A note on the different interpretation of the correlation parameters in the Bivariate Probit and the Recursive Bivariate Probit

This note shows that, if a Bivariate Probit (BP) model is estimated on data arising from a Recursive Bivariate Probit (RBP) process, the resulting BP correlation parameter is a weighted average of the RBP correlation parameter and the parameter associated to the endogenous binary variable. Two corollaries follow this proposition: i) the interpretation of the correlation parameter in the RBP is not the same as in the BP —i.e. the RBP correlation parameter does not necessarily reflect the correlation between the binary variables under study; and ii) a zero correlation parameter in a BP model, usually interpreted as evidence of independence between the binary variables under study, may actually mask the presence of an RBP process.