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

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