  This paper studies nonparametric series estimation and inference for the
effect of a single variable of interest x on an outcome y in the presence of
potentially high-dimensional conditioning variables z. The context is an
additively separable model E[y|x, z] = g0(x) + h0(z). The model is
high-dimensional in the sense that the series of approximating functions for
h0(z) can have more terms than the sample size, thereby allowing z to have
potentially very many measured characteristics. The model is required to be
approximately sparse: h0(z) can be approximated using only a small subset of
series terms whose identities are unknown. This paper proposes an estimation
and inference method for g0(x) called Post-Nonparametric Double Selection which
is a generalization of Post-Double Selection. Standard rates of convergence and
asymptotic normality for the estimator are shown to hold uniformly over a large
class of sparse data generating processes. A simulation study illustrates
finite sample estimation properties of the proposed estimator and coverage
properties of the corresponding confidence intervals. Finally, an empirical
application to college admissions policy demonstrates the practical
implementation of the proposed method.
