Revisiting McFadden’s correction factor for sampling of alternatives in multinomial logit and mixed multinomial logit models

When estimating multinomial logit (MNL) models where choices are made from a large set of available alternatives computational benefits can be achieved by estimating a quasi-likelihood function based on a sampled subset of alternatives in combination with ‘McFadden’s correction factor’. In this paper, we theoretically prove that McFadden’s correction factor minimises the expected information loss in the parameters of interest and thereby has convenient finite (and large sample) properties. That is, in the context of Bayesian estimation the use of sampling of alternatives in combination with McFadden’s correction factor provides the best approximation of the posterior distribution for the parameters of interest irrespective of sample size. As sample sizes become sufficiently large consistent point estimates for MNL can be obtained as per McFadden’s original proof. McFadden’s correction factor can therefore effectively be applied in the context of Bayesian MNL models. We extend these results to the context of mixed multinomial logit models (MMNL) by using the property of data augmentation in Bayesian estimation. McFadden’s correction factor minimises the expected information loss with respect to the augmented individual-level parameters, and in turn also for the population parameters characterising the shape and location of the mixing density in MMNL. Again, the results apply to finite and large samples and most importantly circumvent the need for additional correction factors previously identified for estimating MMNL models using maximum simulated likelihood. Monte Carlo simulations validate this result for sampling of alternatives in Bayesian MMNL models.