  Continuous treatments have posed a significant challenge for causal
inference, both in the formulation and identification of scientifically
meaningful effects and in their robust estimation. Traditionally, focus has
been placed on techniques applicable to binary or categorical treatments with
few levels, allowing for the application of propensity score-based methodology
with relative ease. Efforts to accommodate continuous treatments introduced the
generalized propensity score, yet estimators of this nuisance parameter
commonly utilize parametric regression strategies that sharply limit the
robustness and efficiency of inverse probability weighted estimators of causal
effect parameters. We formulate and investigate a novel, flexible estimator of
the generalized propensity score based on a nonparametric function estimator
that provably converges at a suitably fast rate to the target functional so as
to facilitate statistical inference. With this estimator, we demonstrate the
construction of nonparametric inverse probability weighted estimators of a
class of causal effect estimands tailored to continuous treatments. To ensure
the asymptotic efficiency of our proposed estimators, we outline several
non-restrictive selection procedures for utilizing a sieve estimation framework
to undersmooth estimators of the generalized propensity score. We provide the
first characterization of such inverse probability weighted estimators
achieving the nonparametric efficiency bound in a setting with continuous
treatments, demonstrating this in numerical experiments. We further evaluate
the higher-order efficiency of our proposed estimators by deriving and
numerically examining the second-order remainder of the corresponding efficient
influence function in the nonparametric model. Open source software
implementing our proposed estimation techniques, the haldensify R package, is
briefly discussed.
