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The MDC Procedure

Parameter Heterogeneity: Mixed Logit

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 /
      id pid;

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.

Figure 17.21 Mixed Logit Model Parameter Estimates
The MDC Procedure
Mixed Multinomial Logit Estimates

Parameter Estimates
Parameter DF Estimate Standard
t Value Approx
Pr > |t|
ttime_M 1 -0.5342 0.2184 -2.45 0.0144
ttime_S 1 0.2843 0.1911 1.49 0.1368

Note that the parameter ttime_M corresponds to the constant mean parameter and the parameter ttime_S corresponds to the constant scale parameter of the random coefficient .

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