A theoretical analysis of the cross-nested logit model

The emergence of Intelligent Transportation Systems and the associated technologies has increased the need for complex models and algorithms. Namely, real-time information systems, directly influencing transportation demand, must be supported by detailed behavioral models capturing travel and driving decisions. Discrete choice models methodology provide an appropriate framework to capture such behavior. Recently, the cross-nested logit (CNL) model has received quite a bit of attention in the literature to capture decisions such as mode choice, departure time choice and route choice. In this paper, we develop on the general formulation of the Cross Nested Logit model proposed by Ben-Akiva and Bierlaire (1999) and based on the Generalized Extreme Value (GEV)model. We show that it is basically equivalent to the formulations by Papola (2000) and Wen and Koppelman (2001). We also show that the formulations by Small (1987) and Vovsha (1997) are special cases of this formulation. We formally prove that the Cross-Nested Logit model is indeed a member of the GEV models family. In doing so, we clearly distinguish between conditions that are necessary to be consistent with the GEV theory, from normalization conditions. Finally, we propose a straightforward estimation procedure, based on nonlinear programming. In order to make it operational, we provide the first-derivatives of the log-likelihood function, which are necessary to most optimization procedures.