Operator-Robust Instrumental Variable Regression under Joint Distribution Shifts

Instrumental variable regression identifies causal effects under unobserved confounding, but only when moment conditions hold. However, weak identification increases the sampling error, while invalid instruments and distribution shifts can violate its moment restrictions. Instead of indirectly protecting against perturbations to the data distribution, we directly robustify the identifying moments. By tracing joint distributional perturbations through these moments, we derive a common score that jointly perturbs the treatment operator and outcome moment while preserving their relationship. This pushforward construction creates an operator-robust worst-case criterion and tractable convex average-case surrogates. A special case provides a nonparametric operator analogue of k-class regularization. We characterize how score robustness changes identification and when it preserves the causal target, and bound structural error in terms of identification strength, estimation error, and robustness-induced biases. Across a wide range of synthetic, semi-synthetic, and real-world settings, our method improves over state-of-the-art IV estimators, including under weak identification, invalid instruments, and distribution shift.