Comparison of continuous and discrete representations of unobserved heterogeneity in logit models

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Representing unobserved heterogeneity or taste variations in Marketing Analytic behavioral-choice analysis is receiving increasing attention in the estimation of consumer-choice modeling. The mixed logit (MXL) model, which incorporates random coefficients into the multinomial logit model, has been widely adopted for this purpose. The most commonly adopted method in this context is to assume that the random coefficient follows a continuous, unimodal distribution, and the parameters of the distribution as well as the other parameters for the model can be obtained using maximum simulated likelihood estimation. In this article, we refer to this method as the continuous mixed logit (CMXL) model. This method requires the a priori assumption that the distribution of the random coefficient is continuous and, usually, unimodal. One way to relax this assumption is to estimate the distribution nonparametrically, by assuming a discrete distribution with finite support. We refer to this approach as the discrete mixed logit (DMXL) model. Based on the DMXL model, we propose the mass-point MXL model as one alternative to the continuous-distribution assumption and compare its performance with the latent class logit model (LCLM), also part of the DMXL family. Either model can be used to represent unobserved heterogeneity with a discrete distribution in the parameter space. In this article, we conduct empirical analyses and compare the continuous and discrete representations of unobserved heterogeneity in logit models using simulated data with known parameters and real data with discrete choices. Analysis with simulated data provides insights on the ability to distinguish between continuous and discrete parameter distributions and a better understanding of the goodness-of-fit measures used in evaluating model performance with real data. From the simulation study, we find that when the data is generated from a normal distribution, the CMXL model with the unimodal-distribution assumption is preferred to the DMXL mode. From the real data analysis, we find that the CMXL model fails to recover heterogeneity that is identified by the DMXL model. In conclusion, we suggest that when estimating a random-coefficient MXL model, one should start with a CMXL model, but should not accept a ‘no heterogeneity’ conclusion without estimating a series of DMXL models using either the mass-point MXL model or the LCLM with different starting values.