The order of variables, simulation noise, and accuracy of mixed logit estimates

The simulated choice probabilities in mixed logit models are usually approximated numerically using Halton or random draws from a multivariate mixing distribution for the random parameters. Theoretically, the order in which the estimated variables enter the model should not matter. However, in practice, simulation noise inherent in the numerical procedure leads to differences in the magnitude of the estimated coefficients depending on the arbitrary order in which the random variables are estimated. The problem is exacerbated when a low number of draws are used or if correlation among coefficients is allowed. In particular, the Cholesky factorization procedure, which is used to incorporate correlation into the model, propagates simulation noise in the estimate of one coefficient to estimates of all subsequent coefficients in the model. Ignoring the potential ordering effects in simulated maximum likelihood estimation methods may seriously compromise the ability of replicating the results and can inadvertently influence policy recommendations. We find that better estimation accuracy is achieved with Halton draws using small prime numbers as it is the case for small integrating dimensions; but random draws provide better accuracy than Halton draws from large prime numbers as it is normally the case in high integrating dimensions. With correlation, the standard deviations have very large fluctuations depending on the order of the variables, affecting the conclusions regarding heterogeneity of preferences.