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The MDC Procedure

Nested Logit

The nested logit model (McFadden 1978, 1981) allows partial relaxation of the assumption of independence of the stochastic components of utility of alternatives. In some choice situations, the IIA property holds for some pairs of alternatives but not all. In these situations, the nested logit model can be used if the set of alternatives faced by an individual can be partitioned into subsets such that the IIA property holds within subsets but not across subsets.

In the nested logit model, the joint distribution of the errors is generalized extreme value (GEV). This is a generalization of the type I extreme-value distribution that gives rise to the conditional logit model. Note that all within each subset are correlated with each other. Refer to McFadden (1978, 1981) for details.

Nested logit models can be described analytically following the notation of McFadden (1981). Assume that there are levels, with 1 representing the lowest and representing the highest level of the tree. The index of a node at level in the tree is a pair , where is the index of the adjacent node at level . Thus, the primitive alternatives, at level 1 in the tree, are indexed by vectors , while the alternative nodes at level L are indexed by integers . The choice set is the set of primitive alternatives (at level 1) that belong to branches below the node . The notation is also used to denote a set of indices such that is a node immediately below . Note that is a set with a single element, while represents a choice set containing all possible alternatives. As an example, consider the circled node at level 1 in Figure 17.26. Since it stems from node , , and since it is the second node stemming from , , so that its index is . Similarly, contains all the possible choices below .

Note that while this notation is useful for writing closed-form solutions for probabilities, the MDC procedure allows a more flexible definition of indices. See the NEST statement in the section "Syntax" for more details on how to describe decision trees within the MDC procedure.

Figure 17.26 Node Indices for a Three-Level Tree
Node Indices for a Three-Level Tree

Let denote the vector of observed variables for individual common to the alternatives below node . The probability of choice at level has a closed-form solution and is written


where is the inclusive value (at level ) of the branch below node and is defined recursively as follows:


The inclusive value denotes the average utility that the individual can expect from the branch below . The dissimilarity parameters or inclusive value parameters () are the coefficients of the inclusive values and have values between 0 and 1 if nested logit is the correct model specification. When they all take value 1, the nested logit model is equivalent to the conditional logit model.

At decision level 1, there is no inclusive value; i.e., . Therefore, the conditional probability is


The log-likelihood function at level can then be written


where is an indicator variable that has the value of 1 for the selected choice. The full log-likelihood function of the nested logit model is obtained by adding the conditional log-likelihood functions at each level:


Note that the log-likelihood functions are computed from conditional probabilities when . The nested logit model is estimated using the full information maximum likelihood method.

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