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

\[  P_{i}(j) = \frac{\exp (\mathbf{x}_{ij}\bbeta )}{\sum _{k=1}^{J}\exp (\mathbf{x}_{ik}\bbeta )}  \]

where $\bbeta $ is a vector of coefficients that varies across individuals and $\mathbf{x}_{ij}$ is a vector of exogenous attributes.

For example, you can specify the distribution of the parameter $\beta $ 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 /
   id pid;

Let $\beta ^{m}$ and $\beta ^{s}$ be mean and scale parameters, respectively, for the random coefficient, $\beta $. The relevant utility function is

\[  U_{ij} = \mr {\Variable{ttime} }_{ij}\beta + \epsilon _{ij}  \]

where $\beta = \beta ^{m} + \beta ^{s}\eta $ ($\beta ^{m}$ and $\beta ^{s}$ are fixed mean and scale parameters, respectively). The stochastic component, $\eta $, 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
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 $\beta ^ m$ and the parameter ttime_S corresponds to the constant scale parameter $\beta ^ s$ of the random coefficient $\beta $.