  Jackknife instrumental variable estimation (JIVE) is a classic method to
leverage many weak instrumental variables (IVs) to estimate linear structural
models, overcoming the bias of standard methods like two-stage least squares.
In this paper, we extend the jackknife approach to nonparametric IV (NPIV)
models with many weak IVs. Since NPIV characterizes the structural regression
as having residuals projected onto the IV being zero, existing approaches
minimize an estimate of the average squared projected residuals, but their
estimates are biased under many weak IVs. We introduce an IV splitting device
inspired by JIVE to remove this bias, and by carefully studying this split-IV
empirical process we establish learning rates that depend on generic complexity
measures of the nonparametric hypothesis class. We then turn to leveraging this
for semiparametric inference on average treatment effects (ATEs) on unobserved
long-term outcomes predicted from short-term surrogates, using historical
experiments as IVs to learn this nonparametric predictive relationship even in
the presence of confounding between short- and long-term observations. Using
split-IV estimates of a debiasing nuisance, we develop asymptotically normal
estimates for predicted ATEs, enabling inference.
