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 ith 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
Table 13.9 Actual vs Predicted: Employment and Problem Drinking 
. estat class Probit model for emp
True

(*) 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 гth 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 гth 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 loglikelihood 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 mordered probit model are given in Maddala (1983, pp. 4749).
Illustrative Example: Corporate Bond Rating. This data set is obtained from Baum(2006) by issuing the following command in Stata:
.use http://www. statapress. com/data/imeus/panel84extract, clear
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

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.

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 
Table 13.13 Predicted Bond Rating
. predict pbabc pbaa paa paaa, pr. sum pbabc pbaa paa paaa, separator(0)

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.
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