When the stochastic components of utility are heteroscedastic and independent, you can model the data by using an HEV or a multinomial probit model. The HEV model assumes that the utility of alternative for each individual has heteroscedastic random components,

where the cumulative distribution function of the Gumbel distributed is

Note that the variance of is . Therefore, the error variance is proportional to the square of the scale parameter . For model identification, at least one of the scale parameters must be normalized to 1. The following SAS statements estimate
an HEV model under a unit scale restriction for mode
“1” ():
/* hev with gausslaguerre method */ proc mdc data=newdata; model decision = ttime / type=hev nchoice=3 hev=(unitscale=1, integrate=laguerre) covest=hess; id pid; run;
The results of computation are presented in Figure 18.14 and Figure 18.15.
Figure 18.14: HEV Estimation Summary
Model Fit Summary  

Dependent Variable  decision 
Number of Observations  50 
Number of Cases  150 
Log Likelihood  33.41383 
Maximum Absolute Gradient  0.0000218 
Number of Iterations  11 
Optimization Method  Dual QuasiNewton 
AIC  72.82765 
Schwarz Criterion  78.56372 
Figure 18.15: HEV Parameter Estimates
Parameter Estimates  

Parameter  DF  Estimate  Standard Error 
t Value  Approx Pr > t 
ttime  1  0.4407  0.1798  2.45  0.0143 
SCALE2  1  0.7765  0.4348  1.79  0.0741 
SCALE3  1  0.5753  0.2752  2.09  0.0366 
The parameters SCALE2 and SCALE3 in the output correspond to the estimates of the scale parameters and , respectively.
Note that the estimate of the HEV model is not always stable because computation of the loglikelihood function requires numerical integration. Bhat (1995) proposed the GaussLaguerre method. In general, the loglikelihood function value of HEV should be larger than that of conditional logit because HEV models include the conditional logit as a special case. However, in this example the reverse is true (–33.414 for the HEV model, which is less than –33.321 for the conditional logit model). (See Figure 18.14 and Figure 18.3.) This indicates that the GaussLaguerre approximation to the true probability is too coarse. You can see how well the GaussLaguerre method works by specifying a unit scale restriction for all modes, as in the following statements, since the HEV model with the unit variance for all modes reduces to the conditional logit model:
/* hev with gausslaguerre and unit scale */ proc mdc data=newdata; model decision = ttime / type=hev nchoice=3 hev=(unitscale=1 2 3, integrate=laguerre) covest=hess; id pid; run;
Figure 18.16 shows that the ttime
coefficient is not close to that of the conditional logit model.
Figure 18.16: HEV Estimates with All Unit Scale Parameters
Parameter Estimates  

Parameter  DF  Estimate  Standard Error 
t Value  Approx Pr > t 
ttime  1  0.2926  0.0438  6.68  <.0001 
There is another option of specifying the integration method. The INTEGRATE=HARDY option uses the adaptive Rombergtype integration method. The adaptive integration produces much more accurate probability and loglikelihood function values, but often it is not practical to use this method of analyzing the HEV model because it requires excessive CPU time. The following SAS statements produce the HEV estimates by using the adaptive Rombergtype integration method:
/* hev with adaptive integration */ proc mdc data=newdata; model decision = ttime / type=hev nchoice=3 hev=(unitscale=1, integrate=hardy) covest=hess; id pid; run;
The results are displayed in Figure 18.17 and Figure 18.18.
Figure 18.17: HEV Estimation Summary Using Alternative Integration Method
Model Fit Summary  

Dependent Variable  decision 
Number of Observations  50 
Number of Cases  150 
Log Likelihood  33.02598 
Maximum Absolute Gradient  0.0001202 
Number of Iterations  8 
Optimization Method  Dual QuasiNewton 
AIC  72.05197 
Schwarz Criterion  77.78803 
Figure 18.18: HEV Estimates Using Alternative Integration Method
Parameter Estimates  

Parameter  DF  Estimate  Standard Error 
t Value  Approx Pr > t 
ttime  1  0.4580  0.1861  2.46  0.0139 
SCALE2  1  0.7757  0.4283  1.81  0.0701 
SCALE3  1  0.6908  0.3384  2.04  0.0412 
With the INTEGRATE=HARDY option, the loglikelihood function value of the HEV model, , is greater than that of the conditional logit model, . (See Figure 18.17 and Figure 18.3.)
When you impose unit scale restrictions on all choices, as in the following statements, the HEV model gives the same estimates as the conditional logit model. (See Figure 18.19 and Figure 18.6.)
/* hev with adaptive integration and unit scale */ proc mdc data=newdata; model decision = ttime / type=hev nchoice=3 hev=(unitscale=1 2 3, integrate=hardy) covest=hess; id pid; run;
Figure 18.19: Alternative HEV Estimates with Unit Scale Restrictions
Parameter Estimates  

Parameter  DF  Estimate  Standard Error 
t Value  Approx Pr > t 
ttime  1  0.3572  0.0776  4.60  <.0001 
For comparison, the following statements estimate a heteroscedastic multinomial probit model by imposing a zero restriction on the correlation parameter, . The MDC procedure requires normalization of at least two of the error variances in the multinomial probit model. Also, for identification, the correlation parameters associated with a unit normalized variance are restricted to be zero. When the UNITVARIANCE= option is specified, the zero restriction on correlation coefficients applies to the last choice of the list. In the following statements, the variances of the first and second choices are normalized. The UNITVARIANCE=(1 2) option imposes additional restrictions that . The default for the UNITVARIANCE= option is the last two choices (which would have been equivalent to UNITVARIANCE=(2 3) for this example). The result is presented in Figure 18.20.
The utility function can be defined as

where

/* mprobit estimation */ proc mdc data=newdata; model decision = ttime / type=mprobit nchoice=3 unitvariance=(1 2) covest=hess; id pid; restrict RHO_31 = 0; run;
Figure 18.20: Heteroscedastic Multinomial Probit Estimates
Parameter Estimates  

Parameter  DF  Estimate  Standard Error 
t Value  Approx Pr > t 
Parameter Label 
ttime  1  0.3206  0.0920  3.49  0.0005  
STD_3  1  1.6913  0.6906  2.45  0.0143  
RHO_31  0  0  0  
Restrict1  1  1.1854  1.5490  0.77  0.4499*  Linear EC [ 1 ] 
Note that in the output the estimates of standard errors and correlations are denoted by STD_i and RHO_ij, respectively. In this particular case the first two variances (STD_1 and STD_2) are normalized to one, and corresponding correlations (RHO_21 and RHO_32) are set to zero, so they are not listed among parameter estimates.