The MDC Procedure

Nested Logit Modeling

A more general model can be specified using the nested logit model.

Consider, for example, the following random utility function:

$U_{ij} = {x}_{ij}\beta + \epsilon _{ij}\; \; j=1,\ldots ,3$

Suppose the set of all alternatives indexed by j is partitioned into $K$ nests, $B_1,\ldots ,B_ K$ . The nested logit model is obtained by assuming that the error term in the utility function has the GEV cumulative distribution function

$\exp \left(-\sum _{k=1}^ K\left(\sum _{j\in B_ k}\exp \{ -\epsilon _{ij}/\lambda _ k\} \right)^{\lambda _ k}\right)$

where $\lambda _ k$ is a measure of a degree of independence among the alternatives in nest k. When $\lambda _ k=1$ for all k, the model reduces to the standard logit model.

Since the public transportation modes, 1 and 2, tend to be correlated, these two choices can be grouped together. The decision tree displayed in Figure 18.8 is constructed.

Figure 18.8: Decision Tree for Model Choice

The two-level decision tree is specified in the NEST statement. The NCHOICE= option is not allowed for nested logit estimation. Instead, the CHOICE= option needs to be specified, as in the following statements:

/*-- nested logit estimation --*/
proc mdc data=newdata;
   model decision = ttime /
            type=nlogit
            choice=(mode 1 2 3)
            covest=hess;
   id pid;
   utility u(1,) = ttime;
   nest level(1) = (1 2 @ 1, 3 @ 2),
        level(2) = (1 2 @ 1);
run;

In Figure 18.9, estimates of the inclusive value parameters, INC_L2G1C1 and INC_L2G1C2, are indicative of a nested model structure. See the section Nested Logit and the section Decision Tree and Nested Logit for more details about inclusive values.

Figure 18.9: Two-Level Nested Logit Estimates

The MDC Procedure

Nested Logit Estimates

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
ttime_L1	1	-0.4040	0.1241	-3.25	0.0011
INC_L2G1C1	1	0.8016	0.4352	1.84	0.0655
INC_L2G1C2	1	0.8087	0.3591	2.25	0.0243

The nested logit model is estimated with the restriction INC_L2G1C1 = INC_L2G1C2 by specifying the SAMESCALE option, as in the following statements:

/*-- nlogit with samescale option --*/
proc mdc data=newdata;
   model decision = ttime /
            type=nlogit
            choice=(mode 1 2 3)
            samescale
            covest=hess;
   id pid;
   utility u(1,) = ttime;
   nest level(1) = (1 2 @ 1, 3 @ 2),
        level(2) = (1 2 @ 1);
run;

The estimation result is displayed in Figure 18.10.

Figure 18.10: Nested Logit Estimates with One Dissimilarity Parameter

The MDC Procedure

Nested Logit Estimates

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
ttime_L1	1	-0.4025	0.1217	-3.31	0.0009
INC_L2G1	1	0.8209	0.3019	2.72	0.0066

The nested logit model is equivalent to the conditional logit model if INC_L2G1C1 = INC_L2G1C2 = 1. You can verify this relationship by estimating a constrained nested logit model as shown in the following statements. (See the section RESTRICT Statement for details about imposing linear restrictions on parameter estimates.)

/*-- constrained nested logit estimation --*/
proc mdc data=newdata;
   model decision = ttime /
            type=nlogit
            choice=(mode 1 2 3)
            covest=hess;
   id pid;
   utility u(1,) = ttime;
   nest level(1) = (1 2 @ 1, 3 @ 2),
        level(2) = (1 2 @ 1);
   restrict INC_L2G1C1 = 1, INC_L2G1C2 =1;
run;

The parameter estimates and the active linear constraints for the constrained nested logit model are displayed in Figure 18.11.

Figure 18.11: Constrained Nested Logit Estimates

The MDC Procedure

Nested Logit Estimates

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Parameter Label
ttime_L1	1	-0.3572	0.0776	-4.60	<.0001
INC_L2G1C1	0	1.0000	0
INC_L2G1C2	0	1.0000	0
Restrict1	1	-2.1706	8.4098	-0.26	0.7993*	Linear EC [ 1 ]
Restrict2	1	3.6573	10.0001	0.37	0.7186*	Linear EC [ 2 ]

* Probability computed using beta distribution.

Linearly Independent Active Linear Constraints
1	0	=	-1.0000	+	1.0000	*	INC_L2G1C1
2	0	=	-1.0000	+	1.0000	*	INC_L2G1C2