Circumventing the problem of the scale: discrete choice models with multiplicative error terms

We propose a multiplicative specification of a discrete choice model that renders choice probabilities independent of the scale of the utility. The scale can thus be random with unspecified distribution. The model mostly outperforms the classical additive formulation over a range of stated choice data sets. In some cases, the improvement in likelihood is greater than that obtained from adding observed and unobserved heterogeneity to the additive specification. The multiplicative specification makes it unnecessary to capture scale heterogeneity and, consequently, yields a significant potential for reducing model complexity in the presence of heteroscedasticity. Thus the proposed multiplicative formulation should be a useful supplement to the techniques available for the analysis of discrete choices. There is however a cost to be paid in terms of increased analytical complexity relative to the additive formulations.