  Heterogeneous effect estimation plays a crucial role in causal inference,
with applications across medicine and social science. Many methods for
estimating conditional average treatment effects (CATEs) have been proposed in
recent years, but there are important theoretical gaps in understanding if and
when such methods are optimal. This is especially true when the CATE has
nontrivial structure (e.g., smoothness or sparsity). Our work contributes in
several main ways. First, we study a two-stage doubly robust CATE estimator and
give a generic model-free error bound, which, despite its generality, yields
sharper results than those in the current literature. We apply the bound to
derive error rates in nonparametric models with smoothness or sparsity, and
give sufficient conditions for oracle efficiency. Underlying our error bound is
a general oracle inequality for regression with estimated or imputed outcomes,
which is of independent interest; this is the second main contribution. The
third contribution is aimed at understanding the fundamental statistical limits
of CATE estimation. To that end, we propose and study a local polynomial
adaptation of double-residual regression. We show that this estimator can be
oracle efficient under even weaker conditions, if used with a specialized form
of sample splitting and careful choices of tuning parameters. These are the
weakest conditions currently found in the literature, and we conjecture that
they are minimal in a minimax sense. We go on to give error bounds in the
non-trivial regime where oracle rates cannot be achieved. Some finite-sample
properties are explored with simulations.
