Uniform Inference for Subsampled Moment Regression

We propose a method for constructing a confidence region for the solution to a conditional moment equation. The method is built around a class of algorithms for nonparametric regression based on subsampled kernels. This class includes random forest regression. We bound the error in the confidence region's nominal coverage probability, under the restriction that the conditional moment equation of interest satisfies a local orthogonality condition. The method is applicable to the construction of confidence regions for conditional average treatment effects in randomized experiments, among many other similar problems encountered in applied economics and causal inference. As a by-product, we obtain several new order-explicit results on the concentration and normal approximation of high-dimensional $U$-statistics.