On the covariance structure of the Cross-Nested Logit model

This paper proposes a detailed analysis of the covariance structure of the Cross-Nested Logit model, widely applied thanks to its closed-form and to the expected flexibility of the underlying covariance matrix structure. In more detail, through the numerical computation of actual CNL covariances, some relevant aspects are analysed and addressed. Firstly, the degree of approximation inherent Papola's conjecture [Papola, A., 2004. Some developments of the Cross-Nested Logit model. Transportation Research Part B 38 (14), 833-851] on CNL covariances is explored. Then, it is shown that the CNL model is not generally able to cover the whole domain of the feasible homoskedastic covariance matrices, and that the degree of coverage depends on the adopted CNL nesting structure. In that respect, the issue of finding the most flexible CNL structure is addressed from a theoretical and numerical standpoint, leading to the result that maximum flexibility is reached for a "full" nesting structure (i.e. each alternative belongs to all the groups): a rule-of-thumb is also established for the choice of the number of groups. Moreover, it is shown that, when a covariance matrix is reproducible, there are generally infinite CNL specifications - leading to different choice probabilities - corresponding to that matrix. A direct consequence is that the issue of finding a CNL model specification able to reproduce a given known correlation matrix (relevant in some contexts, e.g. route choice modelling) can be not so practically relevant since choice probabilities are not in a one-to-one correspondence with covariances.