Extending the logit-mixed logit model for a combination of random and fixed parameters

A semi-parametric logit-mixed logit (LML) model to specify the mixing distribution of the preference heterogeneity is the recent advancement in logit-type models. LML provides a general specification for many previous semi-parametric and non-parametric models. Additionally, a special form of the likelihood gradient makes LML computationally more efficient than other parametric models. However, the original LML formulation assumes all utility parameters to be random. This study extends LML to a combination of fixed and random parameters (LML-FR) and motivates such combination in random parameter choice models in general. We further show that the likelihood of LML-FR specification looses its special properties, leading to a much higher estimation time than that of the original LML specification. In an empirical application 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. The comparison of the original LML and LML-FR specifications with other parametric models indicate that LML-FR model outperforms these parametric models in terms of model-fit-criteria (e.g., AIC, BIC, and likelihood), but the same does not hold for the original LML, i.e. when all parameters are assumed random.