Recursive nested extreme value model

The objective of this note is to set out the specification of a quite general member of McFadden’s (1981) GEV family of models and to discuss some of the relevant properties of that model.

The key value of McFadden’s (1981) result was to establish that a large family of models was consistent with the ‘RUM’ concept of utility maximisation. The simplest members of that family were the MNL models, for which this consistency had already been known for a few years. The GEV result was also immediately applied to show the consistency of the ‘tree logit’ model with RUM, a result that was also proved at around the same time by other researchers independently of GEV theory.

Although McFadden indicated further possibilities for the GEV concept, applications of these were slow to materialise, primarily because specialised software was not available and general-purpose software, in particular Gauss, was too slow on the computers of the time to be used for serious modelling. However, in recent years applications have been made: the Paired Comparisons Logit (PCL, Koppelman and Wen, 2000), Cross-Nested Logit (CNL, McFadden, 1981, Vovsha, 1997), Generalised Nested Logit (GNL, Wen and Koppelman, 2000) and Ordered Generalised Extreme Value (OGEV, Small 1987). It is interesting to note that all of these models are of two-level structures: that is, elementary alternatives combine in ‘nests’ to form composite alternatives (in various ways); choice is then represented as a two stage process of choosing between these nests and then within them. The sole exception to this that has been found in the recent literature is Bhat’s (1998) model in which an OGEV is ‘grafted’ onto the bottom of an MNL model to give a three-level structure. In each case the GEV function has been constructed and shown to satisfy McFadden’s requirements, thus showing that the model is consistent with utility maximisation.

In contrast, tree logit models have long been known and even occasionally used in structures with multiple levels. Indefinite structures can be specified and even programmed efficiently (Daly, 1987).

A paper by Dagsvik (1994) indicates that there is a huge variety of GEV models, fitting effectively every possible RUM structure. However, neither Dagsvik nor McFadden indicate how GEV models should be constructed to meet specific requirements.

Recent research appears to be moving away from further exploitation of the GEV family, focussing instead on models of the ‘mixed logit’ family, which are in many cases easier to construct to meet specific requirements, as McFadden (2000) points out. However, GEV models still offer considerable advantages.

− In GEV models it is much quicker to calculate probabilities and their derivatives, making estimation and application much quicker than with other model forms.

− The existence of the GEV function itself is a considerable advantage for evaluation and use in further modelling.

− Several of the GEV models can be proved to converge in transport planning applications with conventional assignment procedures (see Prashker and Bekhor, 1999); it is not clear how far these proofs can be extended to more general GEV forms.

A substantial problem with GEV models is that it is necessary to specify the structure in advance in order to be able to work with them. Thus when the structure is at issue this family is not as convenient as those in which the significance of several structural elements can be tested simultaneously. For these reasons it is desirable to be able to write down GEV models with pre-defined properties.

In this note a model is presented which generalises all of the GEV models presented in the literature listed above, as well as generalising a multi-level tree logit model. This is achieved by specifying a recursive nesting structure that allows cross-nesting.

The Recursive Nested Extreme Value (RNEV) model can be specified with a wide range of cross-elasticity properties to fit a wide range of circumstances. The means by which this can be achieved are discussed in Section 3. The fact that this model generalises several other models means that its properties can be used to derive the properties of its special cases.

A further advantage of this models is that there exists an efficient estimation procedure, based on a generalisation – itself useful – of the tree logit estimation method of Daly (1987). This procedure is also discussed. It offers an efficient estimation method, using first and true second derivatives of the likelihood function, for the RNEV model itself and of course for its special cases. These special cases have been estimated by much less efficient means to date, impairing their applicability substantially. The estimation procedure is outlined in Section 4.