[Paper Review] Causal Inference for Nonlinear Outcome Models with Possibly Invalid Instrumental Variables

Sai Li, Zijian Guo|arXiv (Cornell University)|Oct 19, 2020

Advanced Causal Inference Techniques56 references4 citations

TL;DR

This paper proposes a novel semi-parametric method, SpotIV, for causal inference in nonlinear outcome models with possibly invalid instrumental variables. It introduces new identifiability conditions—dimension reduction and majority rule—that allow consistent estimation of the average structural function and conditional average treatment effect even when most instruments are invalid, establishing asymptotic normality and valid bootstrap confidence intervals.

ABSTRACT

Instrumental variable methods are widely used for inferring the causal effect in the presence of unmeasured confounders. Existing instrumental variable methods for nonlinear outcome models require stringent identifiability conditions. This paper considers a flexible semi-parametric potential outcome model that allows for possibly invalid instruments. We propose new identifiability conditions to identify the causal parameters when the majority of the instrumental variables are valid. We devise a novel inference procedure for a new average structural function and the conditional average treatment effect. We establish the asymptotic normality of the proposed estimators and construct confidence intervals for the causal estimands by bootstrap. The proposed method is demonstrated in large-scale simulation studies and is applied to infer the effect of income on house ownership.

Motivation & Objective

• To address the challenge of causal inference in nonlinear outcome models when instrumental variables may be invalid.
• To relax the stringent identifiability conditions required by existing methods, particularly the assumption of valid control functions.
• To develop a method that remains consistent even when the majority of instruments are invalid, relying only on the majority rule and dimension reduction.
• To provide asymptotically normal estimators for the average structural function and conditional average treatment effect with valid confidence intervals via bootstrap.

Proposed method

• Proposes a three-step inference procedure, SpotIV, for semi-parametric outcome models with possibly invalid IVs.
• Introduces a new identifiability condition based on dimension reduction to handle high-dimensional unmeasured confounders.
• Imposes a majority rule condition, assuming at least half of the instruments are valid, to ensure identification of causal parameters.
• Uses kernel-based estimation with bandwidth selection to approximate conditional expectations and gradients in the estimation of the average structural function.
• Employs a bootstrap procedure to construct asymptotically valid confidence intervals for causal estimands.
• Derives asymptotic normality of the estimators under regularity conditions, including smoothness of the outcome model and kernel bandwidth constraints.

Experimental results

Research questions

• RQ1 Can causal effects be consistently estimated in nonlinear outcome models when some instrumental variables are invalid?
• RQ2 What minimal identifying assumptions are sufficient to identify the average structural function and conditional average treatment effect under potential IV invalidity?
• RQ3 How can the majority rule and dimension reduction be formalized as new identifiability conditions in semi-parametric models?
• RQ4 What is the asymptotic distribution of the proposed estimators under these new conditions?
• RQ5 Can valid confidence intervals be constructed without prior knowledge of which instruments are valid?

Key findings

• The proposed SpotIV method achieves asymptotic normality of the estimators for the average structural function and conditional average treatment effect under the new identifiability conditions.
• The asymptotic variance of the estimators is of the order $ O(1/\sqrt{nh^2}) $, where $ h $ is the bandwidth and $ n $ is the sample size.
• The method remains consistent even when more than half of the instruments are invalid, provided the majority rule and dimension reduction conditions hold.
• Bootstrap-based confidence intervals are shown to be asymptotically valid, enabling reliable inference without prior knowledge of instrument validity.
• Simulation studies demonstrate the method's robustness and accuracy in finite samples under various levels of IV invalidity.
• An empirical application to income and house ownership confirms the method’s practical utility in real-world observational data.