On Deep Instrumental Variables Estimate

Abstract

The endogeneity issue is fundamentally important as many empirical applications may suffer from the omission of explanatory variables, measurement error, or simultaneous causality. Recently, \cite{hllt17} 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.

References in corpus (7)

• Training Very Deep Networks
• Variable selection in nonparametric additive models
• Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks
• Deep Network Approximation for Smooth Functions
• The phase diagram of approximation rates for deep neural networks
• General frequentist properties of the posterior profile distribution
• On the rate of convergence of fully connected very deep neural network regression estimates

Cited by in corpus (6)

• Semiparametric proximal causal inference
• Optimal Nonparametric Inference via Deep Neural Network
• Mostly Harmless Machine Learning: Learning Optimal Instruments in Linear IV Models
• Calibrating multi-dimensional complex ODE from noisy data via deep neural networks
• Minimax optimal high-dimensional classification using deep neural networks
• Optimal Classification for Functional Data