Can scale and coefficient heterogeneity be separated in random coefficients models?

There is growing interest in the notion that a significant component of the heterogeneity retrieved in random coefficients models may actually relate to variations in absolute sensitivities, a phenomenon referred to as scale heterogeneity. As a result, a number of authors have tried to explicitly model such scale heterogeneity, which is shared across coefficients, and separate it from heterogeneity in individual coefficients. This direction of work has in part motivated the development of specialised modelling tools such as the G-MNL model. While not disagreeing with the notion that scale heterogeneity across respondents exists, this paper argues that attempts in the literature to disentangle scale heterogeneity from heterogeneity in individual coefficients in discrete choice models are misguided. In particular, we show how the various model specifications can in fact simply be seen as different parameterisations, and that any gains in fit obtained in random scale models are the result of using more flexible distributions, rather than an ability to capture scale heterogeneity. We illustrate our arguments through an empirical example and show how the conclusions from past work are based on misinterpretations of model results.