Double Machine Learning of Continuous Treatment Effects with Additive Instrumental Variables

Estimating causal effects of continuous treatments is a common problem in practice, for example, in studying average dose-response functions. Classical analyses typically assume that all confounders are fully observed, whereas in real-world applications, unmeasured confounding often persists. In this article, we propose a novel framework for the identification of average dose-response functions using instrumental variables, thereby mitigating bias induced by unobserved confounders. We introduce the concept of a uniform regular weighting function and consider covering the treatment space with a finite collection of open sets. On each of these sets, such a weighting function exists, allowing us to identify the average dose-response function locally within the corresponding region. For estimation, we propose an augmented inverse probability weighted score for continuous treatments with instrumental variables under a debiased machine learning framework, and provide practical guidance for adaptively constructing regular weighting functions from the data. We also propose a falsification test for the additive instrumental variable condition and develop a testing procedure for assessing the validity of regular weighting functions. We further establish the asymptotic properties of the resulting estimators based on kernel regression or empirical risk minimization as well as the theoretical validity of the proposed testing procedures. Finally, we conduct both simulation and empirical studies to assess the finite-sample performance of the proposed methods.