  Maximum simulated likelihood estimation of mixed multinomial logit (MMNL) or
probit models requires evaluation of a multidimensional integral. Quasi-Monte
Carlo (QMC) methods such as shuffled and scrambled Halton sequences and
modified Latin hypercube sampling (MLHS) are workhorse methods for integral
approximation. A few earlier studies explored the potential of sparse grid
quadrature (SGQ), but this approximation suffers from negative weights. As an
alternative to QMC and SGQ, we looked into the recently developed designed
quadrature (DQ) method. DQ requires fewer nodes to get the same level of
accuracy as of QMC and SGQ, is as easy to implement, ensures positivity of
weights, and can be created on any general polynomial spaces. We benchmarked DQ
against QMC in a Monte Carlo study under different data generating processes
with a varying number of random parameters (3, 5, and 10) and
variance-covariance structures (diagonal and full). Whereas DQ significantly
outperformed QMC in the diagonal variance-covariance scenario, it could also
achieve a better model fit and recover true parameters with fewer nodes (i.e.,
relatively lower computation time) in the full variance-covariance scenario.
Finally, we evaluated the performance of DQ in a case study to understand
preferences for mobility-on-demand services in New York City. In estimating
MMNL with five random parameters, DQ achieved better fit and statistical
significance of parameters with just 200 nodes as compared to 1000 QMC draws,
making DQ around five times faster than QMC methods.
