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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
*j* 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

1 |
-0.5342 |
0.2184 |
-2.45 |
0.0144 |

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 .