  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.
