Competing Methods for Representing Random Taste Heterogeneity in Discrete Choice Models
S Hess, IVT-ETH Zurich, CH; M Fosgerau, Danish Transport Research Institute, DK
This paper presents a competition between two highly flexible and innovative methods for representing random taste heterogeneity in discrete choice models.
The representation of random taste heterogeneity has been a prime research interest in the field of discrete choice modelling over recent years. Introducing random taste heterogeneity brings highly valued advantages in model flexibility and the ability of models to fit particular data. There are however also some drawbacks that must be addressed.
Aside from the heightened cost of estimation, the main complication arising with the use of mixture models is the specification of a distribution for those taste parameters that vary randomly across respondents. This specification may fail in essentially two ways as explored in Fosgerau (2005c). First, it may happen that the data, due to range limitations, do not allow one to identify the distribution of interest. This may not show up as a problem in the estimation of the model but can lead to extreme biases in the estimated moments of random parameters. Second, any given distribution may simply not fit the data.
However, essentially all large scale applications rely exclusively on the use of a small set of standard continuous distributions, with a heavy bias towards the Normal. Even though several authors have discussed the potential inappropriateness of the most basic distributions (e.g. Hess et al., 2005a), this situation has not changed significantly. There are good practical reasons for this, as the computational and programming costs of alternative methods may be high.
It is therefore of interest to seek to be able to specify models using mixture distributions that allow the range to be controlled while also yielding sufficient flexibility to fit the data. We further require that flexibility should be scalable such that it is possible to gradually increase the flexibility of the mixture as desired in any given application. This would allow practitioners to start with a standard model and then adapt it to the situation at hand. We finally ask that increased flexibility can be achieved with minimal additional computational cost such that there is hope that the methods will be applied in large scale applications. We summarise these conditions as range control (a), flexibility (b), scalable flexibility (c), and economy (d).
Some effort has gone into advocating the use of discrete mixture models (e.g. Hess et al., 2005b) and non-parametric distributions (Fosgerau 2005a, 2005b 2005c). The distributions afforded by these methods are as flexible as the data allow and also give direct control over the range of the mixture distribution. This meets requirements a to c but not requirement d. With discrete mixtures, the number of mass points required may be excessively high and there may be substantial numerical problems involved. Nonparametric methods are generally very computationally intensive. For these reasons these methods are probably not considered for large scale applications.
Some authors have investigated the use of more advanced continuous distributions such as Johnson SB or Johnson SU. This is a step in the right direction, but the flexibility of such distributions is still not scalable. They are also mostly unimodal, which might not hold for the true distribution to be estimated.
In this paper, we stage a competition between two alternative approaches to the specification of a mixture distribution that both meet our requirements. The competition takes place over a number of matches, where each match is the estimation of a model on simulated datasets comprising a true distribution to be estimated. These distributions are specified by us in advance so as to be challenging estimation problems. We will mimic what a practitioner might do: we will fix the estimation methods without using our a priori knowledge of the true distribution, scale the flexibility as indicated by the data and in each match evaluate which approach performs best in terms of our criteria.
The first of our contenders in the competition is a mixture distribution that is itself a discrete mixture of continuous distributions. In principle, the continuous distributions can be any continuous parametric distributions. However, we fix attention to using the Normal distribution as the base distribution and get a discrete mixture of Normals. This approach is scalable via the number of Normal distributions used and is a straightforward extension of the standard Normal mixture. It can easily accommodate a multimodal distribution.
The second contender is essentially seminonparametric (SNP) in nature and uses a representation of densities from Bierens (2005) that can approximate virtually any continuous distribution. The method, first described in Fosgerau & Bierlaire (2005), uses a base distribution, which is extended with a number of SNP parameters to increase flexibility. Again we fix the base distribution to be the Normal distribution. The approach is simple to implement since the additional parameters enter only through a weight in the likelihood function. It is also scalable since the number of additional parameters can be increased as required and it is very flexible and particularly it can accommodate multimodal distributions. Fosgerau & Bierlaire (2005) investigate the approach as a test of the base distribution against the SNP alternative and find that this test is well able to discriminate true from false base distributions in a number of cases.
Fosgerau, M. 2005a "Specification of a model to measure the value of travel time savings", in European Transport Conference.
Fosgerau, M. 2005b, "Unit income elasticity of the value of travel time savings", in European Transport Conference.
Fosgerau, M. 2005c, "Investigating the distribution of the value of travel time savings", Transportation Research Part B: Methodological, vol. Forthcoming.
Fosgerau, M. & Bierlaire, M. (2005) A practical test for the choice of mixing distribution in discrete choice models.
Hess, S., Bierlaire, M. and Polak, J. W. (2005a), ?Estimation of value of travel time savings using mixed logit models?, Transportation Research: 39A(2-3), 221?236
Hess, S., Bierlaire, M. and Polak, J. W. (2005b), Discrete mixture models with applications to the estimation of value of travel-time savings, CTS Working paper, Centre for Transport Studies, Imperial College London.
Association for European Transport