Comparison of Parametric and Semiparametric Representations of Unobserved Preference Heterogeneity in Logit Models

The recently derived Logit-Mixed Logit (LML) model captures unobserved preference heterogeneity seminonparametrically as a function of polynomials, step functions, and splines. This study contributes twofold in exploring and extending the usefulness of the LML specification. First, we conduct a Monte-Carlo study to analyze the number of required LML parameters to recover the true distributions, and also compare LML performance -- in terms of accuracy, precision, estimation time, and model fit -- with a Mixed Multinomial Logit specification with Normal heterogeneity (MMNL-N). As expected, LML is able to retrieve the underlying Bimodal-Normal (mixture of two Normal distributions), Lognormal, and Uniform distributions much better than the MMNL-N model. In an empirical case study, we also estimate the willingness to pay (WTP) of German consumers for different vehicle attributes when making choices for alternative-fuel cars. LML is able to capture the bimodal nature of WTP for vehicle attributes, which was not possible to retrieve using the MMNL-N model. LML also appears to be more useful for panel data because the computation time is not affected by the number of choice situations. Second, whereas the original LML formulation assumes all utility parameters to be random, we extend the model to a combination of fixed and random parameters (LML-FR). We further show that the gradient of the LML-FR likelihood loses its convenient properties, leading to a much higher estimation time than that of the original LML specification. In an empirical case study about preferences for alternative-fuel vehicles in China, estimation time increased by a factor of 15-20 when introducing fixed parameters to the LML model. Thus, LML is computationally efficient and better than MMNL-N in retrieving the true distribution of the random taste heterogeneity, but loses its computational efficiency feature if any parameter is assumed to be fixed