Local Effects of Continuous Instruments without Positivity

Abstract

Instrumental variables have become a popular study design for the estimation of treatment effects in the presence of unobserved confounders. In the canonical instrumental variables design, the instrument is a binary variable, and most extant methods are tailored to this context. In many settings, however, the instrument is a continuous measure. Standard estimation methods can be applied with continuous instruments, but they require strong assumptions regarding functional form. Moreover, while some recent work has introduced more flexible approaches for continuous instruments, these methods require an assumption known as positivity that is unlikely to hold in many applications. We derive a novel family of causal estimands using a stochastic dynamic intervention framework that considers a range of intervention distributions that are absolutely continuous with respect to the observed distribution of the instrument. These estimands focus on a specific form of local effect but do not require a positivity assumption. Next, we develop doubly robust estimators for these estimands that allow for estimation of the nuisance functions via nonparametric estimators. We use empirical process theory and sample splitting to derive asymptotic properties of the proposed estimators under weak conditions. In addition, we derive methods for profiling the principal strata as well as a method for sensitivity analysis for assessing robustness to an underlying monotonicity assumption. We evaluate our methods via simulation and demonstrate their feasibility using an application on the effectiveness of surgery for specific emergency conditions.