Jacob Nielsen Arendt and Holm Anders Larsen, Probit Models with Dummy Endogenous Regressors (February 2006), available at SSRN: http://ssrn.com/abstract=994189.
This study considers heckit-type approximations useful for a number of different trivariate probit models. They are simple to use and have no convergence problems like full maximum likelihood. Simulations show that a heckit and a least squares approximation perform as well as the trivariate probit estimator in small samples when the degree of endogeneity is not too severe. A simple double-heckit and a heteroskedasticity corrected heckit approximation seem particularly robust and promising for testing exogeneity. The methods are used to estimate the impact of physician advice on physical activity, where the heckit approximations work as well as full maximum likelihood.