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. If the RANDNUM=PSEUDO option is specified in the MODEL statement, pseudo-random numbers are generated; if the RANDNUM=HALTON option is specified, Halton quasi-random sequences are used. 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 in 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 18.21.

Figure 18.21: Mixed Logit Model Parameter Estimates

The MDC Procedure

Mixed Multinomial Logit Estimates

Parameter Estimates
Parameter DF Estimate Standard
Error
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 .