A unified Mixed Logit Framework for Modeling Revealed and Stated Preferences: Formulation and Application to Congestion Pricing Analysis in the San Francisco Bay Area

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A Unified Mixed Logit Framework for Modeling Revealed and Stated Preferences: Formulation and Application to Congestion Pricing Analysis in the San Francisco Bay Area

Chandra R. Bhat

Saul Castelar

Research Report SWUTC/02/167220

Southwest Regional University Transportation Center

Center for Transportation Research

The University of Texas at Austin

Austin, Texas 78712
April 2003

The contents of this report reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein. This document is disseminated under the sponsorship of the Department of Transportation, University Transportation Centers Program, in the interest of information exchange. The U.S. Government assumes no liability for the contents or use thereof.

This report formulates and applies a unified mixed-logit framework for joint analysis of revealed and stated preference data that accommodates a flexible competition pattern across alternatives, scale difference in the revealed and stated choice contexts, heterogeneity across individuals in the intrinsic preferences for alternatives, heterogeneity across individuals in the responsiveness to level-of-service factors, state dependence of the stated choices on the revealed choice, and heterogeneity across individuals in the state dependence effect. The estimation of the mixed logit formulation is achieved using simulation techniques that employ quasi-random Monte Carlo draws. The formulation is applied to examine the travel behavior responses of San Francisco Bay Bridge users to changes in travel conditions. The data for the study are drawn from surveys conducted as part of the 1996 San Francisco Bay Area Travel Study. The results of the mixed logit formulation are compared with those of more restrictive structures on the basis of parameter estimates, implied trade-offs among level-of-service attributes, heterogeneity and state dependence effects, data fit, and substantive implications of congestion pricing policy simulations.
Keywords: Revealed preference, stated preference, mixed logit, quasi-Monte Carlo simulation, state dependence, unobserved heterogeneity, congestion pricing.

This research was funded by the U.S. Department of Transportation through the Southwest Region University Transportation Center. The authors appreciate the useful comments of David Hensher. Ken Vaughn provided the San Francisco Bay area data and clarified data issues. Lisa Weyant helped with typesetting and formatting the report.
This report examines the travel behavior responses of San Francisco Bay Bridge users to changes in travel conditions, including changes in bridge tolls, parking costs, travel times, transit fares, and transit service headway. Several results from the empirical analysis in the paper are noteworthy. First, the results emphasize the advantage of combining revealed preference (RP) and stated preference (SP) data in travel modeling. Using only RP data results in a statistically insignificant cost coefficient, reflecting the limited variation in cost within the RP sample as well as multi-collinearity between time and cost. On the other hand, using only SP data would, in general, result in estimates of alternative-specific constants that are not reflective of the market shares of the alternatives; also using only SP data would not recognize state dependence effects. Joint RP-SP methods are better able to represent trade-offs in level-of-service attributes and also provide efficiency benefits in estimation by recognizing the presence of a common latent preference structure underlying the RP and SP responses. Second, the results indicate substantial unobserved variation (or unobserved heterogeneity) across individuals in overall preferences for alternatives. There is also significant difference in sensitivity in response to level-of-service measures. Between the time and cost sensitivities, there appears to be substantially more taste variation across individuals in time sensitivity than in cost sensitivity. Third, ignoring unobserved heterogeneity and/or state dependence effects leads to an overestimation of time sensitivity; thus, using “cross-sectional” methods of analysis that ignore the repeated-choice nature of SP responses and the dependence of SP responses on RP responses lead to biased estimates of the effects of level-of-service variables in the current empirical context. Fourth, there is a dramatic improvement in data fit when one introduces taste variation. The rho-bar squared value increases from about 0.07 to about 0.53 when unobserved heterogeneity is introduced. Fifth, the results indicate substantial variation across individuals in the state-dependence effect; it appears that both a positive effect (due to factors such as habit persistence, inertia to explore another alternative, or learning combined with risk aversion) or a negative effect (due to, for instance, variety seeking or the latent frustration of the inconvenience associated with a current alternative) may be associated with the influence of current choice on future choices. Sixth, there is a dramatic increase in the estimated scale difference in RP and SP responses when unobserved heterogeneity is accommodated; that is, after accommodating unobserved heterogeneity effects, the error variance in the SP choice context is much lower than in the RP choice context. This result is quite different from those in most earlier RP-SP studies, which have estimated a larger error variance in the SP context than in the RP context or have found the error variances to not be significantly different. These earlier studies have attributed a higher SP error variance to the limited set of attributes in SP experiments or to experimental design effects. Our results suggest that the larger SP variance in earlier studies may have been an artifact of ignoring error correlations across repeated SP choices from the same individual. Thus, it is possible that the SP choice context provides a very focused setting compared to an RP context, with relatively little room for measurement error or imputation of variable values. This, in combination with recent studies that suggest the ability of consumers to systematically evaluate even rather complex hypothetical scenarios (see Louviere and Hensher, 2000), points toward using SP experiments as the main data source for analysis and supplementing with small samples of RP data for anchoring with actual market activity.

The substantive congestion-pricing policy implications on the shares of the various travel alternatives are quite different among the alternative models. The results suggest that the cross-sectional MNL model will provide an overly-optimistic projection of the alleviation in traffic congestion during the peak periods due to congestion-pricing schemes, while the other restrictive models (the cross-sectional error-components model, the panel-data model with unobserved heterogeneity only, and the panel data model with state dependence only) will under-predict the alleviation in peak-period traffic congestion (compared to the general model). The differences between the general model and the restrictive structures are particularly noticeable in their predictions of the increases in the market share of the non-DAP alternatives. These results highlight the need to include (or at least test for) flexible inter-alternative error structures, unobserved heterogeneity, state dependence, and heterogeneity in the state dependence effects within the context of a unified methodological framework to assist informed policy decision-making.




1.1 Inter-Alternative Error Structure 1

1.2 Scale Difference 2

1.3 Unobserved Heterogeneity Effects 3

1.4 State Dependence and Heterogeneity in State Dependence 4

1.5 A Unified RP-SP Modeling Framework 5





5.1 Cross-Sectional Models 19

5.2 Panel Data Models 25




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