Accounting for stochastic variables in discrete choice models

The estimation of discrete choice models requires measuring the attributes describing the alternatives within each individual’s choice set. Even though some attributes are intrinsically stochastic (e.g. travel times) or are subject to non-negligible measurement errors (e.g. waiting times), they are usually assumed fixed and deterministic. Indeed, even an accurate measurement can be biased as it might differ from the original (experienced) value perceived by the individual.Experimental evidence suggests that discrepancies between the values measured by the modeller and experienced by the individuals can lead to incorrect parameter estimates. On the other hand, there is an important trade-off between data quality and collection costs. This paper explores the inclusion of stochastic variables in discrete choice models through an econometric analysis that allows identifying the most suitable specifications. Various model specifications were experimentally tested using synthetic data; comparisons included tests for unbiased parameter estimation and computation of marginal rates of substitution. Model specifications were also tested using a real case databank featuring two travel time measurements, associated with different levels of accuracy.Results show that in most cases an error components model can effectively deal with stochastic variables. A random coefficients model can only effectively deal with stochastic variables when their randomness is directly proportional to the value of the attribute. Another interesting result is the presence of confounding effects that are very difficult, if not impossible, to isolate when more flexible models are used to capture stochastic variations. Due the presence of confounding effects when estimating flexible models, the estimated parameters should be carefully analysed to avoid misinterpretations. Also, as in previous misspecification tests reported in the literature, the Multinomial Logit model proves to be quite robust for estimating marginal rates of substitution, especially when models are estimated with large samples.     

Fundamentals of discrete multimodal choice modelling

Abstract

In this paper an overview is given of the most relevant issues relating to the application of multimodal choice models, with particular emphasis on disaggregate modal split models. The paper considers questions of data, such as type of data, alternative sampling strategies and problems of measurement; and modelling issues, such as model specification and estimation, including a good presentation of the statistical techniques'available. The paper also addresses the aggregation problem, which lies at the heart of one of today's most hotly contested debates: whether to use aggregate or disaggregate models for policy analysis, and in which circumstances.     

Specification of regret-based models of choice behaviour: formal analyses and experimental design based evidence

Travel behaviour analysis has recently witnessed a rapidly growing interest in regret-based models of choice behaviour. Two different model specifications have been introduced in the transportation literature. Chorus et al. (Transportation Research B 42: 1–18, 2008a; in: Proceedings 87th Annual Meeting of the Transportation Research Board, Washington DC, 2008b) specified regret as a (non) linear function of the difference between the best-foregone choice alternative and the chosen alternative. Later, as an approximation to the original specification, Chorus (2010) suggested a logarithm function of utility differences between all choice alternatives, mainly for ease of estimation. This paper makes two contributions to this literature. First, formal analyses are conducted to identify the parameter space where the logarithmic specification becomes theoretically inferior to the original specification. Second, an empirical stated choice study on the choice of shopping centre is conducted to empirically test which specification best describes stated choices. Results suggest that for the collected data the original specification outperforms the new specification. Implications of this finding for the application of regret-based choice models in travel behaviour analysis are discussed.     

Identifying a behaviourally relevant choice set from stated choice data

Stated choice experiments are designed optimally in a statistical sense but not necessarily in a behavioural choice making sense. Statistical designs, and consequently model estimation, assume that the set of alternatives offered in the experiment are processed by respondents with a specific processing strategy. Much has been studied about attribute processing using discrete choice methods in travel choice studies, but this paper focuses more broadly on processing of alternatives in the choice set offered in the experiment. This paper is motivated by the primary idea that the distribution of predicted choice probabilities associated with a set of alternatives defining a given choice set might provide strong evidence on the strategies that agents appear to use when choosing a preferred alternative. In an empirical setting of a choice set of size three, four model specifications are considered including a model for the selection of the best alternative in the full choice set and three variants of a best–worst regime. Using state choice data on road pricing reform, the empirical analysis examines which model specification delivers the most accurate prediction of the chosen alternative. The results suggest which alternatives really matter in choice making and hence the alternatives that might be included in a choice set for model specification.     

Estimating choice models containing attraction variables

The paper presents a family of disaggregate choice models, which are shown to be equivalent to many of the aggregate models commonly used in planning studies. A brief summary is given of the method that has been developed for estimating the parameters of these models. A generalisation is then introduced in which variables representing the attractiveness in terms of size or quantity of each alternative are allowed to enter the model. It is shown that the form in which these variables enter the model requires a more general estimation algorithm than is commonly used, and such an algorithm is presented. A series of practical tests of the new algorithm is described.     

Representing mental maps and cognitive learning in micro-simulation models of activity-travel choice dynamics

This paper develops a model, based on Bayesian beliefs networks, for representing mental maps and cognitive learning into micro-simulation models of activity-travel behavior. Mental maps can be used to address the problem that choice sets in models of travel demand are often ad hoc specified. The theory underlying the model is discussed, a specification is derived and numerical simulation is used to illustrate the properties of the model.     

Development of an indicator to assess the spatial fit of discrete choice models

Discrete choice models are increasingly implemented using geographical data. When this is the case, it may not be sufficient to project market shares accurately, but also to correctly replicate the spatial pattern of choices. Analysts might then be interested in assessing the results of a model’s fit relative to the spatial distribution of the observed responses. While canonical approaches exist for the exploratory spatial analysis of continuous variables, similar tools have not been widely implemented for discrete choice models, where the variable of interest is categorical. For this reason, despite recent progress with spatial models for discrete outcomes, there is still not a simple and intuitive tool to assess the quality of the spatial fit of a discrete choice model. The objective of this paper is to introduce a new indicator of spatial fit that can be applied to the results of discrete choice models. Utility of the indicator is explored by means of numerical experiments and then demonstrated by means of a case study of vehicle ownership in Montreal, Canada.     

Random utility models with implicit availability/perception of choice alternatives for the simulation of travel demand

Random utility models are undoubtedly the most used models for the simulation of transport demand. These models simulate the choice of a decision-maker among a set of feasible alternatives and their operational use requires that the analyst is able to correctly specify this choice-set for each individual.Some early applications basically ignored this problem by assuming that all decision-makers chose from the same pre-specified choice-set. This assumption may be unrealistic in many practical cases and cause significant misspecification problems (P. Stopher, Transportation Journal of ASCE 106 (1980) 427; H. Williams, J. Ortuzar, Transportation Research B 16 (1982) 167).The problem of choice-set simulation has been dealt within the literature following two basically different approaches:

• •simulating the perception/availability of an alternative implicitly in the choice model,
• •simulating the choice-set generation explicitly in a separate model.

The implicit approach is more convenient from an operational point of view, while the explicit one is more appealing from a theoretical point of view.In this paper, a different approach to the modeling of availability/perception of alternatives in the context of random utility model is proposed. This approach is based on the concept of intermediate degrees of availability/perception of each alternative simulated through a model (or “inclusion function”) which in turn is introduced in the systematic utility of standard random utility models.This model, named implicit availability/perception (IAP), may be differently specified depending on assumptions made on the joint distribution of random residuals and the way in which the average degree of availability/perception is modeled.In this paper, a possible specification of the IAP model, based on the assumption of random residual distributed as i.i. Gumbel and with the average degree of availability/perception modeled as a binomial logit, is proposed.The paper also proposes ML estimation models in two cases: in the first, only information on alternatives choices is available, while in the second, this information is complemented with others on variables related to a latent (i.e., non-observable) alternatives availability/perception degree (e.g., information on car availability of decision-maker i used as an indirect measurement of the unknown and non-observable availability/perception degree of alternative car for decision-maker i in a modal split).The proposed specification is tested on mode choice data; the calibration results are compared with those of a similar logit specification with encouraging results in terms of goodness of fit.     

A sequential approach to exploiting the combined strengths of SP and RP data: Application to freight shipper choice

The possibility of and procedure for pooling RP and SP data have been discussed in recent research work. In that literature, the RP data has been viewed as the yardstick against which the SP data must be compared. In this paper we take a fresh look at the two data types. Based on the peculiar strengths and weaknesses of each we propose a new, sequential approach to exploiting the strengths and avoiding the weaknesses of each data source. This approach is based on the premise that SP data, characterized by a well-conditioned design matrix and a less constrained decision environment than the real world, is able to capture respondents' tradeoffs more robustly than is possible in RP data. (This, in turn, results in more robust estimates of share changes due to changes in independent variables.) The RP data, however, represent the current market situation better than the SP data, hence should be used to establish the aggregate equilibrium level represented by the final model. The approachfixes the RP parameters for independent variables at the estimated SP parameters but uses the RP data to establish alternative-specific constants. Simultaneously, the RP data are rescaled to correct for error-in-variables problems in the RP design matrixvis-à- vis the SP design matrix. All specifications tested are Multinomial Logit (MNL) models.The approach is tested with freight shippers' choice of carrier in three major North American cities. It is shown that the proposed sequential approach to using SP and RP data has the same or better predictive power as the model calibrated solely on the RP data (which is the best possible model for that data, in terms of goodness-of-fit figures of merit), when measured in terms of Pearson's Chi-squared ratio and the percent correctly predicted statistic. The sequential approach is also shown to produce predictions with lower error than produced by the more usual method of pooling the RP and SP data.     

Recent developments in the econometrics of panel data analysis

The present paper surveys different issues related to the econometric analysis of panel data. It discusses the controversy over fixed effects vs. random effects models, specification error tests with panel data, the problem of specification of the distribution of initial values in dynamic models, maximum-likelihood estimation of dynamic models, tests for serial correlation with panel data, serial correlation vs. state dependence, multiple equation models with panel data, and errors in variables in panel data. References are also given for certain other miscellaneous problems involving panel data.     

Nested logit models for mixed-mode travel in urban corridors

The nested logit (NL) model is a generalisation of the well-known multinomial logit (MNL) model which copes with its “independence from irrelevant alternatives” problem, at the expense of more difficult calibration and use. Mixed-mode movements (i.e. park-and-ride) are by nature not independent of competing single-mode options and have, therefore, traditionally been inadequately modelled in most empirical applications. This paper reports on the specification, estimation, testing and comparison of MNL and NL models using disaggregate data of work trips in an urban corridor, where choice was among several alternatives including mixed-mode options. It was found that the more general NL model was more adequate, not only in theory but in practice. The paper concludes by comparing the disaggregate NL model with previously calibrated aggregate NL models for the same corridor using a different data set.     

Multidimensional choice model transferability

Transferability studies have focused on the component models of the conventional four-step urban travel forecasting model system. This study extends previous analyses by examining the transferability of models describing multidimensional travel and related choices. In particular, we examine the hypothesis that joint and sequential choice models are equally transferable against the alternative hypotheses that either of the model types is more transferable. Measures of goodness of fit and transfer effectiveness are formulated for sequential choice models to provide a consistent comparison between the joint and sequential models. An empirical analysis is undertaken in the context of joint (multinomial logit) and sequential (nested logit) models of automobile ownership and mode choice to work. This study finds little difference between the transferability of these joint and sequential models. However, this conclusion appears to be dependent on the similarity of the estimation results for the joint and sequential models in this case. These results suggest a need for additional testing in other empirical contexts to identify the relative transferability of joint versus sequential models when the estimation results are distinct.     

Empirical test of a constrained choice discrete model: Mode choice in São Paulo,Brazil

This paper examines the properties and empirically tests a model of discrete choice which incorporates probabilistic choice set generation. Denominated the Parametrized Logit Captivity (PLC) model, it is a generalization of the well-known “dogit” specification. The PLC model is shown to be theoretically and empirically more flexible than the latter. Work mode choice data collected in a 1977 O/D survey in São Paulo, Brazil, is used to obtain parameter estimates, as well as to evaluate consumer reaction to a series of perturbations in travel time, travel cost and income, for both the PLC and Multinomial Logit models. Comparisons between the two specifications are made in terms of statistical fit, reasonableness of predictions and differences in predictions across models.     

A comparative study of aggregate and disaggregate gravity models using Seoul metropolitan subway trip data

Abstract

Existing origin constrained and doubly constrained gravity models have not been compared, theoretically or empirically, in terms of their forecasting power. Due to the newly advanced technology of intelligent transport systems, the expanded data presently available have made various models more comparable in terms of forecasting power. This paper uses archived automatic passenger counting (APC) data for urban rail in the Seoul metropolitan area. The APC data contains information about each trip's origin, destination, ticket type, fare, and distance on a daily basis. The objective of this paper is to compare the goodness-of-fit of aggregate and disaggregate gravity modeling using these data. A Hyman aggregate gravity model is used as the aggregate model without the spatial effect. The disaggregate model adopts a multinomial logit as the destination choice model with the spatial effect. In general, while the formulation of aggregate and disaggregate gravity model models are similar, the calibration and parameter estimation methods of the two models are different. As a result, this empirical study demonstrates that the variation in goodness-of-fit and forecasting power largely depends on the estimation method and selected variables. The forecasting power of the disaggregate modeling approach outperforms that of the aggregate model. This paper further confirms that spatial arrangement plays important roles in gravity modeling.     

Motor fuel choice: An econometric analysis

Concerns about air pollution and energy security have stimulated interest in alternative automotive fuels and in vehicles that can use multiple fuels and combinations of fuels. Consumer behavior in the choice of motor fuel for flexible-fuel vehicles is likely to be a key factor in the creation and stability of markets for new fuels. The sensitivity of fuel choice to fuel prices is investigated here using data on purchases of regular, premium, leaded, and unleaded grades of gasoline. Multinomial logit choice models are estimated for the years 1982, 1983, and 1984. Consumer choices are found to be highly elastic with respect to fuel prices. New fuels will have to be priced within a few cents of existing fuels to capture significant market shares. Consumers also exhibit strong preferences for the fuel type specified by law (unleaded vs. leaded) for their vehicles. Thus, legal restrictions could play an important role in stabilizing alternative fuels markets. The typical consumer is willing to pay 5–10 cents more per gallon for premium grade gasoline. The premium for premium has been increasing as increasing numbers of turbocharged and fuel-injected engines join the fleet, reflecting the fact that consumers are willing to pay more for a fuel if they believe their engines require it.     

Airline competition and market frequency: A comparison of the s-curve and schedule delay models

We compare two common ways of incorporating service frequency into models of airline competition. One is based on the so called s-curve, in which, all else equal, market shares are determined by frequency shares. The other is based on schedule delay—the time difference between when travelers wish to travel and when flights are available. We develop competition models that differ only with regard to which of the above approaches is used to capture the effect of frequency. The demand side of both models is an approximation of a nested logit model which yields endogenous travel demand by including not traveling in the choice set. We find symmetric competitive equilibrium for both models analytically, and compare their predictions concerning market frequency with empirical evidence. In contrast to the s-curve model, the schedule delay model depicts a more plausible relationship between market share and frequency share and accurately predicts observed patterns of supply side behavior. Moreover, the predictions from both models are largely the same if we employ numerical versions of the model that capture real-world aspects of competition. We also find that, for either model, the relationship between airline frequency and market traffic is the same whether frequency is determined by competitive equilibrium, social optimality, or social optimality with a break-even constraint.     

Experimental design influences on stated choice outputs: An empirical study in air travel choice

Discrete choice experiments are conducted in the transport field to obtain data for investigating travel behaviour and derived measures such as the value of travel time savings. The multinomial logit (MNL) and other more advanced discrete choice models (e.g., the mixed MNL model) have often been estimated on data from stated choice experiments and applied for planning and policy purposes. Determining efficient underlying experimental designs for these studies has become an increasingly important stream of research, in which the objective is to generate stated choice tasks that maximize the collected information, yielding more reliable parameter estimates. These theoretical advances have not been rigorously tested in practice, such that claims on whether the theoretical efficiency gains translate into practice cannot be made. Using an extensive empirical study of air travel choice behaviour, this paper presents for the first time results of different stated choice experimental design approaches, in which respective estimation results are compared. We show that D-efficient designs keep their promise in lowering standard errors in estimating, thereby requiring smaller sample sizes, ceteris paribus, compared to a more traditional orthogonal design. The parameter estimates found using an orthogonal design or an efficient design turn out to be statistically different in several cases, mainly attributed to more or less dominant alternatives existing in the orthogonal design. Furthermore, we found that small designs with a limited number of choice tasks performs just as good (or even better) than a large design. Finally, we show that theoretically predicted sample sizes using the so-called S-estimates provide a good lower bound. This paper will enable practitioners in better understanding the potential benefits of efficient designs, and enables policy makers to make decisions based on more reliable parameter estimates.     

Mode choice and travel distance joint models in school trips

We test a copula-based joint discrete–continuous model to unravel mode choice and travel distance decisions in a joint framework for school trips. This framework explicitly accounts for common unobserved factors that may affect both the mode choice and travel distance. Joint estimation of the models makes a significant difference in the effect of travel distance on willingness to walk to school. The absolute value of the travel distance coefficient in the mode choice model increases by 22% when a joint formulation is adopted instead of the conventional single estimations. We find a significant decrease of 19% in the coefficient of travel safety perception in the joint mode choice model compared to the single model. This underscores the impact of model specification, in terms of the variable effect interpretation and policy assessments. The effect magnitude of several policy-sensitive variables is discussed and compared with previous studies. Particularly, we indicate that the probability of walking is reduced by 0.85% due to a 1% increase in travel distance; accordingly, it propels parents to select non-active modes, particularly school bus. This study also demonstrates how addressing parental concerns about travel safety could double the propensity to walk to school.