The MDC Procedure

Example 17.3 Correlated Choice Modeling

Often, it is not realistic to assume that the random components of utility for all choices are independent. This example shows the solution to the problem of correlated random components by using multinomial probit and nested logit.

To analyze correlated data, trinomial choice data (1000 observations) are created using a pseudo-random number generator by using the following statements. The random utility function is

where

```   /*-- generate simulated series --*/
%let ndim = 3;
%let nobs = 1000;

data trichoice;
array error{&ndim} e1-e3;
array vtemp{&ndim} _temporary_;
array lm{6} _temporary_ (1.4142136 0.4242641 0.9055385 0 0 1);
retain nseed 345678 useed 223344;

do id = 1 to &nobs;
index = 0;
/* generate independent normal variate */
do i = 1 to &ndim;
/* index of diagonal element */
vtemp{i} = rannor(nseed);
end;
/* get multivariate normal variate */
index = 0;
do i = 1 to &ndim;
error{i} = 0;
do j = 1 to i;
error{i} = error{i} + lm{index+j}*vtemp{j};
end;
index = index + i;
end;
x1 = 1.0 + 2.0 * ranuni(useed);
x2 = 1.2 + 2.0 * ranuni(useed);
x3 = 1.5 + 1.2 * ranuni(useed);
util1 = 2.0 * x1 + e1;
util2 = 2.0 * x2 + e2;
util3 = 2.0 * x3 + e3;
do i = 1 to &ndim;
vtemp{i} = 0;
end;
if ( util1 > util2 & util1 > util3 ) then
vtemp{1} = 1;
else if ( util2 > util1 & util2 > util3 ) then
vtemp{2} = 1;
else if ( util3 > util1 & util3 > util2 ) then
vtemp{3} = 1;
else continue;
/*-- first choice --*/
x = x1;
mode = 1;
decision = vtemp{1};
output;
/*-- second choice --*/
x = x2;
mode = 2;
decision = vtemp{2};
output;
/*-- third choice --*/
x = x3;
mode = 3;
decision = vtemp{3};
output;
end;
run;
```

First, the multinomial probit model is estimated (see the following program). Results show that standard deviation, correlation, and slope estimates are close to the parameter values. Note that , , , and the parameter value for the variable x is 2.0. (See Output 17.3.1.)

```   /*-- Trinomial Probit --*/
proc mdc data=trichoice randnum=halton nsimul=100;
model decision = x /
type=mprobit
choice=(mode 1 2 3)
covest=op
optmethod=qn;
id id;
run;
```

Output 17.3.1 Trinomial Probit Model Estimation
The MDC Procedure

Multinomial Probit Estimates

Parameter Estimates
Parameter DF Estimate Standard
Error
t Value Approx
Pr > |t|
x 1 1.7987 0.1202 14.97 <.0001
STD_1 1 1.2824 0.1468 8.74 <.0001
RHO_21 1 0.4233 0.1041 4.06 <.0001

The nested model is also estimated based on a two-level decision tree (see the following program). (See Output 17.3.2.) The estimated result (see Output 17.3.3) shows that the data support the nested tree model since the estimates of the inclusive value parameters are significant and are less than 1.

Output 17.3.2 Nested Tree Structure

```   /*-- Two-Level Nested Logit --*/
proc mdc data=trichoice;
model decision = x /
type=nlogit
choice=(mode 1 2 3)
covest=op
optmethod=qn;
id id;
utility u(1,) = x;
nest level(1) = (1 2 @ 1, 3 @ 2),
level(2) = (1 2 @ 1);
run;
```

Output 17.3.3 Two-Level Nested Logit
The MDC Procedure

Nested Logit Estimates

Parameter Estimates
Parameter DF Estimate Standard
Error
t Value Approx
Pr > |t|
x_L1 1 2.6672 0.1978 13.48 <.0001
INC_L2G1C1 1 0.7911 0.0832 9.51 <.0001
INC_L2G1C2 1 0.7965 0.0921 8.65 <.0001

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