One way of modeling unobserved heterogeneity across individuals in their sensitivity to observed exogenous variables is to use the mixed logit model with a random parameters or random coefficients specification. The probability of choosing alternative is written as

where is a vector of coefficients that varies across individuals, and is a vector of exogenous attributes.

For example, you can specify the distribution of the parameter to be the normal distribution.

The mixed logit model uses a Monte Carlo simulation method to estimate the probabilities of choice. There are two simulation methods available. When the RANDNUM=PSEUDO option is given in the MODEL statement, pseudo-random numbers are generated, while the RANDNUM=HALTON option uses Halton quasi-random sequences. The default value is RANDNUM=HALTON.

You can estimate the model with normally distributed random coefficients of ttime with the following SAS statements:

   /*-- mixed logit estimation --*/
   proc mdc data=newdata type=mixedlogit;
      model decision = ttime /
               nchoice=3
               mixed=(normalparm=ttime);
      id pid;
   run;

Let and be mean and scale parameters, respectively, for the random coefficient, . The relevant utility function is

where ; and are fixed mean and scale parameters, respectively. The stochastic component, , is assumed to be standard normal since the NORMALPARM= option is given. Alternatively, the UNIFORMPARM= or LOGNORMALPARM= option can be specified. The LOGNORMALPARM= option is useful when nonnegative parameters are being estimated. The NORMALPARM=, UNIFORMPARM=, and LOGNORMALPARM= variables must be included on the right-hand side of the MODEL statement. See the section Mixed Logit Model for more details. To estimate a mixed logit model by using the transportation mode choice data, the MDC procedure requires the MIXED= option for random components. Results of the mixed logit estimation are displayed in Figure 17.21.