Abstract
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.
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Notes
We focus on random scale heterogeneity alone in this paper, thus not looking at efforts to link scale heterogeneity to characteristics of the respondent, the alternative, or the choice set.
With ξ1 and ξ2 being independent standard normal variates, draws from the distribution of θTT are obtained as \(\mu_{\theta_{\rm TT}}+s_{{11}, \theta}\xi_1, \) while draws from the distribution of θTC are obtained as \(\mu_{\theta_{\rm TC}}+s_{{21}, \theta}\xi_1+s_{{22}, \theta}\xi_2.\) Correlation is allowed for as ξ1 is used for both coefficients, with the covariance between θTT and θTC being given by \(s_{{21}, \theta}s_{{11}, \theta}. \)
In the MMNLC model, draws from θTT are obtained as \(-e^{\mu_{\ln\left(-\theta_{\rm TT}\right)}+s_{11,\ln\left(-\theta\right)}\xi_1}, \) with draws from βTC being obtained as \(-e^{\mu_{\ln\left(-\theta_{\rm TC}\right)}+s_{21,\ln\left(-\theta\right)}\xi_1+s_{22,\ln\left(-\theta\right)}\xi_2},\) with ξ1 and ξ2 once again giving independent standard normal variates. In the S-MMNLC model, draws from α* are obtained as \(e^{\sigma_{\ln\left(\alpha^{\ast}\right)}\xi_3},\) where ξ3 is an additional standard normal variate. The draws for \(\theta_{\rm TT}=\alpha^{\ast}\beta_{\rm TT}^{\ast}\) are thus given by \(-e^{\mu_{\ln\left(-\beta_{\rm TT}^{\ast}\right)}+s_{11,\ln\left(-\beta^{\ast}\right)}\xi_1+\sigma_{\ln\left(\alpha^{\ast}\right)}\xi_3},\) while the draws for \(\theta_{\rm TC}=\alpha^{\ast}\beta_{\rm TC}^{\ast}\) are given by \(-e^{\mu_{\ln\left(-\beta_{\rm TC}^{\ast}\right)}+s_{21,\ln\left(-\beta^{\ast}\right)}\xi_1+s_{22,\ln\left(-\beta^{\ast}\right)}\xi_2+\sigma_{\ln\left(\alpha^{\ast}\right)}\xi_3}.\) Working on the basis of the underlying normal distribution, we can see that the variance of \(\ln\left(-\theta_{\rm TT}\right)\) is equal to \({s_{11,\ln\left(-\beta^{\ast}\right)}}^2+{\sigma_{\ln\left(\alpha^{\ast}\right)}}^2,\) while, for \(\ln\left(-\theta_{\rm TC}\right), \) the variance is given by \({s_{21,\ln\left(-\beta^{\ast}\right)}}^2+{s_{22,\ln\left(-\beta^{\ast}\right)}}^2+{\sigma_{\ln\left(\alpha^{\ast}\right)}}^2. \) Finally, the covariance between \(\ln\left(-\theta_{\rm TT}\right)\) and \(\ln\left(-\theta_{\rm TC}\right)\) is given by \(s_{21,\ln\left(-\beta^{\ast}\right)}s_{11,\ln\left(-\beta^{\ast}\right)}+{\sigma_{\ln\left(\alpha^{\ast}\right)}}^2.\) However, the exact same covariance matrix can be obtained on the basis of a correlated θ alone, as in the MMNLC model, with one less parameter, meaning that the S-MMNLC model is indeed over-specified.
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Acknowledgments
The first author acknowledges the financial support by the Leverhulme Trust, in the form of a Leverhulme Early Career Fellowship. The majority of this work was carried out during a visit by the first author to the Institute of Transport and Logistics Studies at the University of Sydney, which was made possible by a Faculty of Economics and Business Visiting Scholar Grant. The authors would like to thank Kenneth Train and Thijs Dekker for valuable feedback on an earlier version of this paper.
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Hess, S., Rose, J.M. Can scale and coefficient heterogeneity be separated in random coefficients models?. Transportation 39, 1225–1239 (2012). https://doi.org/10.1007/s11116-012-9394-9
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DOI: https://doi.org/10.1007/s11116-012-9394-9