Estimating Mixed Logit with Non-parametric Random Variables
Fabian Bastin, Cinzia Cirillo, Philippe L. Toint, FUNDP, BE
In this paper we develop a non-parametric estimation of random coefficients in mixed logit models. The method is applied to both simulated and real data.
The estimation of random parameters by means of mixed logit models is becoming current practice amongst discrete choice analysts, one of the most straightforward applications being the derivation of willingness to pay distribution over a heterogeneous population. In many practical cases parametric distributions are a priori specified and the parameters for these distributions are estimated. This way to proceed has lead to many practical problems. Firstly, it is difficult to assess which is the more appropriate analytical distribution, secondly unbounded distributions produce often range of values with difficult behavioral interpretation and thirdly little is known about the tails and their effects on the mean of the estimates.
Recently, several research works have being published on that topic. Mass Point Mixed logit models have been explored to avoid prior assumptions. In that context, Dong and Koppelman (2003) use the Bayesian method to recover mass points and their associated probabilities, while Hess et al. (2005) propose discrete mixture of GEV models over a finite set of distinctive support points. In both cases only two mass points along each of the parameters dimensions are used, although they both conclude that the extension to multiple support points is possible.
Hensher (2005) resolves the problem of behaviorally sign changes by imposing a global sign condition on the marginal disutility expression and gives an application on the valuation of travel-time savings for car commuters. Train and Weeks (2004) place distributional assumptions on the willingness to pay and derive the distribution of the coefficients. Fosgerau (2005) employs various non-parametric techniques to investigate the distribution of the travel-time savings from a stated choice experiments. The proposed method does not account for repeated observations and applies only to binomial choices.
Our approach develops the non-parametric methods in a classical context of mixed logit models. The random variables of the objective functions are assumed to be continuous, and we are interested by the inverse cumulating distribution functions. These functions are modeled by means of cubic B-splines with strictly increasing base coefficients, a sufficient condition to construct monotonic (increasing) functions. The problem becomes more complicate because the number of parameters to be estimated increases; however the information on the tails and on the shape of the random variables should help the analyst to find the right parametric distribution for the random parameters (if this exists).
Various methods will be applied to simulated data and the ability to recover both parametric and non-parametric random vectors will be tested. The non-parametric mixed logit model will also used to estimate the willingness to pay for an electric car, whose prototype has been realized and tested in a number of cities in Europe. The data set, which is part of a European study called ?Cybercar? is a Stated Preference experiment conducted in Brussels in 2002. The model presents multiple choices and is estimated on repeated observations.
Association for European Transport