Several causal parameters in short panel data models are scalar summaries of a function called a nested nonparametric instrumental variable regression (nested NPIV). Examples include long term, mediated, and time varying treatment effects identified using proxy variables. However, it appears that no prior estimators or guarantees for nested NPIV exist, preventing flexible estimation and inference for these causal parameters. A major challenge is compounding ill posedness due to the nested inverse problems. We analyze adversarial estimators of nested NPIV, and provide sufficient conditions for efficient inference on the causal parameter. Our nonasymptotic analysis has three salient features: (i) introducing techniques that limit how ill posedness compounds
(ii) accommodating neural networks, random forests, and reproducing kernel Hilbert spaces
and (iii) extending to causal functions, e.g. long term heterogeneous treatment effects. We measure long term heterogeneous treatment effects of Project STAR and mediated proximal treatment effects of the Job Corps.