Data-Driven Uniform Inference for General Continuous Treatment Models via Minimum-Variance Weighting

Ai et al. (2021) studied the estimation of a general dose-response function (GDRF) of a continuous treatment that includes the average dose-response function, the quantile dose-response function, and other expectiles of the dose-response distribution. They specified the GDRF as a parametric function of the treatment status only and proposed a weighted regression with the weighting function estimated using the maximum entropy approach. This paper specifies the GDRF as a nonparametric function of the treatment status, proposes a weighted local linear regression for estimating GDRF, and develops a bootstrap procedure for constructing the uniform confidence bands. We propose stable weights with minimum sample variance while eliminating the sample association between the treatment and the confounding variables. The proposed weights admit a closed-form expression, allowing them to be computed efficiently in the bootstrap sampling. Under certain conditions, we derive the uniform Bahadur representation for the proposed estimator of GDRF and establish the validity of the corresponding uniform confidence bands. A fully data-driven approach to choosing the undersmooth tuning parameters and a data-driven bias-control confidence band are included. A simulation study and an application demonstrate the usefulness of the proposed approach.

💡 Research Summary

This paper develops a comprehensive framework for estimating and performing uniform inference on the general dose‑response function (GDRF) associated with a continuous treatment variable. The GDRF, defined as the minimizer of an expected loss L(Y* − a) for each treatment level t, encompasses many causal effect measures such as the average dose‑response function (ADRF), quantile dose‑response function (QDRF), and various expectiles, depending on the choice of L. Unlike the earlier work of Ai et al. (2021), which imposed a parametric form on the GDRF and estimated the stabilizing weight π(T,X) via a maximum‑entropy approach, the present study treats the GDRF as a fully non‑parametric function of t and introduces a weighted local linear regression estimator that directly targets g(t) and its derivative g′(t).

A central methodological innovation is the construction of minimum‑variance weights. Starting from the moment condition E