Estimation of discrete choice models under various sampling strategies

We first review the impact of various sampling strategies on the estimation of discrete choice models. In particular, consistent estimates of all parameters of a multinomial logit (MNL) model, except the constants, can be obtained from an exogenous sample maximum likelihood (ESML) estimation in the presence of choice-based sampling strategies. Then, we describe the recent estimator proposed by Bierlaire, Bolduc and McFadden (2008) [The estimation of Generalized Extreme Value models from choice-based samples, Transportation Research Part B: Methodological 42(4):381-394] for the estimation of Multivariate Extreme Value models. We show that this property does not hold in general for multivariate extreme value (MEV) models. We propose a consistent ESML estimator for MEV models in this context. We then propose a new and simple weighted conditional maximum likelihood (WCML) estimator for the more general case. Contrarily to the weighted exogenous sample maximum likelihood (WESML) estimator by Manski and Lerman [Manski, C., Lerman, S., 1977. The estimation of choice probabilities from choice-based samples. Econometrica 45, 1977-1988], the new WCML estimator does not require an external knowledge of the market shares. We show that this applies also to the case where alternatives are sampled from a large choice set, and we illustrate the use of the estimator on synthetic and real data.