### Analyzing Multinomial Response Data

Multinomial discrete choice models suffer the same problems with collinearity of the regressors and small sample sizes as linear models. Unordered multinomial discrete choice models can be estimated using a variant of GME for discrete models called GME-D.

Consider the model shown in Golan, Judge, and Perloff (1996). In this model, there are five occupational categories, and the categories are considered a function of four individual characteristics. The sample contains 337 individuals.

```data kpdata;
input job x1 x2 x3 x4;
datalines;
0 1 3 11 1

... more lines ...

```

The dependent variable in this data, `job`, takes on values 0 through 4. Support points are used only for the error terms; so error supports are specified on the MODEL statement.

```proc entropy data=kpdata gmed tech=nra;
model job = x1 x2 x3 x4 / noint
esupports=( -.1 -0.0666 -0.0333 0 0.0333 0.0666 .1 );
run;
```

Figure 13.20: Estimate of Jobs Model by Using GME-D

 Prior Distribution of Parameter T

The ENTROPY Procedure

GME-D Variable Estimates
Variable Estimate Approx Std Err t Value Approx
Pr > |t|
x1_1 1.802572 1.3610 1.32 0.1863
x2_1 -0.00251 0.0154 -0.16 0.8705
x3_1 -0.17282 0.0885 -1.95 0.0517
x4_1 1.054659 0.6986 1.51 0.1321
x1_2 0.089156 1.2764 0.07 0.9444
x2_2 0.019947 0.0146 1.37 0.1718
x3_2 0.010716 0.0830 0.13 0.8974
x4_2 0.288629 0.5775 0.50 0.6176
x1_3 -4.62047 1.6476 -2.80 0.0053
x2_3 0.026175 0.0166 1.58 0.1157
x3_3 0.245198 0.0986 2.49 0.0134
x4_3 1.285466 0.8367 1.54 0.1254
x1_4 -9.72734 1.5813 -6.15 <.0001
x2_4 0.027382 0.0156 1.75 0.0805
x3_4 0.660836 0.0947 6.98 <.0001
x4_4 1.47479 0.6970 2.12 0.0351

Note there are five estimates of the parameters produced for each regressor, one for each choice. The first choice is restricted to zero for normalization purposes. PROC ENTROPY drops the zeroed regressors. PROC ENTROPY also generates tables of marginal effects for each regressor. The following statements generate the marginal effects table for the previous analysis at the means of the variables.

```proc entropy data=kpdata gmed tech=nra;
model job = x1 x2 x3 x4 / noint
esupports=( -.1 -0.0666 -0.0333 0 0.0333 0.0666 .1 )
marginals;
run;
```

Figure 13.21: Estimate of Jobs Model by Using GME-D (Marginals)

 Prior Distribution of Parameter T

The ENTROPY Procedure

GME-D Variable Marginal Effects Table
Variable Marginal Effect Mean
x1_0 0.338758 1
x2_0 -0.0019 20.50148
x3_0 -0.02129 13.09496
x4_0 -0.09917 0.916914
x1_1 0.859883 1
x2_1 -0.00345 20.50148
x3_1 -0.0648 13.09496
x4_1 0.034396 0.916914
x1_2 0.86101 1
x2_2 0.000963 20.50148
x3_2 -0.04948 13.09496
x4_2 -0.16297 0.916914
x1_3 -0.25969 1
x2_3 0.0015 20.50148
x3_3 0.009289 13.09496
x4_3 0.065569 0.916914
x1_4 -1.79996 1
x2_4 0.00288 20.50148
x3_4 0.126283 13.09496
x4_4 0.162172 0.916914

The marginals are derivatives of the probabilities with respect to each variable and so summarize how a small change in each variable affects the overall probability.

PROC ENTROPY also enables the user to specify where the derivative is evaluated, as shown below:

```proc entropy data=kpdata gmed tech=nra;
model job = x1 x2 x3 x4 / noint
esupports=( -.1 -0.0666 -0.0333 0 0.0333 0.0666 .1 )
marginals=( x2=.4 x3=10 x4=0);
run;
```

Figure 13.22: Estimate of Jobs Model by Using GME-D (Marginals)

 Prior Distribution of Parameter T

The ENTROPY Procedure

GME-D Variable Marginal Effects Table
Variable Marginal Effect Mean Marginal Effect
at User Supplied
Values
User Supplied
Values
x1_0 0.338758 1 -0.0901 1
x2_0 -0.0019 20.50148 -0.00217 0.4
x3_0 -0.02129 13.09496 0.009586 10
x4_0 -0.09917 0.916914 -0.14204 0
x1_1 0.859883 1 0.463181 1
x2_1 -0.00345 20.50148 -0.00311 0.4
x3_1 -0.0648 13.09496 -0.04339 10
x4_1 0.034396 0.916914 0.174876 0
x1_2 0.86101 1 -0.07894 1
x2_2 0.000963 20.50148 0.004405 0.4
x3_2 -0.04948 13.09496 0.015555 10
x4_2 -0.16297 0.916914 -0.072 0
x1_3 -0.25969 1 -0.16459 1
x2_3 0.0015 20.50148 0.000623 0.4
x3_3 0.009289 13.09496 0.00929 10
x4_3 0.065569 0.916914 0.02648 0
x1_4 -1.79996 1 -0.12955 1
x2_4 0.00288 20.50148 0.000256 0.4
x3_4 0.126283 13.09496 0.008956 10
x4_4 0.162172 0.916914 0.012684 0

In this example, you evaluate the derivative when x1=1, x2=0.4, x3=10, and x4=0. If the user neglects a variable, PROC ENTROPY uses its mean value.