On the Covariance Structure of Generalized Extreme Value Models
A Papola, V Marzano, University of Naples "Federico II", IT
The paper proposes a systematic and exhaustive analysis of covariances underlying GEV models, and consistently defines some rules-of-thumb for the specification of a GEV model able to reproduce the covariance structure of a given
Random utility theory represents one of the most common and well-established paradigms for the simulation of discrete choice contexts. Several models have been proposed within this theoretical framework, depending on different assumptions about the joint distribution function of decision makers? perceived utilities. A wide class of random utility models, namely the General Extreme Value (GEV) models defined by McFadden?s (1978) theorem, hypothesizes random residuals to follow a multivariate extreme value distribution, assumption leading to interesting and useful model properties. In spite of their diffusion, some fundamental issues concerning the properties of GEV models remain unsolved. Mainly, except for GEV models whose correlation structure can be drawn as a tree (i.e. Multinomial and Nested Logit models), the actual covariance figures underlying whatever network correlation structure (i.e. Cross-Nested Logit and RNEV/Network GEV) cannot be expressed through a closed-form statement. This circumstance prevents addressing some theoretical and operative issues, i.e. whether and how a choice context characterized by a known covariance matrix can be reproduced through a GEV model.
With reference to the former issue, Papola and Marzano (2005) showed, for two elementary CNL structures, that the positive definiteness domain of feasible covariance/correlation matrices is not guaranteed to be covered by any CNL structure. With reference to the latter issue, a CNL able to reproduce approximately a given covariance matrix (according to the bounds of feasible correlations allowed by the model) can be practically specified through either solving a system of linear equations (Papola, 2000) or applying an automatic procedure (Papola and Marzano, 2005). Actually, the exact way to face the problem, suggested by Abbé et al. (2005), seems not to provide for practical applications due to its mathematical complexity.
This paper proposes therefore a systematic analysis of this aspect, generalizing and carrying out the previously mentioned literature contributions. For this aim, a more efficient numerical procedure for the calculation of GEV covariances, different from the one applied by Papola and Marzano (2005) and Abbé et al. (2005), has been firstly developed. In more detail, the procedure is based on an expression of the correlation between pair of random residuals which does not involve the calculation of partial derivatives of the joint cumulative distribution function. Then, starting from the most elementary CNL structure, the covariance figures underlying all its possible generalizations have been investigated, i.e. whether allowing for a higher number of common groups for a pair of alternatives or for all the alternatives to belong simultaneously to all the groups actually influences the coverage of the domain of the feasible covariance matrices. Within this context, the operative importance of the multi-level generalization of the CNL model provided by the Network GEV model has also been investigated.
Therefore, through these analyses the paper provides for a general overview of the covariances underlying network-based (i.e. CNL and Network GEV) GEV models, consistently defining some operative rules-of-thumb for the specification of a GEV model able to face a choice context with a given/expected covariance structure. In detail, this implies defining, from one side, an operative procedure for a given covariance matrix to be reproduced and, from the other side, finding out the CNL/RNEV specification able to allow for the maximum flexibility in the underlying covariances. The results achieved up to now show that, following some specification rules of the model structure pointed out in the paper, the bounds in the covariance matrices reproducible through a CNL model are not so restrictive as suggested by the preliminary results provided by Papola and Marzano (2005).
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