Continuous Choice Model Research Articles

Finite-Mixture Continuous Logit Model: Formulation and Application for Shopping Start Time Choice of Non-Commuters

As a continuous generalization of the multinomial logit (MNL) model, the continuous logit (CL) model can be used for continuous response variables (e.g., departure time and activity duration). However, the existing CL model requires the calculation of numerical integrals to obtain the choice probabilities; it thus takes a long time to estimate the model parameters, particularly when the sample size is large. In this paper, we formulate the finite-mixture CL (FMCL) model as a new continuous choice model by combining the finite-mixture method and the CL model, in which the continuous distributional function of the finite mixture is embedded in the CL model. As a result, the individual choice probability can be obtained directly by computing the probability density of the continuous distribution function; this avoids calculation of the integral but still obeys the random utility maximization (RUM) principle. Simulation experiments are conducted to demonstrate the capability of the model. In an empirical study, the proposed model is applied for non-commuters’ shopping activity start time using the expectation-maximization (EM) algorithm based on Shanghai Household Travel Survey data. The results show that the FMCL model developed in this paper can greatly reduce the model estimation time (10,048 observations requiring only 3 min) of the CL model, and the model also has a more intuitive interpretation of model coefficients, directly reflecting variable effects on time-of-day choice. These two advantages can greatly enhance the practical value of the proposed modeling method.

A multiple discrete continuous extreme value choice (MDCEV) model with a linear utility profile for the outside good recognizing positive consumption constraints

A variant of the traditional multiple discrete-continuous extreme value (MDCEV) model that obviates the need to have budget information, labeled as the Lγ-profile MDCEV model, has been proposed recently. This new model structure breaks the strong linkage between the discrete and continuous choice dimensions of decision-making. But recent studies show that this Lγ-profile model may not work well in situations when, even if the budget is unobserved, the budget is known to be finite and small in magnitude. The reason is that the formulation, while ensuring the positivity of consumptions of the inside goods (that may or may not be consumed), does not guarantee, within the model formulation and estimation itself, the positivity of the consumption of the essential outside good. In this paper, we develop a formulation based on a reverse Gumbel structure for the stochastic terms in the utility functions of alternatives that develops a closed-form probability expression, while also accommodating the positivity requirement for the outside good. The ability of our proposed Budget-based Reverse Generalized Lγ-profile model (labeled the BR-GLγ-profile model) to recover true underlying model parameters is assessed. Our results clearly point to the benefit of employing the proposed model (relative to extant linear outside utility profile models in the literature) in empirical contexts when there is reason to believe that a finite ceiling applies to the budget (even if the budget is unobserved) or if the budget is actually available. In the latter case when the budget is available, our proposed model is a serious contender to the traditional γ-profile-MDCEV model and will generally outperform the traditional γ-profile-MDCEV when the consumption share of the outside good is high.

Discrete-Continuous Dynamic Choice Models: Identification and Conditional Choice Probability Estimation

This paper develops a general framework for models, static or dynamic, in which agents simultaneously make both discrete and continuous choices. I show that such models are nonparametrically identified. Based on the constructive identification arguments, I build a novel two-step estimation method in the lineage of Hotz and Miller (1993) but extended to discrete and continuous choice models. The method is especially attractive for complex dynamic models because it significantly reduces the computational burden associated with their estimation. To illustrate my new method, I estimate a dynamic model of female labor supply and consumption.

Impacts of trade facilitation on modal choice and international trade flows

Over the last decade, the use of air transport to facilitate global merchandise trade has grown significantly, driven by the increasing demand for ‘just in time’ delivery of intermediate goods as well as due to the growing importance of trade between more distant countries. In this paper we analyze the effect of improved global logistics and trade facilitation on transport mode choice in international trade using the Global Trade Analysis Project (GTAP) model of global trade which we modify to incorporate modal choice. To parameterize this model, we develop a continuous modal choice model and estimate the associated elasticities of substitution between transport modes. We find that modal substitution elasticities have values ranging from 0.57 to 2.14, indicating significant response to changes in the relative cost of different modes of goods transport. The quality of logistics infrastructure is also found to influence modal choice in international trade with improved logistics performance generally leading to increased reliance on air transportation.We validate the modified and parameterized GTAP model by simulating the impact of improvements in logistics, measured as a change in the Logistics Performance Index (LPI). We find that improvement in LPI in the poorest countries of the world reduces the overall cost of transport and amount of services required to transport a given product along a given route by a given mode. Importantly, the reduction in modal cost of transport results in modal substitution. We use the framework to explore the consequences of an improvement in low income country logistics performance for GDP and economic welfare.

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.

A new closed form multiple discrete-continuous extreme value (MDCEV) choice model with multiple linear constraints

Traditional multiple-discrete continuous choice models that have been formulated and applied in recent years consider a single linear resource constraint, which, when combined with consumer preferences, determines the optimal consumption point. However, in reality, consumers may face multiple resource constraints, such as those associated with time, money, and storage capacity. Ignoring such multiple constraints and instead using a single constraint can, and in general will, lead to poor data fit and inconsistent preference estimation, which can then have a serious negative downstream effect on forecasting and welfare/policy analysis. Unlike earlier attempts to address this multiple constraint situation, we formulate a new multiple-constraint (MC) multiple discrete continuous extreme value (MDCEV) model (or the MC-MDCEV model) that retains a closed-form probability structure and is as simple to estimate as the MDCEV model with one constraint. We achieve this by assuming a type-I extreme value distribution for the error term in its minimization form in the baseline utility preference of each good rather than a maximization form as in Bhat's (2005; 2008) original MDCEV formulation. The statistical foundation of the proposed model is based on the fact that the difference between a minimal type-I extreme value random variable with scale σ and the weighted sum of the exponential of standardized minimal type-I extreme value random variables (scaled up by σ) leads to an apparently new multivariate distribution that has an elegant and closed-form survival distribution function. Results from a simulation experiment show that our proposed model substantially outperforms single-constraint models; the results also highlight the serious mis-estimation that is likely to occur if only a subset of active constraints is used. The proposed model is applied to a case of week-long activity participation where individuals are assumed to maximize their utility from time-use subject to time and money budgets.It is hoped that our proposed simple closed-form multi-constraint MDCEV model will contribute to a new direction of application possibilities and to new research into situations where consumers face multiple constraints within a multiple discrete-continuous choice context.

What do Consumers Consider Before They Choose? Identification from Asymmetric Demand Responses

AbstractConsideration set models generalize discrete-choice models by relaxing the assumption that consumers consider all available options. Determining which options were considered has previously required either survey data or restrictions on how attributes affect consideration or utility. We provide an alternative route. In full-consideration models, choice probabilities satisfy a symmetry property analogous to Slutsky symmetry in continuous-choice models. This symmetry breaks down in consideration set models when changes in characteristics perturb consideration. We show that consideration probabilities are constructively identified from the resulting asymmetries. We validate our approach in a lab experiment where consideration sets are known and then apply our framework to study a “smart default” policy in Medicare Part D, wherein consumers are automatically reassigned to lower-cost prescription drug plans with the option of opting out. Full-consideration models imply that such a policy will be ineffective because consumers will opt out to avoid switching costs. Allowing for inattention, we find that defaulting all consumers to lower-cost options produces negligible welfare benefits on average, but defaulting only consumers who would save at least $300 produces large benefits.

Modeling workers’ daily out-of-home maintenance activity participation and duration

Abstract This paper applies a multiple discrete–continuous choice model to determine worker’s decisions on interdependencies of maintenance activity participation and duration over different time periods in a worker’s day. To accommodate household effect, each time period is further defined according to whether the maintenance activities were conducted independently or jointly with other household members. Using the household survey data from the New York metropolitan area, the worker’s participation and duration decisions were jointly estimated for four time periods, including the morning commute, midday, evening commute, and after evening commute. To account for the censored nature of the duration data, a “censored system of equations” proposed by Shonkwiler and Yen was adopted and modified to increase asymptotic efficiency of the parameter estimates. Despite simplicity of the model and its ease of implementation, the estimation results offer valuable insights into the effects of socio-demographics, land use density, transportation service, and work duration characteristics on worker’s daily scheduling of maintenance activities and travel in a highly urbanized environment.

Identification of Random Coefficient Latent Utility Models

This paper provides nonparametric identification results for random coefficient distributions in perturbed utility models. We cover discrete and continuous choice models. We establish identification using variation in mean quantities, and the results apply when an analyst observes aggregate demands but not whether goods are chosen together. We require exclusion restrictions and independence between random slope coefficients and random intercepts. We do not require regressors to have large supports or parametric assumptions.

The relationship between length of stay and land transportation mode in the tourism sector: A discrete–continuous framework applied to Swiss data

This study aims to investigate the relationship between travel mode and length of stay decisions in the context of domestic trips in Switzerland, formulating a discrete–continuous choice model. The adoption of this method allows us to explicitly recognize the interrelated nature of such decisions while testing and measuring the underlying correlation effects. The data source employed for the empirical analysis refers to the Household Budget Survey, which gathers information with regard to the 642 trips undertaken by Swiss resided tourists in 2010 and in particular to the most recent nonbusiness sojourn in this regard. The estimated correlation parameters reveal that tourists are likely to stay longer at the destination when travelling by train and appear to be less inclined to spend more extended sojourns when heading to the destination by car.

Autoregressive continuous logit: Formulation and application to time-of-day choice modeling

In this paper, we formulate the autoregressive continuous logit model as a novel continuous choice model capable of representing correlations across alternatives in the continuous spectrum. We formulate this model by considering two approaches: combining a discrete-time autoregressive process of order one (i.e., a linear stochastic difference equation) with the continuous logit model, which leads to the discrete-time autoregressive continuous logit; and combining a continuous-time autoregressive process (i.e., a linear stochastic differential equation), known in the stochastic process literature as Ornstein-Uhlenbeck process with the continuous logit model, which leads to the continuous-time autoregressive continuous logit. The autoregressive nature of the model is in its error structure, allowing correlations in unobserved heterogeneity. For both approaches, we study their properties numerically. We also compare both approaches to highlight their relation to each other.

Exit dynamics of start-up firms: Structural estimation using indirect inference

We estimate by means of indirect inference a structural economic model where firms’ exit and investment decisions are the solution to a discrete–continuous stochastic dynamic programming problem. Our method solves the main difficulty of simulation-based inference in structural discrete–continuous choice models, namely that the simulated trajectories are discontinuous functions of the structural parameters. Estimating the model on all start-up firms in the Norwegian manufacturing sector, we find that if the expected value of continuing production is persistently low relative to the expected value of exit, the firm has a high probability to exit.

GISwaps

This article describes how continuous GIS-MCDM problems are commonly managed by combining some weighting method based on pairwise comparisons of criteria with an aggregation method. The reliability of this approach may be questioned, though. First, assigning weights to criteria, without taking into consideration the actual consequences or values of the alternatives, is in itself controversial. Second, the value functions obtained by this approach are in most cases linear, which is seldom the case in reality. The authors present a new method for GIS-MCDM in continuous choice models based on Even Swaps. The method is intuitive and easy to use, based on value trade-offs, and thus not relying on criteria weighting. Value functions obtained when using the method may be linear or non-linear, and thereby are more sensitive to the characteristics of the decision space. The performed case study showed promising results regarding the reliability of the method in GIS-MCDM context.

Passenger booking timing for low-cost airlines: A continuous logit approach

Low-cost airlines irregularly launch ticket promotions, where fares may differ by day of the week and departure dates. The timing for purchasing air tickets is thus closely associated with fares. Airlines should analyze booking timing for tickets and increase the accuracy of demand forecasts to achieve efficient use of airplane seating and plan for promotional activities. Therefore, this study develops a continuous choice model for timing decision making regarding bookings. A trigonometric function is used and interacts with trip and socioeconomic characteristics to capture the heterogeneous preferences of air passengers. Data were collected from low-cost airline passengers traveling from Taiwan to Singapore. The results of the continuous multinomial logit model indicate that lower fares increase the number of bookings and heterogeneous preferences in booking timing are present. Some travelers tend to book flights earlier than the other groups: these are the price-sensitive customers.

Functional forms and price elasticities in a discrete continuous choice model of the residential water demand

AbstractDuring recent decades, water demand estimation has gained considerable attention from scholars. From an econometric perspective, the most used functional forms include log‐log and linear specifications. Despite the advances in this field and the relevance for policymaking, little attention has been paid to the functional forms used in these estimations, and most authors have not provided justifications for their selection of functional forms. A discrete continuous choice model of the residential water demand is estimated using six functional forms (log‐log, full‐log, log‐quadratic, semilog, linear, and Stone‐Geary), and the expected consumption and price elasticity are evaluated. From a policy perspective, our results highlight the relevance of functional form selection for both the expected consumption and price elasticity.

CCP Estimation of Dynamic Discrete/Continuous Choice Models with Generalized Finite Dependence and Correlated Unobserved Heterogeneity

This paper investigates conditional choice probability es timation of dynamic structural discrete and continuous choice models. I extend the concept of finite dependence in a way that accommodates non-stationary, irreducible transition probabilities. I show that under this new definition of finite dependence, one-peri od dependence is obtainable in any dynamic structural model. This finite dependence prop erty also provides a convenient and computationally cheap representation of the optimality conditions for the continuous choice variables. I allow for a general form of discrete-valued unobserved heterogeneity in utilities, transition probabilities, an d production functions. The unobserved heterogeneity may be correlated with the observable state variables. I show the estimator is root-n‐asymptotically normal. I develop a new and computationally cheap algorithm to compute the estimator. I apply our method to estimate a model of education and labor supply choices to investigate properties of t he distribution of returns to education, using data from the National Longitudinal Survey of Youth 1979.

A Microeconomic Theory–Based Latent Class Multiple Discrete–Continuous Choice Model of Time Use and Goods Consumption

A microeconomic theory–based multiple discrete–continuous choice model was developed to accommodate ( a) both time allocation and goods consumption as decision variables in the utility function, ( b) both time and money budget constraints governing the activity participation and goods consumption decisions, ( c) a finite probability of zero consumption and zero time allocation (i.e., corner solutions), and ( d) technical constraints in the form of minimum consumption levels for any good that would be consumed and minimum time allocation for any activity conducted. The proposed model was applied in the form of a latent class model (to consider heterogeneity) on a Dutch data set to understand the determinants of weekly time use and goods consumption behavior.

Structural estimation of continuous choice models: Evaluating the EGM and MPEC

In this paper, I evaluate the performance of two recently proposed approaches to solving and estimating structural models: The Endogenous Grid Method (EGM) and Mathematical Programming with Equilibrium Constraints (MPEC). Monte Carlo simulations confirm that both the EGM and MPEC have advantages relative to standard methods. The EGM proved particularly robust, fast and straightforward to implement. Approaches trying to avoid solving the model numerically, therefore, seem to be dominated by these approaches.

Using economic and other performance measures to evaluate a municipal drought plan

This paper explores the welfare costs associated with drought plan transactions between a public municipal water agency, the El Dorado Irrigation District (EID) in California in the USA and its customers. The EID imposes a tiered pricing plan for municipal customers, which was analyzed as a discrete continuous choice (DCC) model by water users within a climate driven water evaluation and planning (WEAP) model of the EID water system. The DCC is subsequently used to estimate the compensating variation (CV) measure of the loss of consumer welfare in the case where a customer does not receive water that matches a preferred level of demand. In addition to monetized welfare loss, we look at other metrics of performance such as reservoir storage and hydropower generation. For the drought-of-record under full build-out, results show that the welfare loss to EID customers from the imposed drought plan is far less than if no drought plan were in place. This suggests that current consumption is well beyond essential needs and, without a drought plan, water shortages in the later period of a drought would generate a much greater welfare loss. Most of the cost associated with the drought plan is born by EID in the form of reduced revenues.

Combining Stated Preference and Revealed Preference Methods for the Valuation of Non-market Goods

Abstract In this paper, we introduce the combining stated preference and revealed preference methods which is the state-ofthe- art method for the valuation of non-market goods. Revealed preference methods and stated preference (SP) methods have both been applied by economists in valuing non-market goods; however both methods have inherent weaknesses. In order to exploit the strengths of the various approaches while minimizing their weaknesses, combining the two methods has become an important methodological option. Up to now, a growing number of literatures of studying combing the two types of data have evolved, and researchers developed three types of models to estimate combining revealed preference and stated preference data in academic fields: discrete choice model, continuous choice model and mixed choice model. Combining stated and revealed preference methods have been developed fast especially in the transport study field in which ideas could also be leant for environmental economic studies.