Individual-specific posterior distributions from Mixed Logit models: Properties, limitations and diagnostic checks

doi:10.1016/j.jocm.2020.100224

Journal of Choice Modelling

Volume 36, September 2020, 100224

https://doi.org/10.1016/j.jocm.2020.100224 Get rights and content

Highlights

•
There exists some risk and limitations when using individual-specific estimates.
•
The properties of the individual-specific estimates from the MIXL model are reviewed.
•
A Monte Carlo experiment was conducted to study the behavior of some diagnostic checks.
•
Some reasonable guidelines for the correct use of the individual-specific estimates are provided.

Abstract

Individual-specific posterior distributions are an attractive tool for disentangling the tastes for each person in the sample. However, there exists some risks and certain limitations regarding their use. This study reviews and summarizes the theoretical literature about the individual-specific posterior distributions derived from the Mixed Logit model, focusing on their properties, limitations and common pitfalls. It also reviews and analyzes the behavior of some diagnostic checks proposed in the literature for the reliability of such estimates in applied works using Monte Carlo experiments. Finally, this article provides reasonable guidelines for the correct use of individual-specific posterior distributions.

Introduction

Since the Revelt and Train (2000)'s work and Train (2009)'s book, the use of conditional estimators derived from conditional posterior distributions has been an attractive tool for disentangling the tastes for each individual in the sample. Conditional estimates, put simply, allow us to know something about respondents' tastes based on their previous choices providing their most likely location on the population distribution.

Due to the attractiveness of eliciting the tastes for each person, individual-specific estimates (or conditional estimates) derived from the Mixed Logit (MIXL) Model have received a lot of attention from the applied literature in different fields. For example, they have been used to compute willingness-to-pay (WTP) measures at the individual level (Sillano and de Dios Ortúzar, 2005; Greene et al., 2005; Hensher et al., 2003, 2006; Hess, 2007; Sandorf et al., 2017; Dumont et al., 2015), to analyze the spatial dependence of tastes (Campbell et al., 2009; Budziński et al., 2018; Abildtrup et al., 2013), to retrieve individual-specific attribute processing strategies (Hess and Hensher, 2010), to create clusters or segments of individuals (Richter and Pollitt, 2018; Huber and Train, 2001) and for predicting the future behavior of the individuals (Train, 2009; Dumont et al., 2015).

Unfortunately, there exists some risks and certain limitations regarding the use of conditional estimates. For example, Revelt and Train (2000) have already shown that individual-specific estimates are consistent if and only if, given a fixed number of individuals, the number of choice situations increases without bound.¹ That is, we need several choice situations per individual to learn something about the preferences of each respondent: if we could observe infinitely how individuals react to changes in attributes and their choices in terms of these changes, in theory, we could elucidate the specific tastes of individuals. However, there exist published articles using individual-specific estimates with a very low number of choice situations without reporting any type of diagnostic check to analyze whether the estimates are reliable to be used in a second-step procedure.

In addition, it seems that there is still some confusion regarding the relationship between the variance of the individual-specific estimates and the variance of the population distribution of the random parameters. Some researchers use individual-specific estimates arguing that they show more plausible results in terms of their domain. For example, it is common to find in the applied literature claims such that “conditional estimates show more reasonable estimates or comments such as “using conditional means sign violations of the coefficients disappears”, especially when computing individual-specific WTP. But, as briefly mentioned by Daly et al. (2012), such claims ignore the fact that the variance of the conditional mean will be lower than the variance of the population distribution of the random parameters (or unconditional population), specially when the number of choice situations per individual is low. In other words, the apparent better fit of the individual estimates in terms of sign and values might be an artifact of the statistical behavior of the conditional estimates when the number of choice situations is not large enough.

Another problem not yet analyzed in depth, and pointed out by Hess (2010), is the potential impact of the misspecification of the parametric population distribution on the individual estimates. As argued by Hess (2010), researchers should analyze the impact of assumptions made for the unconditional distributions on the shape of the conditional distributions. Failing to choose the most adequate distribution for the random parameters might invalidate the use of conditional estimates and lead to misleading conclusions.

Thus, the first objective of this paper is to review and summarize the properties, limitations and common pitfalls when using individual-specific estimates in the context of the MIXL model and reinforce their understanding. Relying on the previous work of Revelt and Train (2000), Daly et al. (2012) and Hess (2010), this study revisits the theoretical properties of individual-specific estimator regarding to their consistency and the relationship between the conditional and unconditional distribution of tastes. The second objective is to provide reasonable guidelines for the correct use of individual-specific tastes under a well-specified specification and analyze the potential problems under misspecification, extending the previous work done by Hess (2010). To achieve this goal, the study extends the Monte Carlo experiment carried out by Revelt and Train (2000) by analyzing the behavior of the conditional estimates under misspecification and by including new measures as diagnostic tools.

The rest of the paper is organized as follows. Section 2 briefly reviews the MIXL model. Section 3 summarizes the main theoretical properties of the individual-specific estimates. Section 4 explains the Monte Carlo setup, whereas Section 5 presents the results. Finally, Section 6 discusses the results and concludes.

Section snippets

The Mixed Logit model

Assume that each individual $(i = 1, \dots, N)$ faces a choice among J alternatives in each of T choice situations.² Then, the utility associated with each alternative $j = 1, \dots, J$ for individual i in choice situation t is: $U_{i j t} = x_{i j t}' β_{i} + ε_{i j t},$ where $x_{i j t}$ is a $K \times 1$ vector of attributes of the alternatives. The important characteristic of the MIXL model is the vector of tastes $β_{i}$ , which is assumed to vary across

Estimator

The individual-specific parameters can be obtained by deriving the individual's conditional distribution based on the choices observed for that individual using Bayes's formula (Revelt and Train, 2000; Train, 2009). Explicitly, the conditional distribution for individual i is given by: $h (β_{i} | y_{i}, X_{i}, θ) = \frac{f (y_{i} | X_{i}, β_{i}) g (β_{i} | Ω)}{f (y_{i} | X_{i}, Ω)} = \frac{f (y_{i} | X_{i}, β) g (β_{i} | Ω)}{\int f (y_{i} | X_{i}, β_{i}) g (β_{i} | Ω) d β_{i}} .$

Thus, $h (β_{i} | y_{i}, X_{i}, θ)$ is the conditional (on the observed choices) distribution of the individual i's tastes, whereas $g (β_{i} | Ω)$ is the unconditional

Monte Carlo setup

To understand how well the diagnostic measures for individual-specific estimates perform under a well-specified model and different number of choice situations, a Monte Carlo experiment with a similar setup as in Revelt and Train (2000) is carried out.⁸ Specifically, it is assumed that the true utility for individual i when choosing alternative j in choice occasion t is given by: $U_{i j t} = β_{1} x_{1}$

Revisiting Revelt and Train (2000)

Similar in spirit to Revelt and Train (2000)'s work, this Section analyzes the behavior of the conditional means and the diagnostic measures under a well-specified model.

Table 1 presents the Monte Carlo results averaged over S, where S is the total number of samples for which the SML converged.⁹ The estimations were carried out using package gmnl in R (Sarrias and

Conclusion

The MIXL model has been a revolution in the field of discrete choice models during the last two decades due to its capability to accommodate heterogeneity in tastes by assuming that the marginal utility coefficients are distributed randomly across respondents. However, in some practical cases, just knowing that a coefficient varies across individuals is not enough (Revelt and Train, 2000; Hess, 2010). Given this, individual-specific estimators have been gaining more attention from researchers.

CRediT authorship contribution statement

Mauricio Sarrias: Conceptualization, Methodology, Software, Writing - original draft, Funding acquisition.

Declaration of competing interest

I declare that I do not have conflict of interest.

Acknowledgements

All persons who have made substantial contributions to the work reported in the manuscript (e.g., technical help, writing and editing assistance, general support), but who do not meet the criteria for authorship, are named in the Acknowledgements and have given us their written permission to be named. If we have not included an Acknowledgements, then that indicates that we have not received substantial contributions from non-authors.

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Citation Excerpt :
Second, though flexible DCMs structures enable modelers to capture individual preference, the results are defined by a specific distribution of coefficients, for example, Gaussian-distributed coefficients in a mixed logit model (MXL). Many researchers have already pointed out the risk of failing to choose adequate distributions for random coefficients (Hess, 2010; Sarrias, 2020). Third, DCMs are stochastic estimation approaches, generating demand functions that are non-linear and non-convex in the explanatory variables (Ljubić and Moreno, 2018; Pacheco et al., 2021).
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^☆: This research is based upon work supported by FONDECYT Grant, 11160104.

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Individual-specific posterior distributions from Mixed Logit models: Properties, limitations and diagnostic checks☆

Highlights

Abstract

Introduction

Section snippets

The Mixed Logit model

Estimator

Monte Carlo setup

Revisiting Revelt and Train (2000)

Conclusion

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgements

Ecol. Econ.

J. Econom.

Transportmetrica: Transport. Sci.

J. Air Transport. Manag.

J. Choice. Model.

Transp. Res. Part B Methodol.

Energy Econ.

J. Choice. Model.

J. Choice. Model.

Using geographically weighted choice models to account for the spatial heterogeneity of preferences

J. Agric. Econ.

Using choice experiments to explore the spatial distribution of willingness to pay for rural landscape improvements

Environ. Plann.

Assuring finite moments for willingness to pay in random coefficient models

Transportation