# On Deep Instrumental Variables Estimate

arXiv: https://arxiv.org/abs/2004.14954 · HTML: https://mlsanity.com/paper/2004.14954/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2020.

## Built on by (most cited first, 14 of 14)

- [Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closedness](https://mlsanity.com/paper/2302.05404/) (2023)
- [Source Condition Double Robust Inference on Functionals of Inverse Problems](https://mlsanity.com/paper/2307.13793/) (2023)
- [Stochastic Online Instrumental Variable Regression: Regrets for Endogeneity and Bandit Feedback](https://mlsanity.com/paper/2302.09357/) (2023)
- [Calibrating multi-dimensional complex ODE from noisy data via deep neural networks](https://mlsanity.com/paper/2106.03591/) (2021)
- [Deep Feature Screening: Feature Selection for Ultra High-Dimensional Data via Deep Neural Networks](https://mlsanity.com/paper/2204.01682/) (2022)
- [Nonparametric Instrumental Variable Regression through Stochastic Approximate Gradients](https://mlsanity.com/paper/2402.05639/) (2024)
- [Deep Neural Network Classifier for Multi-dimensional Functional Data](https://mlsanity.com/paper/2205.08592/) (2022)
- [Robust Deep Neural Network Estimation for Multi-dimensional Functional Data](https://mlsanity.com/paper/2205.09604/) (2022)
- [Semiparametric Regression for Spatial Data via Deep Learning](https://mlsanity.com/paper/2301.03747/) (2023)
- [Off-Policy Evaluation for Action-Dependent Non-Stationary Environments](https://mlsanity.com/paper/2301.10330/) (2023)
- [Multiclass classification for multidimensional functional data through deep neural networks](https://mlsanity.com/paper/2305.13349/) (2023)
- [Nonparametric regression with modified ReLU networks](https://mlsanity.com/paper/2207.08306/) (2022)
- [Machine learning the first stage in 2SLS: Practical guidance from bias decomposition and simulation](https://mlsanity.com/paper/2505.13422/) (2025)
- [NetworkNet: A Deep Neural Network Approach for Random Networks with Sparse Nodal Attributes and Complex Nodal Heterogeneity](https://mlsanity.com/paper/2604.11673/) (2026)

## Abstract (arXiv)

The endogeneity issue is fundamentally important as many empirical applications may suffer from the omission of explanatory variables, measurement error, or simultaneous causality. Recently, citehllt17 propose a "Deep Instrumental Variable (IV)" framework based on deep neural networks to address endogeneity, demonstrating superior performances than existing approaches. The aim of this paper is to theoretically understand the empirical success of the Deep IV. Specifically, we consider a two-stage estimator using deep neural networks in the linear instrumental variables model. By imposing a latent structural assumption on the reduced form equation between endogenous variables and instrumental variables, the first-stage estimator can automatically capture this latent structure and converge to the optimal instruments at the minimax optimal rate, which is free of the dimension of instrumental variables and thus mitigates the curse of dimensionality. Additionally, in comparison with classical methods, due to the faster convergence rate of the first-stage estimator, the second-stage estimator has a smaller (second order) estimation error and requires a weaker condition on the smoothness of the optimal instruments. Given that the depth and width of the employed deep neural network are well chosen, we further show that the second-stage estimator achieves the semiparametric efficiency bound. Simulation studies on synthetic data and application to automobile market data confirm our theory.