A Consistent Variance Estimator for 2SLS When Instruments Identify Different LATEs

Under treatment effect heterogeneity, an instrument identifies the instrument-specific local average treatment effect (LATE). If a regression model is estimated by the two-stage least squares (2SLS) using multiple instruments, then 2SLS is consistent for a weighted average of different LATEs. In practice, a rejection of the overidentifying restrictions test can indicate that there are more than one LATE. What is often overlooked in the literature is that the postulated moment condition evaluated at the 2SLS estimand does not hold unless those LATEs are the same. If so, the conventional heteroskedasticity-robust variance estimator would be inconsistent. However, 2SLS standard errors based on the conventional variance estimator have been reported even when the overidentifying restrictions test is rejected. I propose a consistent estimator for the asymptotic variance of 2SLS by using the result of Hall and Inoue (2003) on misspecified moment condition models. This can be used to correctly calculate the standard errors regardless of whether there are more than one LATE or not