  In theory, two-stage least squares (TSLS) identifies a weighted average of
covariate-specific local average treatment effects (LATEs) from a saturated
specification without making parametric assumptions on how available covariates
enter the model. In practice, TSLS is severely biased when saturation leads to
a number of control dummies that is of the same order of magnitude as the
sample size, and the use of many, arguably weak, instruments. This paper
derives asymptotically valid tests and confidence intervals for an estimand
that identifies the weighted average of LATEs targeted by saturated TSLS, even
when the number of control dummies and instrument interactions is large. The
proposed inference procedure is robust against four key features of saturated
economic data: treatment effect heterogeneity, covariates with rich support,
weak identification strength, and conditional heteroskedasticity.
