# Semiparametric Estimation with Data Missing Not at Random Using an Instrumental Variable

arXiv: https://arxiv.org/abs/1607.03197 · HTML: https://mlsanity.com/paper/1607.03197/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2016.

## Built on by (most cited first, 9 of 9)

- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [Sufficient Identification Conditions and Semiparametric Estimation under Missing Not at Random Mechanisms](https://mlsanity.com/paper/2306.06443/) (2023)
- [Off-Policy Evaluation Under Nonignorable Missing Data](https://mlsanity.com/paper/2507.06961/) (2025)
- [Multi-armed Bandits with Missing Outcome](https://mlsanity.com/paper/2411.05661/) (2024)
- [Response to Discussions of "Causal and Counterfactual Views of Missing Data Models"](https://mlsanity.com/paper/2510.14694/) (2025)
- [AI-Generated Measurements for Identification and Inference with Missing Data: A Weak Shadow Variable Approach](https://mlsanity.com/paper/2602.16061/) (2026)
- [Identifiable Deep Latent Variable Models for MNAR Data](https://mlsanity.com/paper/2603.24771/) (2026)
- [A Practical Upper Bound on Selection Bias Effects in Medical Prediction Models](https://mlsanity.com/paper/2606.00563/) (2026)
- [Off-Policy Evaluation for Missingness-Aware Policies in MDPs with Rewards Missing Not at Random](https://mlsanity.com/paper/2606.20206/) (2026)

## Abstract (arXiv)

Missing data occur frequently in empirical studies in health and social sciences, often compromising our ability to make accurate inferences. An outcome is said to be missing not at random (MNAR) if, conditional on the observed variables, the missing data mechanism still depends on the unobserved outcome. In such settings, identification is generally not possible without imposing additional assumptions. Identification is sometimes possible, however, if an instrumental variable (IV) is observed for all subjects which satisfies the exclusion restriction that the IV affects the missingness process without directly influencing the outcome. In this paper, we provide necessary and sufficient conditions for nonparametric identification of the full data distribution under MNAR with the aid of an IV. In addition, we give sufficient identification conditions that are more straightforward to verify in practice. For inference, we focus on estimation of a population outcome mean, for which we develop a suite of semiparametric estimators that extend methods previously developed for data missing at random. Specifically, we propose inverse probability weighted estimation, outcome regression-based estimation and doubly robust estimation of the mean of an outcome subject to MNAR. For illustration, the methods are used to account for selection bias induced by HIV testing refusal in the evaluation of HIV seroprevalence in Mochudi, Botswana, using interviewer characteristics such as gender, age and years of experience as IVs.