## Circumventing the Problem of the Scale: Discrete Choice Models with Multiplicative Error Terms

### Authors

M Fosgerau, Danish Transport Research Institute, DK; M Bierlaire, EPFL, CH

### Description

We propose a multiplicative specification of a discrete choice model that renders choice probabilities independent of the scale of the utility. The model outperforms the classical additive formulation over a range of data sets.

### Abstract

Discrete choice models have been a major part of the transport analysts' toolbox for decades. These models are able to accommodate diverse requirements and they have a firm theoretical foundation in

utility theory.

Discrete choice models with additive independent error

terms pose the problem that the scale of the error terms is not identified. Earlier models assumed the problem away by requiring the scale to be constant. Later contributions have allowed the scale to

vary across data sets and individuals.

We propose instead a multiplicative specification of discrete choice models that circumvent the problem by making the scale irrelevant. It can thus be random and have any distribution. This specification is applicable in situations where we have a priori information about the sign of the systematic utility.

The multinomial logit (MNL) model has been very successful, due to its computational and analytical tractability. Later, generalized extreme value (GEV) models and mixtures of MNL and GEV models have

gained popularity due to their flexibility and theoretical results relating these models to random utility maximization.

So far, most applications of these models have used an additive specification where the random utility for an alternative is specified as U_i = V_i + mu * e_i, and where usually V_i= beta x_i and mu is a scale parameter such that the e are i.i.d.

The additive specification is computationally convenient, which may explain its systematic use.

The basic formulation of MNL and GEV models assume that mu is constant across the population, and can therefore be arbitrarily normalised. This assumption is strong, and a number of techniques to relax it have

been developed in the literature.

The additive specification is however not required by utility theory. There are alternative formulations which cannot be ruled out a priori. In this paper we investigate a multiplicative specification, which is

the natural alternative to the additive specification.

With this model it is the relative differences that matter. If V_i is linear in travel time then the effect on choice probabilities of a 10 minute difference in travel times depends on the length of the trip under the multiplicative specification. A 10 minute difference under the additive specification has constant effect on choice probabilities regardless of whether it relates to a very short or a

very long journey. Thus using the multiplicative specification may reduce the need for segmentation and may hence be able to use data more efficiently.

This is similar to the common practice in econometrics of expressing most variables in regressions in logs. Applying logs in the regression context removes the scale from the data, such that the errors for small and large values of the independent variables have the same variance.

We compare the additive to the multiplicative specification on a number of datasets, using otherwise the same specifications of V_i.

We systematically obtain a very large improvement in the goodness of fit. In all cases, the improvement is greater than the one obtained from including random parameters in the additive specification.

#### Publisher

Association for European Transport