  We introduce two data-driven procedures for optimal estimation and inference
in nonparametric models using instrumental variables. The first is a
data-driven choice of sieve dimension for a popular class of sieve two-stage
least squares estimators. When implemented with this choice, estimators of both
the structural function $h_0$ and its derivatives (such as elasticities)
converge at the fastest possible (i.e., minimax) rates in sup-norm. The second
is for constructing uniform confidence bands (UCBs) for $h_0$ and its
derivatives. Our UCBs guarantee coverage over a generic class of
data-generating processes and contract at the minimax rate, possibly up to a
logarithmic factor. As such, our UCBs are asymptotically more efficient than
UCBs based on the usual approach of undersmoothing. As an application, we
estimate the elasticity of the intensive margin of firm exports in a
monopolistic competition model of international trade. Simulations illustrate
the good performance of our procedures in empirically calibrated designs. Our
results provide evidence against common parameterizations of the distribution
of unobserved firm heterogeneity.
