A Specification Procedure of the CNL Model Reproducing Any Homoschedastic Covariance Matrix
A Papola, V Marzano, University of Naples, IT
This paper deals with a theoretical and numerical analysis of the covariances underlying the CNL model and consistently addresses an operative specification of the model structure and parameters able to reproduce the desired covariances.
This paper deals with a theoretical and numerical analysis of the covariances underlying the Cross-Nested Logit (CNL) model and consistently addresses an operative specification of the model structure and parameters able to reproduce the desired covariances.
The CNL model represents so far the GEV model which allows the more general structure of the covariance matrix. As known, its application requires the specification of the covariance structure and of degrees of membership; therefore, from an operative point of view, the knowledge of the functional relationship between the model structure and parameters from one hand and the underlying covariance matrix from the other hand represents a preliminary condition for an effective application of the CNL model to any choice context. Unfortunately, this functional relationship is actually unknown; this circumstance is obviously shared by all the generalizations of the CNL such as the Network GEV and by the Path-Multilevel Logit recently proposed in literature. Accordingly to the importance of this issue, a work has been recently proposed in literature by Papola (2004), which proposed a conjecture about the relationship between covariances and parameters of the CNL model.
Starting from this point, this paper aims to propose two main contributions. Firstly, covariances among alternatives in a CNL model are numerically computed on the basis of the cumulative distribution function of the random residuals derived from the GEV generating function of CNL, in order to check how the conjecture mentioned above could be approximated. Some preliminary results indicates Papola?s conjecture to represent a very good approximation when the degrees of membership tends to their boundary values (0/1) and to provide an acceptable overestimation of the covariances themselves in the other cases. Secondly, being demonstrated the reasonable approximation of this conjecture, a specification of the model structure and parameters and a consequent operative procedure for implementing this specification directly from the desired covariance matrix are proposed. In detail, the basic idea is to establish a link between the elements of the Cholesky factorization of the covariance matrix and the degrees of membership of the CNL model, demonstrating how this factor analysis approach could lead directly to Papola?s conjecture. In other words, given a desired covariance matrix consistent with the homoschedasticity of the CNL model (and with the bounds provided by the Gumbel distribution), it is sufficient to express it as the product of a lower triangular matrix and its transposed to obtain all the elements needed for the computation of the degrees of membership and for the definition of the corresponding CNL structure. Some preliminary performance analyses of the proposed specification procedure show very interesting results.
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