Equivalence between the GNL models and entropy models: Equivalence in disaggregate level

This paper provides a proof that the parameter estimation problem in the generalized nested logit (GNL) models used in marketing science and transportation planning fields, is equivalent to information minimization problems with constraints in disaggregate level. To be specific, equivalence between log-likelihood maximization estimation problems of the GNL model and information minimization problems is proved using the two-stage optimization problem. In this problem, parameters in definite utility functions and allocation parameters correspond to an alternative level and similarity parameters correspond to a nest level. As part of the process to provide the proof, we show that constraints on allocation parameters are naturally considered in the log-likelihood maximization estimation problem of the GNL model. Using the properties of allocation parameters, we show new methods for parameter estimation in the GNL model, which rectify the heuristic methods proposed in Vovsha. First, we propose an estimation method using parameters in two-stage estimation as initial values in simultaneous estimation. Second, we propose a method using the primal-dual interior point method which utilizes duality in each stage of two-stage estimation in the GNL model. The GNL model includes the multinomial logit, the nested logit, the cross-nested logit and the pairwise combination logit models, and equivalence between all these models and entropy models are proved in this paper.