International Choice Modelling Conference, International Choice Modelling Conference 2015

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Allowing Heterogeneity in the Sensitivity to Exogenous Variables in the Multiple Discrete-Continuous Probit Model
Sebastian Astroza, Chandra R Bhat

Last modified: 18 May 2015

Abstract


There are several approaches to understanding the decision process when consumers have to choose an alternative from a set and then determine the amount of the chosen alternative to consume. Classical discrete choice models assume that alternatives are mutually exclusive and only one alternative can be chosen. Alternatively, multiple discrete-continuous (MDC) models expand the decision by allowing consumers to choose multiple alternatives at the same time, along with the continuous dimension of the amount of consumption. MDC models have been applied not only in the case of consumer brand choice and purchase quantity, but also in contexts such as household vehicle types and usage, recreational destination choice and number of trips, land-use type and intensity, and stock portfolio selection choice and investment amounts.

The MDC model that has dominated the recent literature is based on a utility maximization framework that assumes a non-linear (but increasing and continuously differentiable) utility function to accommodate the relationship between the decreasing marginal utility (satiation) and the increasing investment in an alternative. The optimal consumption quantities are obtained by writing the Karush-Kuhn-Tucker (KKT) first-order conditions of the utility function with respect to the investment quantities. The most general utility form for this KKT approach was proposed by Bhat (2008). In Bhat's utility function form, stochasticity is introduced in the baseline preference for each alternative to acknowledge the presence of unobserved factors that may impact the utility of each alternative. The typical distributions used for the kernel stochastic error term (across alternatives) are the generalized extreme value (GEV) distribution (see Bhat, 2008; Pinjari, 2011; Castro et al., 2012) and the multivariate normal distribution (see Kim et al., 2002 and Bhat et al., 2013b). The first distribution leads to a closed-form MDC GEV model structure, while the second leads to an MDC probit (MDCP) model structure.

Researchers have also introduced random structures for the coefficients on the exogenous variables that allow heterogeneity (across individuals) in the sensitivity to exogenous variables in discrete choice models. We have three potential approaches to introduce randomness in the response coefficients. The first approach consists of using continuous random structures for the coefficients on the exogenous variables. Within this approach, the most common assumption is that the random response coefficients are realizations from a multivariate normal distribution. But his can lead to a misspecification problem if some other non-normal distribution characterizes the taste heterogeneity for one or more coefficients (see Train, 1998; Amador et al., 2005; Train and Sonnier, 2005; Hensher et al., 2005; Fosgerau, 2005; Greene et al., 2006; Balcombe et al., 2009; and Torres et al., 2011). The second approach uses a discrete distribution for the response coefficients. This approach leads to the familiar latent class model with an endogenous segmentation that allocates individuals probabilistically to segments as a function of exogenous variables (see Bhat, 1997; Greene and Hensher, 2003; Train, 2008; Bastin et al., 2010; Cherchi et al., 2009; and Sobhani et al., 2013). The problem with this approach, however, is that homogeneity in response is assumed within each latent class. The third approach uses a hybrid semi-parametric approach that combines a continuous response surface for the coefficients with a latent class approach (see, for example, Campbell et al., 2010; Bujosa et al., 2010; Greene and Hensher, 2013; and Xiong and Mannering, 2013). In this approach, the response coefficients are assumed to be realizations of a discrete mixture of multivariate normal distributions. That is, the relationship between the propensity variable and exogenous variables is assumed to belong to one of several latent (discrete) classes. Within each of these classes, the coefficients are drawn from a continuous multivariate normal distribution. The resulting finite discrete mixture of normal (FDMN) model generalizes the heterogeneity form because the normally distributed random parameters approach and the latent class approach consist of special cases-the first approach resulting when there is only one latent class and the second resulting when the multivariate normal distribution becomes degenerate within each latent class.

Several earlier studies have included heterogeneity in the sensitivity to exogenous variables in the MDC context. Bhat et al. (2013a) proposed an estimation approach for the MDCP model that allows taste variation through the inclusion of random parameters and applied the model to analyze recreational long-distance travel. On the same topic of recreational travel, Kuriyama et al. (2010) proposed a latent class KKT model based on the linear expenditure system with translated constant elasticity of substitution utility functions proposed by Hanemman (1978). Sobhani et al. (2013) and Wafa et al. (2014) use a latent class approach with the MDCEV kernel structure. In Sobhani et al. (2013), the authors propose an estimation approach combining the full information maximum likelihood and the expectation maximization approaches. The latent class MDCEV model is applied to study non-workers' daily decisions regarding vehicle type and usage in conjunction with activity type and accompaniment choice decisions. Wafa et al. (2014) proposed a latent class MDCEV model to study the spatial transferability of activity-travel models.

In this paper, we propose an FDMN version of the MDCP model. To our knowledge, this is the first such formulation and application of an MDCP model in the econometric literature. We also propose the use of Bhat's (2011) maximum approximate composite marginal likelihood (MACML) inference approach for the estimation. This approach is computationally efficient and does not involve simulations. The proposed MACML approach involves only univariate and bivariate cumulative normal distribution function evaluations in the likelihood function, regardless of the number of alternatives or segments in the latent classification. Using the National Household Travel Survey data set, the model is applied to analyze the participation and time allocation of non-workers in out-of-home activities by activity purpose. The results provide insights into the demographic and other factors that influence individuals' preferences for different activities, and show that the FDMN MDCP model is able to identify different segments of the sample, each one of them with different effects of the exogenous variables on time allocation and activity participation. Implications of the results at a time of substantial demographic changes in the US population are highlighted and discussed.

 

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