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 ®ts over logit that are highly signi®cant, and show large heterogeneity in respondents' preferences for alternative-fuel vehicles. The e ects 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 di culties 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. Ó

We discuss the development of predictive choice models that go beyond the random utility model in its narrowest formulation. Such approaches incorporate several elements of cognitive process that have been identified as important to the choice process, including strong dependence on history and context, perception formation, and latent constraints. A flexible and practical hybrid choice model is presented that integrates many types of discrete choice modeling methods, draws on different types of data, and allows for flexible disturbances and explicit modeling of latent psychological explanatory variables, heterogeneity, and latent segmentation.Both progress and challenges related to the development of the hybrid choice model are presented.

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We derive an expression for the critical stock price for the American put. We start by expressing the put price as an integral involving first-passage probabilities. This approach yields intuition for Merton's result for the perpetual put. We then consider the finite-lived case. Using~1! the fact that the put value ceases to depend on time when the critical stock price is reached and~2! the result that an American put equals a European put plus an early-exercise premium, we derive the critical stock price. We approximate the critical-stock-price function to compute accurate put prices.KIM~1990!, JACKA~1991!, AND CARR, JARROW, AND MYNENI~1992! showed that the American put price is equal to the corresponding European put price plus an integral representing the early-exercise premium. However, to use their approach one needs to know the critical stock price, which may be defined in three equivalent ways.1. It is the value of the stock price at which one is indifferent between exercising and not exercising the put. 2. It is the highest value of the stock price for which the value of the put is equal to the exercise price less the stock price. 3. It is the highest value of the stock price at which the put value does not depend on time to maturity.The second definition has ordinarily been used to find the critical stock price, but it requires solving an integral equation numerically. If it is optimal to exercise the put immediately, then its value cannot depend on how much time remains to maturity. Hence, we have the third definition, which, we believe, is new to the literature. We show that the third definition leads to an analytic expression for the critical stock price. Our result is important not only because it provides a better way to get American put prices but also