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• W1509263037 abstract "Introduction: Recent work in the fields of transport, environmental economics and marketing (Daly et al. 2014; Guevara and Ben-Akiva 2013a, 2013b; von Haefen and Domanski 2013; Keane and Wasi 2012) has renewed the attention towards the issue of sampling of alternatives in discrete choice modelling. McFadden (1978) proved that, after applying a sampling correction to the utility function, consistent parameter estimates can be obtained by estimating a basic multinomial logit model over a subset of randomly sampled alternatives. The major benefit of sampling of alternatives is that significant reductions in estimation time can be attained at relatively low costs. Most empirical applications nowadays apply model specifications beyond the multinomial logit model. Multivariate Extreme Value (MEV) type of models, such as the nested logit model (Daly, 1987), and mixed logit models (Revelt and Train, 1998) have become state of the art and more recently hybrid choice models (Ben-Akiva et al. 2002) have increased in popularity. Guevara and Ben-Akiva (2013a) and (2013b) proved that McFadden's result partly carries over to these advanced model structures. Consistent parameter estimates can be obtained when, in addition to the McFadden correction term, the analyst also corrects for the imperfect representation of respectively the LogSum (MEV) and individual level parameters (mixed logit) in the sampled model. Bayesian estimation methods have also increased in popularity in the discrete choice modelling literature (Daziano et al., 2013; Dekker et al., 2014). Bayesian estimation of discrete choice models is particularly fruitful when latent constructs are included in the likelihood function. Significant reductions in estimation time over classical (simulated) maximum likelihood approaches are typically obtained due to augmenting latent constructs, such as individual level parameters in mixed logit models (Tanner and Wong, 1987; Train, 2009). The computational benefits of sampling of alternatives and Bayesian estimation arise in alternative parts of the estimation procedure. Their joint implementation may initiate additional time savings and enable the implementation of more flexible discrete choice models in large-scale (travel)-demand models. A reduction in the gap between the type of discrete choice models applied in most (small-scale) Stated Preference survey and (large-scale) Revealed Preference studies can be foreseen. Objectives and contributions: McFadden (1978) established the asymptotic sampling properties for the parameters of interest. Indeed, when sample sizes become sufficiently large the Bernstein-von Mises theorem (Train, 2009) applies and consistency of Bayesian parameter estimates under sampling of alternatives can be established. The interest of this paper is, however, in the performance of McFadden’s correction factor in the domain of finite sample sizes. First, I extend the work of Keane and Wasi (2012) by proving that for a given set of parameters and under positive conditioning, the McFadden correction factor minimises the expected information loss when moving from the true likelihood to the quasi (sampled) likelihood function. This result holds for any sample size. Ex ante, the corrected sampling likelihood thus provides the best approximation of the true likelihood. For Bayesian estimation this is an important result since it implies that there is minimal (expected) information loss in the `sampled’ posterior relative to the true posterior density. In other words, by including the McFadden correction factor the sampled model is able to most accurately describe the the posterior density. Sampling of alternatives can therefore be applied in a Bayesian context and for small sample sizes. Second, I show that the correction factors developed by Guevara and Ben-Akiva (2013b) for mixed logit models are not necessary in a Bayesian mixed logit model based on augmented individual level parameter estimates. The result follows directly from the fact that the McFadden correction factor provides the most accurate individual-level posterior and therefore also the most accurate representation of the posterior of the hyper-parameters of the mixing density. This result is particularly interesting since it confirms why Azaiez (2010), Keane and Wasi (2012), Lemp and Kockelman (2012), Guevara and Ben-Akiva (2013b), and von Haefen and Domanski (2013) all find that a naive approach to sampling of alternatives in mixed logit models performs relatively well. Monte Carlo simulations In the simulation part of the paper, sampling of alternatives and Bayesian estimation are combined using estimation approaches with and without the McFadden correction factor. Of particular interest is the sensitivity of parameter estimates to alternative sampling protocols. Figure 1 provides preliminary results of 100 different sampling runs for a panel mixed logit model where 1,000 respondents each make 5 choices over 100 alternatives. Even at low sampling rates (5%), the McFadden correction factor appears to provide a rather accurate approximation of the parameter estimates based on the full sample. The size of the bias and mean square errors does not appear to increase when comparing the Bayesian approach against the classical maximum likelihood approach. As expected, coverage improves as more alternatives are sampled. In all, these results provide sufficient support to pursue this line of research. References: Azaiez, I. (2010). Sampling of alternatives for logit mixture models. Master Thesis EPFL Lausanne, http://transp-or.epfl.ch/documents/masterTheses/AZAIEZ10.pdf Ben-Akiva, M., Mcfadden, D., Train, K., Walker, J., Bhat, C., Bierlaire, M., Bolduc, D., Boersch-Supan, A., Brownstone, D., Bunch, D., Daly, A., De Palma, A., Gopinath, D., Karlstrom, A., and Munizaga, M. (2002). Hybrid choice models: Progress and challenges. Marketing Letters , 13(3), 163-175. Daly, A. (1987). Estimating `tree' logit models. Transportation Research Part B: Methodological , 21(4), 251-267. Daly, A. Hess, S. and Dekker T. (2014). Practical solutions for sampling alternatives in large scale models,  Transportation Research Record, forthcoming. Daziano, R.A., Miranda-Moreno, L., and Heydari, S. (2013). Computational bayesian statistics in transportation modeling: From road safety analysis to discrete choice. Transport Reviews , 33(5), 570-592. Dekker, T., Hess, S., Arentze, T., and Chorus, C. (2014). Incorporating needs-satisfaction in a discrete choice model of leisure activities. Journal of Transport Geography , 38, 66 -74. Guevara, C.A. and Ben-Akiva, M. (2013a), Sampling of alternatives in multivariate extreme value (MEV) models, Transportation Research Part B , 48, 31–52. Guevara, C.A. and Ben-Akiva, M. (2013b), Sampling of alternatives in logit mixture models, Transportation Research Part B , 58, 185-198. von Haefen, R.H. and Domanski, A. (2013), Estimating mixed logit models with large choice sets. Paper presented at the 3 rd International Choice Modelling Conference , Sydney, July 2013. Keane, M.  and Wasi, N.  (2012), Estimation of discrete choice models with many alternatives using random subsets of the full choice set: with an application to demand for frozen pizza. www.economics.ox.ac.uk/materials/papers/12755/2012-W13.pdf Lemp, J.D. and Kockelman, K.M. (2012), Strategic sampling for large choice sets in estimation and application. Transportation Research Part A: Policy and Practice , 46(3), 602-613. McFadden, D.L. (1978), Modelling the choice of residential location, in Karlqvist, A., Lundqvist, L., Snickars, F. and Weibull, J., Spatial interaction theory and residential location, North-Holland, pp. 75-96. Revelt, D. and Train, K. (1998). Mixed logit with repeated choices: Households' choices of appliance efficiency level. The Review of Economics and Statistics , 80(4), 647-657. Tanner, M.A. and Wong, W.H. (1987). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82(398), 528-540. Train, K. (2009), Discrete Choice Methods with Simulation, second edition, Cambridge University Press, Cambridge, MA." @default.
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