 The ENTROPY Procedure

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;

... 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 12.20 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 1.885831 1.3463 1.40 0.1622
x2_1 -0.00308 0.0152 -0.20 0.8400
x3_1 -0.17823 0.0872 -2.04 0.0418
x4_1 1.03477 0.6952 1.49 0.1376
x1_2 0.187073 1.2612 0.15 0.8822
x2_2 0.019307 0.0144 1.34 0.1804
x3_2 0.003994 0.0817 0.05 0.9611
x4_2 0.269913 0.5747 0.47 0.6389
x1_3 -4.47704 1.6245 -2.76 0.0062
x2_3 0.025457 0.0164 1.55 0.1214
x3_3 0.235925 0.0969 2.43 0.0155
x4_3 1.25614 0.8294 1.51 0.1308
x1_4 -9.56275 1.5573 -6.14 <.0001
x2_4 0.026694 0.0154 1.73 0.0840
x3_4 0.650215 0.0930 6.99 <.0001
x4_4 1.444471 0.6912 2.09 0.0374

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 12.21 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 0.331414 1
x2_0 -0.00186 20.50148
x3_0 -0.02079 13.09496
x4_0 -0.09815 0.916914
x1_1 0.853981 1
x2_1 -0.00343 20.50148
x3_1 -0.0644 13.09496
x4_1 0.034867 0.916914
x1_2 0.853874 1
x2_2 0.000952 20.50148
x3_2 -0.04913 13.09496
x4_2 -0.16118 0.916914
x1_3 -0.25376 1
x2_3 0.001479 20.50148
x3_3 0.008944 13.09496
x4_3 0.064326 0.916914
x1_4 -1.78551 1
x2_4 0.00286 20.50148
x3_4 0.125377 13.09496
x4_4 0.160136 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 12.22 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 0.331414 1 -0.10594 1
x2_0 -0.00186 20.50148 -0.00202 0.4
x3_0 -0.02079 13.09496 0.010599 10
x4_0 -0.09815 0.916914 -0.13625 0
x1_1 0.853981 1 0.472546 1
x2_1 -0.00343 20.50148 -0.0032 0.4
x3_1 -0.0644 13.09496 -0.04401 10
x4_1 0.034867 0.916914 0.172753 0
x1_2 0.853874 1 -0.06716 1
x2_2 0.000952 20.50148 0.004337 0.4
x3_2 -0.04913 13.09496 0.014829 10
x4_2 -0.16118 0.916914 -0.07592 0
x1_3 -0.25376 1 -0.16652 1
x2_3 0.001479 20.50148 0.000628 0.4
x3_3 0.008944 13.09496 0.009385 10
x4_3 0.064326 0.916914 0.026551 0
x1_4 -1.78551 1 -0.13292 1
x2_4 0.00286 20.50148 0.000261 0.4
x3_4 0.125377 13.09496 0.009199 10
x4_4 0.160136 0.916914 0.012866 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.

Note: This procedure is experimental. Previous Page | Next Page | Top of Page