  We study identification and estimation of the average treatment effect in a
correlated random coefficients model that allows for first stage heterogeneity
and binary instruments. The model also allows for multiple endogenous variables
and interactions between endogenous variables and covariates. Our
identification approach is based on averaging the coefficients obtained from a
collection of ordinary linear regressions that condition on different
realizations of a control function. This identification strategy suggests a
transparent and computationally straightforward estimator of a trimmed average
treatment effect constructed as the average of kernel-weighted linear
regressions. We develop this estimator and establish its
$\sqrt{n}$--consistency and asymptotic normality. Monte Carlo simulations show
excellent finite-sample performance that is comparable in precision to the
standard two-stage least squares estimator. We apply our results to analyze the
effect of air pollution on house prices, and find substantial heterogeneity in
first stage instrument effects as well as heterogeneity in treatment effects
that is consistent with household sorting.
