Designed Quadrature to Approximate Integrals in Maximum Simulated Likelihood Estimation

TL;DR

This paper introduces designed quadrature (DQ), a novel integral approximation method that outperforms traditional quasi-Monte Carlo techniques in maximum simulated likelihood estimation, especially in complex models with fewer nodes and lower computation time.

Contribution

The paper presents designed quadrature (DQ), a new integral approximation method that is more efficient and easier to implement than existing QMC and SGQ methods, ensuring positive weights and requiring fewer nodes.

Findings

DQ outperforms QMC in diagonal covariance scenarios.

DQ achieves better model fit with fewer nodes in full covariance models.

DQ is approximately five times faster than QMC in case studies.

Abstract

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.