Doubly generalized logit: A closed-form discrete choice model system with multivariate generalized extreme value distributed utilities

Abstract

In this study, we generalize our previous q-generalized multinomial logit model (Nakayama & Chikaraishi, 2015), in which the heteroscedastic variance and flexible shape of the utility distribution are considered, by allowing for statistical dependency of alternatives. This is achieved by introducing the q-generalization of McFadden’s multivariate Gumbel distribution. Thus, the logit model is doubly generalized; 1) each utility follows the generalized extreme value distribution that includes the Gumbel, Weibull, and Fréchet distributions; and 2) the utility distribution is multivariate, and therefore, a nested or cross-nested structure and dependency of alternatives are allowed. The proposed doubly generalized logit model system allows for deriving new closed-form discrete choice models such as the q-generalized nested logit model and q-generalized cross-nested logit (CNL) model. Furthermore, the model system includes conventional logit models such as the multinomial logit, nested logit, and CNL models as special cases as well as the new generalized logit models, while retaining a closed-form expression. We empirically confirm that the goodness-of-fit of the proposed model could be substantially better compared to that of the conventional models in some cases, though the degree of improvements varies across cases.