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A theoretical analysis of the cross-nested logit model

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  • Michel Bierlaire

Abstract

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. %The CNL model is an extension of the Nested Logit model, providing %more flexibility at the cost of some complexity in the model formulation. 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 equivalent to the formulations byby Papola (2004) 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 prove consistency with the GEV theory, from normalization conditions. Finally, we propose to estimate the model with non-linear programming algorithms, instead of heuristics proposed in the literature. In order to make it operational, we provide the first derivatives of the log-likelihood function, which are necessary to such optimization procedures. Copyright Springer Science+Business Media, LLC 2006

Suggested Citation

  • Michel Bierlaire, 2006. "A theoretical analysis of the cross-nested logit model," Annals of Operations Research, Springer, vol. 144(1), pages 287-300, April.
    • Handle: RePEc:spr:annopr:v:144:y:2006:i:1:p:287-300:10.1007/s10479-006-0015-x
    DOI: 10.1007/s10479-006-0015-x
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    References listed on IDEAS

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    1. Michel Bierlaire & Tsippy Lotan & Philippe Toint, 1997. "On The Overspecification of Multinomial and Nested Logit Models Due to Alternative Specific Constants," Transportation Science, INFORMS, vol. 31(4), pages 363-371, November.
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    3. Swait, Joffre, 2001. "Choice set generation within the generalized extreme value family of discrete choice models," Transportation Research Part B: Methodological, Elsevier, vol. 35(7), pages 643-666, August.
    4. Papola, Andrea, 2004. "Some developments on the cross-nested logit model," Transportation Research Part B: Methodological, Elsevier, vol. 38(9), pages 833-851, November.
    5. Langche Zeng, 2000. "A Heteroscedastic Generalized Extreme Value Discrete Choice Model," Sociological Methods & Research, , vol. 29(1), pages 118-144, August.
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