Residual Balancing for Non-Linear Outcome Models in High Dimensions
Isaac Meza
TL;DR
The paper tackles high-dimensional causal inference with nonlinear outcomes by extending Approximate Residual Balancing (ARB) to generalized linear models. It shows that ignoring the link function's curvature induces bias in nonlinear settings and introduces a second-order balancing correction that weights covariates by $\psi'(\cdot)$ and their outer products by $\psi''(\cdot)$. A concrete optimization problem yields balancing weights that ensure $\sqrt{n}$-consistency and asymptotic normality for the ATT, without estimating a propensity score, and cross-fitting provides robust inference. The authors extend the framework to a smooth single-index model (SIM) with unknown link, detailing a three-fold splitting scheme and nonparametric derivative estimation. Overall, the work delivers a principled, practically implementable method for valid inference in high-dimensional nonlinear causal models, broadening the applicability of ARB to more complex outcome structures.
Abstract
We extend the approximate residual balancing (ARB) framework to nonlinear models, answering an open problem posed by Athey et al. (2018). Our approach addresses the challenge of estimating average treatment effects in high-dimensional settings where the outcome follows a generalized linear model. We derive a new bias decomposition for nonlinear models that reveals the need for a second-order correction to account for the curvature of the link function. Based on this insight, we construct balancing weights through an optimization problem that controls for both first and second-order sources of bias. We provide theoretical guarantees for our estimator, establishing its $\sqrt{n}$-consistency and asymptotic normality under standard high-dimensional assumptions.