Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression
This work addresses the problem of nonparametric instrumental variable regression for researchers in econometrics and machine learning, providing theoretical guarantees for a data-adaptive deep learning method, though it is incremental as it builds on prior DFIV work.
The paper proves that the deep feature instrumental variable (DFIV) regression algorithm achieves the minimax optimal learning rate for target functions in Besov spaces under standard assumptions, and shows it outperforms fixed-feature methods by maintaining optimality for functions with low spatial homogeneity and being more data-efficient in Stage 1.
We provide a convergence analysis of deep feature instrumental variable (DFIV) regression (Xu et al., 2021), a nonparametric approach to IV regression using data-adaptive features learned by deep neural networks in two stages. We prove that the DFIV algorithm achieves the minimax optimal learning rate when the target structural function lies in a Besov space. This is shown under standard nonparametric IV assumptions, and an additional smoothness assumption on the regularity of the conditional distribution of the covariate given the instrument, which controls the difficulty of Stage 1. We further demonstrate that DFIV, as a data-adaptive algorithm, is superior to fixed-feature (kernel or sieve) IV methods in two ways. First, when the target function possesses low spatial homogeneity (i.e., it has both smooth and spiky/discontinuous regions), DFIV still achieves the optimal rate, while fixed-feature methods are shown to be strictly suboptimal. Second, comparing with kernel-based two-stage regression estimators, DFIV is provably more data efficient in the Stage 1 samples.