## Consistency of Nested Logit Models with Utility Maximization

### Authors

J. Nicolás Ibáñez, ITS, University of Leeds, UK

### Description

Procedure to build nested logit models that split the random parts of the alternatives in two components and allowing for a less restrictive explanation of how higher than one inclusive values are consistent with random utility maximizing theory

### Abstract

This paper provides the minimum set of necessary and sufficient conditions for testing the global consistency of preference models with random utility maximization (RUM), that is, with preferences that can be described by an absolutely continuous, proper, non-defective and, at least, translationally invariant distribution (Marschak, 1960; Block & Marschak, 1960; McFadden, 1981; Daly, 2004). In so doing we discard some of the conditions, added to the previous set, considered by other authors such as Daly & Zachary (1976), McFadden (1981), Börsch-Supan (1990) or Koning & Ridder (1994, 2003) in their necessary and sufficient sets of conditions to guarantee compatibility with RUM.

Particularly, in relation with the condition of symmetrical probability systems included in all these sets, though but not in ours, we offer a consistent proof of why the latter is implied by our minimum set, specifying the lowest number of assumptions to guarantee it, translating it into the context of differences of random terms, and without making necessary the identification of a probability system with the gradient of a social surplus function as in McFadden (1981) or with the use of a GEV function as in McFadden (1978). Moreover, considering the relation of this symmetry with the integrability of demand systems, we show how it affects the negative semi-definite character of the Slutsky matrix governing such systems.

Additionally, we address the issue of local compatibility of a probability system with random utility maximisation, which has received its major impulse from studies such as Börsch-Supan (1990) and Koning & Ridder (1994, 2003): on one hand, we review the main inconsistency posed by the former, due to not guaranteeing the independency of the analysis from the alternative taken as reference to calculate the random terms? differences; on the other hand, and regarding the latter, we point out some inconsistencies present in their theorems to proof that a probability system is locally compatible with RUM, mainly, that the expected value of the differences of the random terms participating in such system do depend on the alternative of reference considered in the analysis.

Moving to the applications of the theoretical results contain in the paper, we show how one of the direct consequences of imposing our three conditions on a widely spread particular discrete choice model, the nested logit one originally developed in Williams (1977), leads to consider as inexact some of the results present in the literature about relaxing the range for the inclusive values or dissimilarity parameters measuring the allowed dependences between alternatives in such nested logit model. Hence, we study the properties to be met by the parameters of two-level and three-level nested logit models in order to guarantee their compliance with the postulates of RUM theory. In so doing we show that the conditions stated in the literature that permit dissimilarity parameters greater than one are in fact necessary but not sufficient conditions to comply with RUM, since they do not guarantee themselves a proper distribution for the random part of the utilities of the alternatives faced by individuals under such models. Particularly, we consider as misleading the relaxations of Herriges & Kling (1996) and Gil-Moltó & Risa (2004).

Thus, applying the previously derived minimum set of conditions to ensure compatibility with RUM, we present more restrictive, but sufficient, conditions to be met by the dissimilarity parameters of these nested logit models in order to assure that they comply with RUM, and conclude that such parameters have to be in the [0,1] interval if these nested logit models are considered to be consistent with RUM.

Besides, we present a sequential implementation of the postulates of RUM more general than the one in Ben-Akiva & Lerman (1985) and in order to relax the range for dissimilarity parameters for a two-level nested logit model with any number of branches and up to two alternatives per branch (or aggregation of alternatives), noting that the extension to a higher number of levels and/or number of alternatives per branch is straightforward. We start considering that the model probabilities are the result of two (not independent) choice processes: first, the process to choose an alternative given its belonging to a particular branch, and second, the choice of one of the total of available branches. This analysis, which relies on considering an split of the utilities? random terms in two components, one associated to a branch level and another one to the alternative or elemental level, permits, in a two-level nested logit with up to two alternatives per branch and unlimited number of branches, that the dissimilarity parameters are in the range [0, 1.28], with each value in this range indicating a different pattern of correlation between the random terms involved in the model.

Thus, we think that both the review carried out of the different material in the literature about ensuring the global and local compatibility of discrete choice modelling with stochastic utility maximisation and the particularisation to the study of nested logit models might offer some value for practitioners when applying RUM theory, in general, or nested logit modelling, in particular, to study transport behaviour.

Keywords: Nested logit, discrete choice, random utility maximization

#### Publisher

Association for European Transport