Simultaneous inference for Best Linear Predictor of the Conditional Average Treatment Effect and other structural functions

This paper provides estimation and inference methods for a structural function, such as Conditional Average Treatment Effect (CATE), based on modern machine learning (ML) tools. We assume that such function can be represented as a conditional expectation = of a signal , where is the unknown nuisance function. In addition to CATE, examples of such functions include regression function with Partially Missing Outcome and Conditional Average Partial Derivative. We approximate by a linear form , where is a vector of the approximating functions and is the Best Linear Predictor. Plugging in the first-stage estimate into the signal , we estimate via ordinary least squares of on . We deliver a high-quality estimate of the pseudo-target function , that features (a) a pointwise Gaussian approximation of at a point , (b) a simultaneous Gaussian approximation of uniformly over x, and (c) optimal rate of convergence of to uniformly over x. In the case the misspecication error of the linear form decays sufficiently fast, these approximations automatically hold for the target function instead of a pseudo-target . The first stage nuisance parameter is allowed to be high-dimensional and is estimated by modern ML tools, such as neural networks, -shrinkage estimators, and random forest. Using our method, we estimate the average price elasticity conditional on income using Yatchew and No (2001) data and provide uniform confidence bands for the target regression function.