  Under treatment effect heterogeneity, an instrument identifies the
instrument-specific local average treatment effect (LATE). With multiple
instruments, two-stage least squares (2SLS) estimand is a weighted average of
different LATEs. 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, and 2SLS standard errors based on
such estimators would be incorrect. I derive the correct asymptotic
distribution, and propose a consistent asymptotic variance estimator by using
the result of Hall and Inoue (2003, Journal of Econometrics) on misspecified
moment condition models. This can be used to correctly calculate the standard
errors regardless of whether there is more than one LATE or not.
