Independence, homoskedasticity and existence in random utility models

Introduction

Random utility models are often characterised by descriptions such as ‘homoskedastic’ or ‘independent’ in the utilities of the alternatives. However these descriptions do not have meaning in any absolute sense and must therefore be used with care. It is the main aim of this paper to demonstrate this point and discuss the issues it raises. In particular, the discussion leads into a consideration of the circumstances under which the models can be said to exist.

The paper gives a definition of random utility models and goes on the define a large sub-class of those models, the additive stimulus models, on which the main discussion of the paper is focussed. The area of discussion is further specified by relating the probability statement, which is the main form in which the model is estimated and used, to the utility and utility difference distributions. New concepts are introduced of indistinguishability and almost-indistinguishability, which can be used in assessing discrete choice models. The paper then shows how a reasonable notion of model structure can be interpreted in terms of utility difference distributions for a class of indistinguishable models.

The discussion of the independence of the utility distributions of the alternatives is based on the concepts introduced in the early parts of the paper. This discussion shows that many indistinguishable models exist for which the correlation of the utility functions is radically different. A following discussion goes on to show that the notion of heteroskedasticity is similarly incapable of clear definition, even within classes of indistinguishable models.

The final main section discusses the issue of existence, finding that it is quite difficult to ensure that models actually represent a ‘real’ situation, although it is seen as important that the models actually ‘exist’ in some sense.. An error components approach, whether using purely probit models or substituting a logit kernel appears a useful approach to maintaining the ‘reality’ of the model.