Reducing simulation bias in mixed logit model estimation

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 log 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 savings and market shares). In this paper, we explicitly estimate simulation bias in the context of mixed logit models 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 quasi-Monte Carlo techniques as long as standard deviations are computed. Numerical costs can be neglected when using Monte Carlo draws and even when advanced strategies as the adaptive sample methodology are in use.