  The purpose of this paper is to provide guidelines for empirical researchers
who use a class of bivariate threshold crossing models with dummy endogenous
variables. A common practice employed by the researchers is the specification
of the joint distribution of the unobservables as a bivariate normal
distribution, which results in a bivariate probit model. To address the problem
of misspecification in this practice, we propose an easy-to-implement
semiparametric estimation framework with parametric copula and nonparametric
marginal distributions. We establish asymptotic theory, including root-n
normality, for the sieve maximum likelihood estimators that can be used to
conduct inference on the individual structural parameters and the average
treatment effect (ATE). In order to show the practical relevance of the
proposed framework, we conduct a sensitivity analysis via extensive Monte Carlo
simulation exercises. The results suggest that the estimates of the parameters,
especially the ATE, are sensitive to parametric specification, while
semiparametric estimation exhibits robustness to underlying data generating
processes. We then provide an empirical illustration where we estimate the
effect of health insurance on doctor visits. In this paper, we also show that
the absence of excluded instruments may result in identification failure, in
contrast to what some practitioners believe.
