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

The logit-mixed logit (LML) model, which allows the analyst to semi-parametrically specify the mixing distribution of preference heterogeneity, is a very recent advancement in logit-type choice models. In addition to generalize many previous semi- and non-parametric specifications, LML is computationally very efficient due to a computationally-convenient likelihood equation that does not require computation of choice probabilities in iterative optimization. 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 the LML-FR specification loses its special properties, leading to a much higher estimation time. In an empirical application about preferences for alternative fuel vehicles in China, estimation time increased by a factor of 20-40 when introducing fixed parameters. Despite losses in computation efficiency, we show that the flexibility of LML-FR is essential for retrieving eventual multimodality of mixing distributions