New results on the asymptotic and finite sample properties of the MaCML approach to multinomial probit model estimation

TL;DR

This paper investigates the properties of the MaCML approach for multinomial probit model estimation, revealing issues with consistency, bias, and performance variability across different approximation methods in finite samples and asymptotic scenarios.

Contribution

It provides a critical evaluation of the MaCML method, highlighting its limitations and the influence of various factors on estimation accuracy, thus guiding future research and application choices.

Findings

MaCML may produce inconsistent estimators for certain Gaussian approximations.

Bias in parameter estimates can be substantial, though implied probabilities are less affected.

Performance of approximation methods varies with system, ordering, and stopping criteria.

Abstract

In this paper the properties of the maximum approximate composite marginal likelihood (MaCML) approach to the estimation of multinomial probit models (MNP) proposed by Chandra Bhat and coworkers is investigated in finite samples as well as with respect to asymptotic properties. Using a small illustration example it is proven that the approach does not necessarily lead to consistent estimators for four different types of approximation of the Gaussian cumulative distribution function (including the Solow-Joe approach proposed by Bhat). It is shown that the bias of parameter estimates can be substantial (while typically it is small) and the bias in the corresponding implied probabilities is small but non-negligible. Furthermore in finite sample it is demonstrated by simulation that between two versions of the Solow-Joe method and two versions of the Mendell-Elston approximation no method dominates the others in terms of accuracy and numerical speed. Moreover the system to be estimated, the ordering of the components in the approximation method and even the tolerance used for stopping the numerical optimization routine all have an influence on the relative performance of the procedures corresponding to the various approximation methods. Jointly the paper thus points towards eminent research needs in order to decide on the method to use for a particular estimation problem at hand.