Local Effects of Continuous Instruments without Positivity

TL;DR

This paper introduces a new approach for estimating local causal effects with continuous instruments that avoids the strong positivity assumption, using stochastic interventions and doubly robust estimators.

Contribution

It develops a novel family of causal estimands based on stochastic interventions that do not require positivity, along with doubly robust estimators and methods for sensitivity analysis.

Findings

Estimators are asymptotically normal under weak conditions.

Simulation studies demonstrate estimator performance.

Application shows practical feasibility in medical treatment evaluation.

Abstract

Instrumental variables are 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. In many settings, however, the instrument is continuous. Standard estimation methods can be applied with continuous instruments, but they require strong assumptions. While recent work has introduced more flexible estimation approaches, these methods require a positivity assumption that is implausible in many applications. We derive a novel family of causal estimands using stochastic dynamic interventions that allows a range of intervention distributions that are continuous with respect to the observed distribution of the instrument. These estimands focus on a specific 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 of sensitivity analysis. We evaluate our methods via simulation and demonstrate their feasibility using an application on the effectiveness of surgery for specific emergency conditions.