  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.
