Biases in Maximum Simulated Likelihood Estimation of Bivariate Models

This paper finds that the maximum simulated likelihood (MSL) estimator produces substantial biases when applied to the bivariate normal distribution. A specification of the random parameter bivariate normal model is considered, in which a direct comparison between the MSL and maximum likelihood (ML) estimators is feasible. The analysis shows that MSL produces biased results for the correlation parameter. This paper also finds that the MSL estimator is biased for the bivariate Poisson-lognormal model, developed by Munkin and Trivedi (1999. “Simulated Maximum Likelihood Estimation of Multivariate Mixed-Poisson Regression Models, with Application.” The Econometrics Journal 2: 29–48). A simulation study is conducted, which shows that MSL leads to serious inferential biases, especially large when variance parameters in the true data generating process are small. The MSL estimator produces biases in the estimated marginal effects, conditional means and probabilities of count outcomes.

Designed quadrature to approximate integrals in maximum simulated likelihood estimation

Maximum simulated likelihood estimation of mixed multinomial logit models requires evaluation of a multidimensional integral. Quasi-Monte Carlo (QMC) methods such as Halton sequences and modified Latin hypercube sampling are workhorse methods for integral approximation. Earlier studies explored the potential of sparse grid quadrature (SGQ), but SGQ 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 space. We benchmarked DQ against QMC in a Monte Carlo and an empirical study. DQ outperformed QMC in all considered scenarios, is practice-ready and has potential to become the workhorse method for integral approximation.

BAYESIAN INFERENCE BASED ONLY ON SIMULATED LIKELIHOOD: PARTICLE FILTER ANALYSIS OF DYNAMIC ECONOMIC MODELS

We note that likelihood inference can be based on an unbiased simulation-based estimator of the likelihood when it is used inside a Metropolis–Hastings algorithm. This result has recently been introduced in statistics literature by Andrieu, Doucet, and Holenstein (2010, Journal of the Royal Statistical Society, Series B, 72, 269–342) and is perhaps surprising given the results on maximum simulated likelihood estimation. Bayesian inference based on simulated likelihood can be widely applied in microeconomics, macroeconomics, and financial econometrics. One way of generating unbiased estimates of the likelihood is through a particle filter. We illustrate these methods on four problems, producing rather generic methods. Taken together, these methods imply that if we can simulate from an economic model, we can carry out likelihood–based inference using its simulations.