Normalization and correlation of cross-nested logit models

Demand analysis is more and more critical in the transportation context. Discrete choice models methodology provide an appropriate framework to capture the behaviour of the actors of transportation systems and, consequently, to forecast the demand. Recently, the cross-nested logit (CNL) model has received significant attention in the literature to capture decisions such as mode choice (Vovsha, 1997), departure time choice (Small, 1987) and route choice (Vovsha and Bekhor, 1998). Its general structure is appealing. Indeed, this model has a closed form probability formula, and allows for a wide range of correlation structures to be modeled. As shown by Bierlaire (forthcoming), various instances of the Cross-Nested Logit (CNL) model have been proposed in the literature. They are more or less the same, some being more specific as they constrain some parameters to fixed values. The issue of normalization of these models is not easy in practice. We provide a detailed analysis of this problem, and show that the normalization proposed by Wen and Koppelman (2001) is indeed correct. We emphasize the relation between the parameters of the CNL and the Alternative Specific Constants. In the second part of the paper, we analyze the correlation structure of the CNL. We show that the conjecture by Papola (2004) is erroneous. In fact, Papolas result is based on the assumption that the relation between the underlying NL correlations and the overall CNL correlations is linear. We show in the paper that this relation is actually based on a maximum. We propose a method to define a CNL with a given correlation structure. This is particularly important in applications such as route choice analysis, where the correlation of the alternatives can be somehow obtained from the network topology. We illustrate our method on a simple example, where we compare the correct results with Papolas conjecture.