Estimation Performance of Low Discrepancy Sequences in Stated Preferences

A standard assumption in stated preferences modelling is the independence between repeated responses from an interviewee. The rationale for this strong assumption is that the mathematical treatment of the choices becomes rather cumbersome when dependency is incorporated in the analysis. Assuming dependence between the various responses given by one individual is known as the repeated observations problem and its mathematical formulation is similar to that of the autocorrelation in panel data surveys. These type of problems can only be studied with the aid of flexible models such as the multinomial probit (MNP) or mixed logit (MXL). However, these models need the computation of a multidimensional integral to obtain values for the choice probabilities. The integration process is usually complex. Therefore, approximation methods must be applied –typically simulation, to evaluate the choice probabilities. The standard simulation approach relies on the Monte Carlo (MC) method, which basically replaces a continuous average (the integral) by a discrete average over a set of points randomly distributed within the region of integration. The numerical analysis literature shows various procedures to choose smart points from a deterministic series instead of random realizations. This type of points are known as low discrepancy sequences (LDS). However, the use of these techniques in econometrics is rather limited and recent, and consequently there are several open questions to be answered before the use of LDS becomes a standard. The evidence found in the fields of mathematics and physics indicates that a LDS called the Sobol sequence, would be a superior alternative to the more known Halton sequences, especially for large dimensions. Nevertheless, the Sobol sequences (to the knowledge of the authors) have not been tested yet in transportation. This paper compares the MC simulation method in three versions: traditional, Halton based, and Sobol based for the estimation of the MXL.