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.