Neural Proximal Causal Inference for Longitudinal Data with Unmeasured Confounding

Unmeasured confounders pose a fundamental challenge in estimating causal effects of time-varying treatments from observational data. Proximal Causal Inference (PCI) addresses this through proxy variables, but estimation under PCI requires solving recursively coupled ill-posed integral equations. Existing longitudinal methods make this tractable only through parametric models that cannot capture nonlinear trajectory dynamics. We introduce Neural-PCI, a deep learning estimation framework grounded in established proxy-based identification and, to our knowledge, the first implementable neural method for the full $T$-stage recursion. A joint Maximum Moment Restriction (MMR) formulation enforces all $T$ moment conditions through a single closed-form kernel loss, and yields three semi-parametric estimators with complementary robustness properties. We show that stage-wise sequential estimation incurs error growth exponential in the time horizon $T$, while our joint formulation gives additive error aggregation that scales linearly in $T$ on average. Experiments on synthetic data and the semi-synthetic MIMIC-III benchmark show that Neural-PCI consistently outperforms parametric PCI, latent variable, and sequential-randomization baselines across horizons.