Multiple Discrete-continuous Model Research Articles

A new closed-form two-stage budgeting-based multiple discrete-continuous model

In this paper, we propose a multiple discrete-continuous (MDC) model approach that (a) does not need the total budget to be observed or predetermined, (b) allows for any finite or not-so-finite budget over the entire set of inside and outside goods, and (c) preserves a strong endogenous utility-theoretic link between inside good consumptions and the budget allocated to the inside goods (that is, to the product group of interest). We show that our proposed model, including a fractional MDC model at the lower level linked up to a Tobit model for the budget allocation to the inside goods, is strictly consistent with a two-stage budgeting utility theoretic structure. As importantly, by using reverse Gumbel distributional assumptions for the stochastic terms in the model system, we derive an incredibly simple closed-form model that, to our knowledge, is a first of its kind in the econometric literature. In doing so, we formally introduce a new distribution, which we label as the minLogistic distribution, to the statistical literature, and derive the properties of the distribution that is then used in the forecasting stage of the proposed model. An application of the proposed model to investigate the household vehicle fleet composition and usage demonstrates its potential relative to an unlinked and exogenously developed budget for the inside goods. The proposed model has the potential to open up a whole new world of MDC applications in general, and particularly for those cases with an unobserved total budget over the inside and outside goods.

How did the COVID-19 Pandemic Impact the Location and Duration of Work Activities? A Latent Class Time-Use Study

We investigate the variation in location and time allocated to work during the COVID-19 pandemic. Data from the American Time Use Survey (2019-2020) was modeled using a latent-class multiple discrete-continuous model. Two main segments of individuals who suffered different impacts from the pandemic on their work arrangements were identified. Men, young adults, and less educated individuals with lower-mid-income were often unable to transfer work into a residential setting, showing a reduction in work opportunities. Women, middle-aged, highly educated, and high-income individuals were prone to an increase in total hours worked when substituting out-of-home work with at-home work, showing productivity loss.

Extending the Multiple Discrete Continuous (MDC) modelling framework to consider complementarity, substitution, and an unobserved budget

Many decisions can be represented as interrelated discrete and continuous choices, i.e. what and how much to choose from a set of finite alternatives (incidence and quantity of consumption). In the last twenty years, several models of Karush–Kuhn–Tucker demand systems have been developed and used to study these kinds of decisions. While strongly grounded in economic theory, most of these models have two limitations: they require specifying a budget, and usually omit any complementarity effects. In this paper, we propose two extensions to the Multiple Discrete Continuous (MDC) modelling framework: (i) an MDC model including explicit complementarity and substitution effects, and (ii) an MDC model with complementarity, substitution that requires no budget definition. Model (ii) relies on the hypothesis that total expenditure on the alternatives under consideration is small compared to the overall budget. This allows using a linear utility function for the numeraire good, leading to a likelihood function without the budget or numeraire good in it. The lack of a budget is specially useful when forecasting, as it avoids cascading errors due to an inaccurate budget specifications. The inclusion of complementarity and substitution effects enriches the interpretability of the models, while the resulting functional form avoids theoretical issues present in previous formulations. Alongside the derivation of the models, we discuss their main properties and propose an efficient forecasting algorithm for (ii). We also report four applications to datasets about time use, household expenditure, supermarket scanner data, and trip generation. Free estimation code for both models is made available online.

Multiple discrete-continuous choice models with additively separable utility functions and linear utility on outside good: Model properties and characterization of demand functions

In this paper, we enhance the current understanding of the properties of multiple discrete-continuous (MDC) choice models with additively separable (AS), independent and identically distributed (IID) utility functions, and linear utility form on the essential outside good. First, we highlight that the prior implementations of this model in the literature ignore primal feasibility conditions related to the budget constraint and the essential nature of outside good in formulating the model likelihood function. Second, we evaluate the suitability and performance of the model for alternative consumption patterns relative to the budget. In addition, we provide a systematic comparison of the performance of MDC choice models with the linear outside good utility form (i.e., the Lγ-profile model) and those with the non-linear outside good utility form (i.e., the NLγ-profile model). Third, for the Lγ-profile model with infinite budgets (i.e., when a very small proportion of the budget is allocated to inside goods), we derive the distributions of the resulting optimal demand functions and analytic expressions for the corresponding first and second moments, and identify a property that makes it easy to estimate the utility function parameters of an inside alternative even when consumption data is not available for other alternatives. In addition, perhaps for the first time in the literature, we show how an independent system of Tobit models can be derived as a restricted version of the utility-theoretic Lγ-profile MDC model structure. Finally, we apply the model for an empirical analysis of expenditure patterns of leisure trips from a domestic tourism survey sample of households in India.

Multiple discrete-continuous choice models with bounds on consumptions

This paper derives a multiple discrete–continuous (MDC) choice model formulation with constraints that specify upper bounds on consumption. To do so, considering the conventional utility maximization problem of a consumer, the Karush-Kuhn-Tucker (KKT) conditions are laid out for the MDC model with a general set of linear constraints that include inequalities. Subsequently, we derive a model with constraints that accommodate upper bounds on consumptions and an additive utility structure that accommodates lower bounds on consumptions. The likelihood expression for the proposed model takes a closed form. Furthermore, we extend the formulation to impose bounds on an MDC choice model with activity episode-level choice alternatives that accommodates a logical ordering among different episodes of an activity. The proposed models are derived for two different specifications of the outside good utility – (1) nonlinear utility with respect to consumption and (2) linear utility with respect to consumption. The proposed models are applied to an empirical context to analyze activity-level as well as episode-level activity participation and time allocation while considering bounds on time allocations. Empirical results suggest that the models that consider upper bounds on consumption offer a better fit to data, avoid predictions of unrealistically large time allocations, and result in overall better predictions than those from models without bounds. The proposed models are useful in situations, such as microsimulation models of travel demand, where it is crucial to avoid unrealistically large predictions.

Managing Members, Donors, and Member-Donors for Effective Nonprofit Fundraising

Nonprofit organizations (NPOs) play a central role in many economies in the form of private entities serving a public purpose. Strengthening the fundraising capabilities of NPOs can have a large impact on their survival and effective functioning. NPOs typically elicit financial contributions through multiple forms of giving, such as donation and membership. These options enable individuals to express their altruism by giving in one or multiple forms. The authors develop a utility-based multiple discrete-continuous model that provides insights into potentially large differences in individuals’ giving behaviors. Through Bayesian Gaussian processes, the model also incorporates changes in givers’ preferences for forms of giving. The authors apply their model to five years of individual giving data. They find that the effects of lifetime, recency, seasonality, and appeals on donation and membership options change nonmonotonically over time and in distinctive ways. The authors demonstrate that the model estimates help predict who will give in more than one form in the future as well as build appeal targeting strategies. The model also shows that fundraising attempts should emphasize participation rather than amount, and that long-lapsed members are still worth pursuing for renewal, whereas long-lapsed donors are less productive for repeat giving.

Investigating the Effects of Excise Taxes, Public Usage Restrictions, and Antismoking Ads Across Cigarette Brands

The prevalence of strong brands such as Coca-Cola, McDonald’s, Budweiser, and Marlboro in “vice” categories has important implications for regulators and consumers. While researchers in multiple disciplines have studied the effectiveness of antitobacco countermarketing strategies, little attention has been given to how brand strength may moderate the efficacy of tactics such as excise taxes, usage restrictions, and educational advertising campaigns. In this research, the authors use a multiple discrete-continuous model to study the impact of antismoking techniques on smokers’ choices of brands and quantities. The results suggest that although cigarette excise taxes decrease smoking rates, these taxes also result in a shift in market share toward stronger brands. Market leaders may be less affected by tax policies because their market power allows strong brands such as Marlboro to absorb rather than pass through increased taxes. In contrast, smoke-free restrictions cause a shift away from stronger brands. In terms of antismoking advertising, the authors find minimal effects on brand choice and consumption. The findings highlight the importance of considering brand asymmetries when designing a policy portfolio on cigarette tax hikes, smoke-free restrictions, and antismoking advertising campaigns.

A multiple discrete continuous model of time use that accommodates non-additively separable utility functions along with time and monetary budget constraints

This paper proposes a multiple discrete continuous (MDC) choice modelling framework with the following features for analyzing individuals’ time use decisions: (a) accommodates non-additively separable (NAS) utility functions with respect to time allocation across different non-work activities, (b) considers both time and money constraints by integrating the two constraints into a single economic constraint, and (c) allows corner solutions for (i.e., non-participation in) discretionary, non-work activities. The proposed framework is applied to analyze weekly time use of employed individuals in Netherlands using a data set derived from the Time use and Consumption survey of the Longitudinal Internet Studies for the Social Sciences (LISS) panel. The empirical results of the proposed model are compared to those from two simpler models – (a) a model that ignores the monetary budget constraint but employs the NAS utility function and (b) a model that considers both time and money budget constraints but uses a simpler, additively separable utility function. The results suggest that considering both time and money constraints, when combined with the NAS utility function, facilitates in capturing rich substitution patterns in time allocation to non-work activities, in addition to revealing demographic heterogeneity in satiation effects. Further, the proposed model offers a superior goodness of fit when compared to that from the simpler models that ignore either the monetary constraint or the NAS utility function. Finally, the model results provide important policy insights for reducing demographic inequities in time allocation among employed individuals.

A new flexible multiple discrete–continuous extreme value (MDCEV) choice model

Traditional multiple discrete–continuous (MDC) models generally predict the continuous consumption quantity component reasonably component well, but not necessarily the discrete choice component. In this paper, we propose, for the first time, a new flexible closed-form MDCEV model that breaks the tight linkage between the discrete and continuous choice dimensions of the traditional MDC models. We do so by (1) employing a linear utility function of consumption for the first outside good (which removes the continuous consumption quantity of the outside good from the discrete consumption decision, and also helps in forecasting), and (2) using separate baseline utilities for the discrete and continuous consumption decisions. In the process of proposing our new formulation, we also revisit two issues related to the traditional MDC model. The first relates to clarification regarding the identification of the scale parameter of the error terms, and the second relates to the probability of the discrete choice component of the traditional MDC model (that is, the multivariate probability of consumption or not of the alternatives). We show why the scale parameter of the error terms is indeed estimable when a γ-profile is used, as well as show how one may develop a closed-form expression for the discrete choice consumption probability. The latter contribution also presents a methodology to estimate pure multiple discrete choice models without the need for information on the continuous consumptions. Finally, we also develop forecasting procedures for our new MDC model structure.We demonstrate an application of the proposed model to the case of time-use of individuals. In a comparative empirical assessment of the fit from the proposed model and from the traditional MDCEV models, our proposed model performs better in terms of better predicting both the discrete outcome data as well as the continuous consumptions.

A multivariate hurdle count data model with an endogenous multiple discrete–continuous selection system

This paper proposes a new econometric formulation and an associated estimation method for multivariate count data that are themselves observed conditional on a participation selection system that takes a multiple discrete–continuous model structure. This leads to a joint model system of a multivariate count and a multiple discrete–continuous selection system in a hurdle-type model. The model is applied to analyze the participation and time investment of households in out-of-home activities by activity purpose, along with the frequency of participation in each selected activity. The results suggest that the number of episodes of activities as well as the time investment in those activities may be more of a lifestyle- and lifecycle-driven choice than one related to the availability of opportunities for activity participation.

Multiple Discrete-Continuous Model of Activity Participation and Time Allocation for Home-Based Work Tours

Activity-based travel demand models use the notion of tours or trip chains as the fundamental building blocks of daily traveler activity-travel patterns. Travelers may undertake a variety of tours during the course of a day, and each tour may include one or more stops where individuals participate in and devote time to the pursuit of activities. This paper presents a framework capable of simulating the complete composition of a tour and offers an approach to model the mix of activities and the time allocated to various activities in a tour. Embedded in the framework is a multiple discrete-continuous extreme value modeling component that was used to model the simultaneous decisions of participating in one or more activities in the course of a tour and of allocating time to each of the activities in the tour. The model was estimated with travel survey data collected in 2008 in the Greater Phoenix Metropolitan Area in Arizona. Validation and policy simulation exercises were conducted to examine the efficacy of the model. The model was found to perform well in replicating tour patterns in the estimation sample and responded in a behaviorally intuitive manner in the context of a policy sensitivity test.

A two-level multiple discrete-continuous model of time allocation to virtual and physical activities

This article develops a model to study individuals’ activity choice and time allocation in the physical and virtual spaces. Based on the recent advancement in modeling multiple discrete-continuous choices, this article develops a so-called two-level nested multiple discrete-continuous model. The model and its estimation algorithm are tested by a case study in Hong Kong in which the activity diaries of 330 respondents in the physical and virtual spaces within 24 h were collected. The model can be used to investigate the bi-directional interactions between activities and time use in physical and virtual spaces and their implications for physical travel. It may also be applied to analyse the impacts of social, economical and spatial factors on participation in and time allocation to virtual and physical activities. The model may also be applied to study transportation, marketing and tourism problems involving two-level multiple discrete-continuous choices.

The multiple discrete-continuous extreme value (MDCEV) model: Role of utility function parameters, identification considerations, and model extensions

Many consumer choice situations are characterized by the simultaneous demand for multiple alternatives that are imperfect substitutes for one another. A simple and parsimonious multiple discrete-continuous extreme value (MDCEV) econometric approach to handle such multiple discreteness was formulated by Bhat (2005) [Bhat, C.R., 2005. A multiple discrete-continuous extreme value model: formulation and application to discretionary time-use decisions. Transportation Research Part B 39(8), 679–707]. within the broader Kuhn–Tucker (KT) multiple discrete-continuous economic consumer demand model of Wales and Woodland (1983) [Wales, T.J., and Woodland, A.D., 1983. Estimation of consumer demand systems with binding non-negativity constraints. Journal of Econometrics 21(3), 263–85]. This paper examines several issues associated with the MDCEV model and other extant KT multiple discrete-continuous models. Specifically, the paper proposes a new utility function form that enables clarity in the role of each parameter in the utility specification, presents identification considerations associated with both the utility functional form as well as the stochastic nature of the utility specification, extends the MDCEV model to the case of price variation across goods and to general error covariance structures, discusses the relationship between earlier KT-based multiple discrete-continuous models, and illustrates the many technical nuances and identification considerations of the multiple discrete-continuous model structure through empirical examples. The paper also highlights the technical problems associated with the stochastic specification used in the KT-based multiple discrete-continuous models formulated in recent Environmental Economics papers.

A multiple discrete-continuous model for independent- and joint-discretionary-activity participation decisions

This paper presents a multiple discrete-continuous econometric structure to model the daily time-investment decisions of couples in solo- and joint-discretionary activities incorporating intra-personal and inter-personal inter-dependencies. The empirical model was estimated using data from the 2000 Bay Area Travel Survey. The results indicate evidence of the positive impact of vehicle availability on independent activity participation and the negative impacts of the presence of children and mandatory time investments on the joint discretionary-activity engagement of the spouses. In addition, we also find the mandatory- and maintenance-activity-participation characteristics of the spouse to influence the discretionary activity choices of individuals. Finally, the analysis also indicates a strong impact of common unobserved factors on the decisions of couples. From a policy analysis perspective, these results imply that demand-management actions directly impacting one adult could also result in changes to the activity patterns of his/her spouse and to changes in joint activity participation characteristics.