A Monte Carlo experiment to analyze the curse of dimensionality in estimating random coefficients models with a full variance–covariance matrix Elisabetta Cherchi a,⇑ , Cristian Angelo Guevara b,1 a Technical University of Denmark, Department of Transport, Bygningstorvet 116 Vest 2800 Kgs. Lyngby, Denmark b Universidad de los Andes, Facultad de Ingeniería y Ciencias Aplicadas, San Carlos de Apoquindo 2200, Santiago, Chile article info Article history: Received 29 August 2011 Accepted 10 October 2011 Keywords: Curse of dimensionality Random coefficients models Estimation methods Monte Carlo experiments abstract When the dimension of the vector of estimated parameters increases, simulation based methods become impractical, because the number of draws required for estimation grows exponentially with the number of parameters. In simulation methods, the lack of empirical identification when the number of parameters increases is usually known as the ‘‘curse of dimensionality’’ in the simulation methods. We investigate this problem in the case of the random coefficients Logit model. We compare the traditional Maximum Simulated Likelihood (MSL) method with two alternative estimation methods: the Expectation–Max- imization (EM) and the Laplace Approximation (HH) methods that do not require simula- tion. We use Monte Carlo experimentation to investigate systematically the performance of the methods under different circumstances, including different numbers of variables, sam- ple sizes and structures of the variance–covariance matrix. Results show that indeed MSL suffers from lack of empirical identification as the dimensionality grows while EM deals much better with this estimation problem. On the other hand, the HH method, although not being simulation-based, showed poor performance with large dimensions, principally because of the necessity of inverting large matrices. The results also show that when MSL is empirically identified this method seems superior to EM and HH in terms of ability to recover the true parameters and estimation time. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The Mixed Multinomial Logit (MMNL) is the most popular model to account for random coefficients. Its strong point is in its simple theoretical formulation, which features a mixture of the standard Multinomial Logit (MNL) integrated out over a density of coefficients. The MMNL has been known since the early 1980 but, despite its simple formulation, it has become fully applicable only recently with the advent of simulation techniques and more computational capabilities. The most widely used technique to estimate the MMNL model is the Maximum Simulated Likelihood (MSL) method (see, e.g. Train, 2009). Although very practical, simulation techniques show some drawbacks. A critical problem is that the MSL estimators are downward biased for a finite number of draws. This problem results from the fact that the MSL method considers a nonlinear transformation of an unbiased estimator of the choice probabilities (Börsch-Supan and Hajivassiliou, 1993). As reported by 0191-2615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2011.10.006 ⇑ Corresponding author. Tel.: +45 45256549; fax: +45 45936533. E-mail addresses: elich@transport.dtu.dk (E. Cherchi), aguevara@uandes.cl (C.A. Guevara). 1 Tel.: +56 24129364; fax: +56 24129642. Transportation Research Part B 46 (2012) 321–332 Contents lists available at SciVerse ScienceDirect Transportation Research Part B journal homepage: www.elsevier.com/locate/trb