Probit models: Regression parameter estimation using the ML principle despite misspecification of the correlation structure

In this paper it is shown that using the maximum likelihood (ML)
principle for the estimation of multivariate probit models leads to
consistent and normally distributed pseudo maximum likelihood
regression parameter estimators (PML estimators) even if the `true'
correlation structure of the responses is misspecified. As a
consequence, e.g. the PML estimator of the random effects probit
model may be used to estimate the regression parameters of a model
with any `true' correlation structure. This result is independent
of the kind of covariates included in the model. The results of a
Monte Carlo experiment show that the PML estimator of the
independent binary probit model is inefficient relative to the PML
estimator of the random effects binary panel probit model and two
alternative estimators using the `generalized estimating equations'
approach proposed by Liang and Zeger (1986), if the `true'
correlations are high. If the `true' correlations are low, the
differences between the estimators are small in finite samples and
for the models used. Generally, the PML estimator of the random
effects probit panel model is very efficient relative to the GEE
estimators. Therefore, if correlations between binary responses
cannot be ruled out and the `true' structure of association is
unknown or infeasible to estimate, the PML estimator of the random
effects probit model is recommended.