This paper considers mixed, or random coefficients, multinomial logit (MMNL) models for discrete response, and establishes the following results: Under mild regularity conditions, any discrete choice model derived from random utility maximization has choice probabilities that can be approximated as closely as one pleases by a MMNLmodel. Practical estimation of a parametric mixing family can be carried out by Maximum Simulated Likelihood Estimation or Method of Simulated Moments, and easily computed instruments are provided that make the latter procedure fairly efficient. The adequacy of a mixing specification can be tested simply as an omitted variable test with appropriately defined artificial variables. An application to a problem of demand for alternative vehicles shows that MMNL provides a flexible and computationally practical approach to discrete response analysis.

: Mixed logit models, also called random-parameters or error-components logit, are a generalization of standard logit that do not exhibit the restrictive "independence from irrelevant alternatives" property and explicitly account for correlations in unobserved utility over repeated choices by each customer. Mixed logits are estimated for households' choices of appliances under utility-sponsored programs that offer rebates or loans on high-efficiency appliances. JEL Codes: C15, C23, C25, D12, L68, L94, Q40 2 Mixed Logit with Repeated Choices: Households' Choices of Appliance Efficiency Level 1. Introduction Mixed logit (also called random-parameters logit) generalizes standard logit by allowing the parameter associated with each observed variable (e.g., its coefficient) to vary randomly across customers. The moments of the distribution of customer-specific parameters are estimated. Variance in the unobserved customer-specific parameters induces correlation over alternatives in the ...

We estimate random-parameter logit models of anglers' choice of fishing site. The models generalize logit by allowing coefficients to vary randomly over anglers rather than being fixed. The models do not exhibit the restrictive "independence from irrelevant alternatives property" of logit and can represent any substitution pattern. Estimation explicitly accounts for the fact that the variation in coefficients over anglers induces correlation in unobserved utility over trips by the same angler. Willingness-to-pay for improved fish stock and the value to anglers of specific sites are calculated from the models and compared with the estimates obtained from a standard logit model.

The mixed logit model is considered to be the most promising state of the art discrete choice model currently available. Increasingly researchers and practitioners are estimating mixed logit models of various degrees of sophistication with mixtures of revealed preference and stated choice data. It is timely to review progress in model estimation since the learning curve is steep and the unwary are likely to fall into a chasm if not careful. These chasms are very deep indeed given the complexity of the mixed logit model. Although the theory is relatively clear, estimation and data issues are far from clear. Indeed there is a great deal of potential mis-inference consequent on trying to extract increased behavioural realism from data that are often not able to comply with the demands of mixed logit models. Possibly for the first time we now have an estimation method that requires extremely high quality data if the analyst wishes to take advantage of the extended behavioural capabilities of such models. This paper focuses on the new opportunities offered by mixed logit models and some issues to be aware of to avoid misuse of such advanced discrete choice methods by the practitioner.

ome detail more recent developments in the economic theory of choice, and modifications to this theory that are being forced by experimental evidence from cognitive psychology. I will close with a survey of statistical methods that have developed as part of the research program on economic choice behavior. Science is a cooperative enterprise, and my work on choice behavior reflects not only my own ideas, but the results of exchange and collaboration with many other scholars. 1 First, of course, is my co-laureate James Heckman, who among his many contributions pioneered the important area of dynamic discrete choice analysis. Nine other individuals who played a major role in channeling microeconometrics and choice theory toward their modern forms, and had a particularly important influence on my own work, are Zvi Griliches, L.L. Thurstone, Jacob Marschak, Duncan Luce, Danny Kahneman, Amos Tversky, Moshe Ben-Akiva, Charles Manski, and Kenneth Train. A gallery of their p

forthcoming, Econometrica We apply recent econometric advances to study the distribution of commuters ’ preferences for speedy and reliable highway travel. Our analysis applies mixed logit to combined revealed and stated preference data on commuter choices of whether to pay a toll for congestion-free express travel. We find that motorists exhibit high values of travel time and reliability and substantial heterogeneity in those values. We suggest that road pricing policies designed to cater to such varying preferences can improve efficiency and reduce the disparity of welfare impacts compared with recent pricing experiments.

ABSTRACT: We compare multinomial logit and mixed logit models for data on California households ' revealed and stated preferences for automobiles. The stated preference (SP) data elicited households ' preferences among gasoline, electric, methanol, and compressed natural gas vehicles with various attributes. The mixed logit models provide improved fits over logit that are highly significant, and show large heterogeneity in respondents ' preferences for alternative-fuel vehicles. The effects of including this heterogeneity are demonstrated in forecasting exercises. The alternative-fuel vehicle models presented here also highlight the advantages of merging SP and revealed preference (RP) data. RP data appear to be critical for obtaining realistic body-type choice and scaling information, but they are plagued by multicollinearity and difficulties with measuring vehicle attributes. SP data are critical for obtaining information about attributes not available in the marketplace, but pure SP models with these data give implausible forecasts. 1

The generalized nested logit model is a new member of the generalized extreme value family of models. The GNL provides a higher degree of flexibility in the estimation of substitution or cross-elasticity between pairs of alternatives than previously developed generalized extreme value models. The generalized nested logit model includes the paired combinatorial logit and cross-nested logit models as special cases. It also includes the product differentiation model, which represents the elasticity structure associated with multi-dimensional choices, and the ordered generalized extreme value model, which represents the elasticity structure associated with ordered alternatives, as special cases. The generalized nested logit model includes the two-level nested logit model as a special case and can approximate closely multi-level nested logit models. The generalized nested logit model accommodates differential cross-elasticity among pairs of alternatives through the fractional allocation of each alternative to a set of nests, each of which has a distinct logsum or dissimilarity parameter. An empirical example of intercity mode choice confirms the statistical superiority of the generalized nested logit model to the paired combinatorial logit, cross-nested logit and nested logit models and indicates important differences in cross-elasticity relationships across pairs of alternatives.

An individual's intrinsic mode preference and responsiveness to level-of-service variables affects her or his travel mode choice for a trip. The mode preference and responsiveness will, in general, vary across individuals based on observed (to an analyst) and unobserved (to an analyst) individual characteristics. The current paper formulates a multinomial-logit based model of travel mode choice that accommodates variations in mode preferences and responsiveness to level-of-service due to both observed and unobserved individual characteristics. The model parameters are estimated using a maximum simulated log-likelihood approach. The model is applied to examine urban work travel mode choice in a multiday sample of workers from the San Francisco Bay area. 1Introduction Most work travel mode choice models are based on the random utility maximization (RUM) framework of microeconomic theory. The RUM framework assumes that an individual's choice of mode on any choice occasion is a reflection of underlying indirect utilities associated with each of the available modes and that the individual selects the alternative which provides

Abstract: With flexible models of customers ’ choices among products and services, we estimate the tastes (part-worths) of each sampled customer as well as the distribution of tastes in the population. First, maximum likelihood procedures are used to estimate the distribution of tastes in the population using the pooled data for all sampled customers. Then, the distribution of tastes of each sampled customer is derived conditional on the observed data for that customer and the estimated population distribution of tastes (accounting for uncertainty in the population estimates.) The procedure provides the same type of information and is similar in spirit to hierarchical Bayes (HB.) The procedure is computationally attractive when it is easier to calculate the likelihood function for the population parameters than to draw from the posterior distribution of parameters as needed for HB. We apply the method to data on residential customers ’ choice among energy suppliers in conjoint-type experiments. The estimated distribution of tastes provides practical information that is useful for suppliers in designing their offers. The conditioning for individual customers is found to differentiate customers effectively for marketing purposes and to improve considerably the predictions in new situations. Acknowledgements: We have benefited from comments and suggestions by Greg Allenby, Joel Huber, Rich Johnson, Daniel McFadden, and Peter Rossi. Of course, we alone are responsible for our representations and conclusions. The data for this analysis were collected by the Electric Power Research Institute (EPRI.) We are grateful to Ahmad Faruqui and EPRI for allowing us to use the data and present the results publicly. Andrew Goett and Kathleen Hudson, who had previously used these data, provided us data files in easily useable form, which saved us a considerable amount of time. For interested readers, software to estimate mixed logits is available free from Train’s web site at