Binary outcomes, OLS, 2SLS and IV probit

For a binary outcome Y, generated by a simple threshold crossing model with a single exogenous normally distributed explanatory variable X, the OLS estimator of the coefficient on X in a linear probability model is a consistent estimator of the average partial effect of X. Even in this very simple setting, we show that when allowing for X to be endogenously determined, the 2SLS estimator, using a normally distributed instrumental variable Z, does not identify the same causal parameter. It instead estimates the average partial effect of Z, scaled by the coefficient on Z in the linear first-stage model for X, denoted γ1, or equivalently, it estimates the average partial effect of the population predicted value of X, Zγ1. These causal parameters can differ substantially as we show for the normal Probit model, which implies that care has to be taken when interpreting 2SLS estimation results in a linear probability model. Under joint normality of the error terms, IV Probit maximum likelihood estimation does identify the average partial effect of X. The two-step control function procedure of Rivers and Vuong can also estimate this causal parameter consistently, but a double averaging is needed, one over the distribution of the first-stage error V and one over the distribution of X. If instead a single averaging is performed over the joint distribution of X and V, then the same causal parameter is estimated as the one estimated by the 2SLS estimator in the linear probability model. The 2SLS estimator is a consistent estimator when the average partial effect is equal to 0, and the standard Wald test for this hypothesis has correct size under strong instrument asymptotics. We show that, in general, the standard weak instrument first-stage F-test interpretations do not apply in this setting.