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

Table 13.9 Actual vs Predicted: Employment and Problem Drinking

. 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%

Correctly classified

90.79%

Подпись: . dprobit f dsex ags26l educ 2 educ 3 age drace inc Probit regression, reporting marginal effects Number of obs = 5768 LR chi2(7) = 964.31 Prob > chi2 = 0.0000 Log likelihood = 1561.1312 Pseudo R2 = 0.2360
Подпись: f dF/dx Std. Err. z P> z x-bar [95% Conf. Interval] dsex* .0302835 .0069532 5.40 0.000 .256415 .016655 .043912 ags26l* -.1618148 .0066629 -13.22 0.000 .377601 -.174874 -.148756 educ 2* .0022157 .0090239 0.24 0.808 .717753 -.015471 .019902 educ 3* .0288636 .0140083 2.45 0.014 .223994 .001408 .056319 age -.0065031 .0007644 -16.65 0.000 32.8024 -.008001 -.005005 drace* -.0077119 .0055649 -1.45 0.146 .773232 -.018619 .003195 inc .0002542 .000241 1.06 0.289 12.8582 -.000218 .000727 obs. P .1137309 pred. P .0367557 (at xbar)

(*) 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.

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

Подпись: . ologit rating83c ia83 dia Ordered logistic regression Number of obs = 98 LR chi2(2) = 11.54 Prob > chi2 = 0.0021 Log likelihood = -127.27146 Pseudo R2 = 0.0434

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.

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