Springer Texts in Business and Economics

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

. estat class Probit model for emp

(*) 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:

.use http://www. stata-press. 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.

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.

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

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.