### Multivariate Normal Utility Function

Consider the random utility function

where

The correlation coefficient () between and represents commonly neglected attributes of public transportation modes, 1 and 2. The following SAS statements estimate this trinomial probit model:

/*-- homoscedastic mprobit --*/
proc mdc data=newdata;
model decision = ttime /
type=mprobit
nchoice=3
unitvariance=(1 2 3)
covest=hess;
id pid;
run;


The UNITVARIANCE=(1 2 3) option specifies that the random component of utility for each of these choices has unit variance. If the UNITVARIANCE= option is specified, it needs to include at least two choices. The results of this constrained multinomial probit model estimation are displayed in Figure 18.12 and Figure 18.13. The test for ttime = 0 is rejected at the 1% significance level.

Figure 18.12: Constrained Probit Estimation Summary

The MDC Procedure

Multinomial Probit Estimates

Model Fit Summary
Dependent Variable decision
Number of Observations 50
Number of Cases 150
Log Likelihood -33.88604
Log Likelihood Null (LogL(0)) -54.93061
Number of Iterations 8
Optimization Method Dual Quasi-Newton
AIC 71.77209
Schwarz Criterion 75.59613
Number of Simulations 100
Starting Point of Halton Sequence 11

Figure 18.13: Multinomial Probit Estimates with Unit Variances

The MDC Procedure

Multinomial Probit Estimates

Parameter Estimates
Parameter DF Estimate Standard
Error
t Value Approx
Pr > |t|
ttime 1 -0.2307 0.0472 -4.89 <.0001
RHO_21 1 0.4820 0.3135 1.54 0.1242