Approximation Issues in Simulation-based Estimation of Random Coefficients Models
S Hess, ITS, University of Leeds, UK
This paper discusses an important issue in the context of approximations in the simulation-based estimation of random coefficients models.
Mixed Logit models have established themselves as the preferred approach for the representation of taste heterogeneity in the analysis of travel behaviour. While inherently flexible, the estimation of these models can be costly due to reliance on simulation to approximate the integrals representing the choice probabilities. In the conventional approach, this integration takes place at the level of individual respondents, such that, in the case of data with multiple observations per respondent, the single MNL choice probability used as the integrand is replaced by a product of individual MNL choice probabilities, representing the sequence of choices for that respondent. This requires a certain adaptation of the estimation code, with the simulation now also being carried out over sequences of choices rather than individual choices. A number of estimation packages that are based entirely on cross-sectional formulations are in fact restricted to using an approximation to this approach, such that it is still individual choice probabilities that are simulated, but with the same draws (simulation points) being used across choice probabilities (observations) for the same respondent. While this approach can have certain speed advantages in estimation, the consistency with the assumed specification has never been discussed in detail in the existing literature. Furthermore, the fact that most packages now use the ?correct? approach has led to reduced interest in this issue. However, with recent developments in the field, there is renewed reason for an investigation of this issue. Indeed, Hess & Rose have recently made the case for a modelling framework which accommodates variations in tastes across observations for the same respondent (intra-respondent) in addition to variations in tastes across respondents. In mathematical terms, this equates to a model in which part of the integration is carried out at the level of individual respondents with a remainder being carried out at the level of individual choices. The simulation-based estimation of such a model presents a formidable task. In this context, two approximations to the ?correct? specification arise. In the first approach, which relates to the above discussion, all simulation is carried out at the level of individual choice situations (such that the simulated term is an individual choice probability), but any variations across respondents (as opposed to within respondents) is accommodated by using the same draws across observations for the concerned coefficients. In the second approximation, all simulation is carried out at the level of individual respondents (such that the simulated term is a product of choice probabilities), but any variations within respondents is accommodated by using separate draws across observations for the concerned coefficients. As in the above discussion of the more conventional case (inter-respondent heterogeneity only), the quality of the approximation of either approach is not clear a priori. However, unlike in the conventional scenario, the computational savings of the approximations are very significant in the case of a model with intra-respondent as well as inter-respondent taste heterogeneity. The purpose of this paper is then to examine the consistency of the various approximation approaches across these two different scenarios (inter-respondent heterogeneity only versus a mixture of inter and intra-respondent heterogeneity). Along with theoretical discussions, we present evidence from a range of different estimations using a combination of real-world and simulated data. Initial insights suggest that the approximations are consistent but not efficient.
Association for European Transport