The Recurrent Problem of Parametrising Nested Logit Models
A Daly, RAND Europe, UK
Despite many efforts, the appropriate parametrisation of nested logit models remains confused and confusing. The paper brings help.
Nested logit models are widely used for travel demand analysis, chiefly using ?tree? structures but occasionally generalising these to allow cross-nesting. An important part of the justification for these models is that they can be derived from a theory of individual utility maximisation (although other theoretical bases also exist). This justification gives the models the familiar Random Utility (RU) basis by exploiting the concept of error in the utility function.
The nested logit model allows asymmetry among the alternatives even at an individual level, thus avoiding reliance on an assumption of ?independence from irrelevant alternatives?. This generalisation is obtained by introducing additional parameters into the structure which express the ratio of the standard errors of utility between alternatives in different subsets. However, there is no unique way to parametrise these ratios.
Different parametrisations have existed since the models were originally shown to be consistent with RU but these were frequently ignored or overlooked. More recently, a series of papers have identified two main parametrisations, called RU1 and RU2, not very originally, and have indicated some of the main issues. However, some of these papers contain important errors and there are further difficulties in ensuring consistency with RU in practical contexts.
The presentation therefore presents the issues and the RU1 and RU2 parametrisations. The way in which RU tests apply to these is explained, along with the conclusion, not made explicit in the literature, that RU2 is always RU-consistent but that RU1 may require constraints or extensions to the model to make it so.
The tests that have been applied to the models involve hypothetical changes to the (indirect) conditional utility of choosing each alternative. This does not help much in practice, of course, because we need to work with measurable variables, such as time and cost. But these variables raise further issues when they are included in the model in non-linear form and these issues are still the subject of research. The issues will be discussed briefly.
Finally, the paper discusses a further issue of normalisation that arises in cross-nested models, where the ?allocation parameters? that specify the way in which the cross-nesting operates require constraint, but the literature is again inconsistent. An effective normalisation is proposed and explained.
It is hoped that the paper will reduce the uncertainty and confusion surrounding this issue, which persist despite the efforts that have been made to reduce them.
Association for European Transport