Multivariate Normal Utility Function :: SAS/ETS(R) 12.3 User's Guide

Multivariate Normal Utility Function

Consider the random utility function

$U_{ij} = \mr {\Variable{ttime} }_{ij}\beta + \epsilon _{ij},\; \; j=1,2,3$

where

$\left[ \begin{array}{c}\epsilon _{i1} \\ \epsilon _{i2} \\ \epsilon _{i3} \\ \end{array} \right] \sim N \left( \mathbf{0} ,\left[ \begin{array}{ccc} 1 & \rho _{21} & 0 \\ \rho _{21} & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \right)$

The correlation coefficient ( $\rho _{21}$ ) between $U_{i1}$ and $U_{i2}$ 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
Maximum Absolute Gradient	0.0002380
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