2823726 .1304381 .0466343 .7528986

The /cut1 to /cut3 give the estimated thresholds of the ratings categories using a logit specifi­cation. The first one is not significant, while the other two are. Note that the 95% confidence interval for these thresholds overlap.

We can predict the probability of achieving this rating using the predict command naming the values we want it to take for each category, see Table 13.13. The average of these predictions is pretty close to the actual frequencies observed in each category.