Reducing simulation bias in mixed logit model estimation

Abstract

Maximum simulated likelihood (MSL) procedure is generally adopted in discrete choice analysis to solve complex models without closed mathematical formulation. This procedure differs from the maximum likelihood simply because simulated probabilities are inserted into the Log-Likelihood (LL) function. The LL function to be maximized is the sum of the logarithm of the expected choice probabilities; since the logarithmic operation is a nonlinear transformation bias is then introduced. The simulation bias depends on the number of draws that are used in the simulation and on the sample size. Although the asymptotic properties of the MSL estimator are well known, the question is how simulation bias affects parameters estimation and therefore the main outcomes of choice models (for instance value of travel time and market shares). In this paper, we estimate explicitly the simulation bias in mixed logit parameter estimation, using Taylor expansion and we correct the log-likelihood objective function during the maximization process. The method is developed in the context of Monte Carlo simulation. We report significant error reduction on the final objective value but also on the optimal parameters. The method could be extended to randomized quasi-Monte Carlo techniques as long as standard deviations of simulated choice probabilities are calculated. Computation costs can be neglected when using Monte Carlo draws and even when advanced strategies such as adaptive sampling methodology are in use.