## Consistent Estimation of Route Choice Models with Link Specific Random Costs

### Authors

A Karlstrom, Royal Institute of Technology, SE; M Fosgerau, Danish Technical University, DK

### Description

In this paper we will for the first time develop an algorithm for (consistent) Maximum-Likelihood estimation of a route choice model with random link specific cost which is computational feasible for realistic applications.

### Abstract

In this paper we present a consistent estimator for a route choice model in a discrete choice framework. To have a realistic substitution pattern across paths, it is important to allow for a correlation structure across paths that overlap. In the literature, there are two different approaches to allow for a flexible correlation structure in a path-based route choice model. First, in logit-based approaches one starts with a discrete choice model in the class of Multivariate Extreme Value (MEV) models. Within the MEV framework, which also can be termed nested logit framework, it is possible to explicitly allow for rather general a correlations structure. Still, the ability for an MEV model to represent a general correlation structure are limited. Yet another possiblity is to use a simple MEV model, such as the standard Multinomial Logit (MNL) model, and define well-chosen corrections to account for the correlation structure across paths.

A second approach is to more explicitly model the error structure on paths by using link specific errors. This will induce a natural correlation structure across paths, since overlapping paths explicitly share a common error strucutre on the overlapping links.

One may also allow for the variance of the error component to be dependent on the link length, such that there is a higher variance for longer links than for shorter. If one can assume that random errors are normal distributed, then the utility for paths will be normal distributed and one is left with a path-based multinomial probit model to estimate.

Models with link specific costs are both theoretical elegant and convenient from the perspective of applied traffic assignment. Indeed, the probit-based route choice model with random link costs were early proposed by Daganzo (1977) and Daganzo and Sheffi (1977). Unfortunately, estimating such models imposes a real challange, since then estimation of such a model is associated with solving multidimensional integrals, and there exists only a few examples in the literature, typically when the number of paths are quite limited.

For real-sized networks, probit-based models quickly impose a comutational challange. First of all, the set of possible choice are typically huge and impractical or impossible to enumerate. One would need to know the size of the choice set to be able to estimate a multinomial probit path-based model. Therefore, previously one has to resorted to path sampling. Bekhor et al (2002) samples (ad hoc) 50 "good" routes for each origin-destination pair and then estimate a multinomial probit model.

However, using the choice sampling approach, one is left with two problems. The first is how to sample "`good" paths. A second problem is that the estimator is not consistent. That is, there is no known correction to bring probit estimates to be consistent in the presence of choice set sampling. In fact, in all discrete choice models except for the standard MNL, choice set sampling will present a risk for bias, see Nerella and Bhat (2004) for a quantitative assessment.

In this paper we will for the first time develop an algorithm for Maximum-Likelihood estimation of a route choice model with random link specific cost which is computational feasible for realistic applications. The model is path-based, such that we allow the individual to - in principle - consider all paths from origin to destination. In contrast with other algorithms, however, we arrive at a consistent estimator although we do not have to sample paths. We demonstrate that our model is feasible even in realistic real-world applications.

#### Publisher

Association for European Transport