Estimating Random Coefficient Logit Models with Full Covariance Matrix: Comparing Performance of Mixed Logit and Laplace Approximation Methods

In the mixed logit model, random coefficients are estimated with the use of simulation methods to approximate the integral of the likelihood over the density of the coefficients. Although the model is very efficient, when it takes into account the full variance–covariance matrix of the coefficients, estimation problems may well arise and the simulation methods become impracticable as the number of coefficients increases–-the well-known curse of dimensionality. With simulated data in this research, the classical simulation approach of the random coefficient mixed logit model is compared with a new method proposed by Harding and Hausman, which is based on the Laplace approximation of the probability integrals to avoid simulation. The comparison carried out in this research differs from that of Harding and Hausman in two ways: (a) observed choices are used instead of observed probabilities and (b) the potential effect of the curse of dimensionality is formally explored by means of synthetic data. Contrary to Harding and Hausman's results, these experiments show that mixed logit is not only capable of estimating the variance–covariance matrix, but when both methods were estimable, it also always outperforms the Laplace approximation method. Estimates for the variance–covariance matrix obtained with both methods are, for almost all cases studied, remarkably poor. As expected, the Laplace approximation method is estimable for a larger number of random coefficients, arguably because the curse of dimensionality makes simulation in mixed logit impracticable. The paper concludes with a discussion of potential lines of improvement in the development of methods to estimate random coefficient models with full variance–covariance matrices.