  We develop results for the use of Lasso and Post-Lasso methods to form
first-stage predictions and estimate optimal instruments in linear instrumental
variables (IV) models with many instruments, $p$. Our results apply even when
$p$ is much larger than the sample size, $n$. We show that the IV estimator
based on using Lasso or Post-Lasso in the first stage is root-n consistent and
asymptotically normal when the first-stage is approximately sparse; i.e. when
the conditional expectation of the endogenous variables given the instruments
can be well-approximated by a relatively small set of variables whose
identities may be unknown. We also show the estimator is semi-parametrically
efficient when the structural error is homoscedastic. Notably our results allow
for imperfect model selection, and do not rely upon the unrealistic "beta-min"
conditions that are widely used to establish validity of inference following
model selection. In simulation experiments, the Lasso-based IV estimator with a
data-driven penalty performs well compared to recently advocated
many-instrument-robust procedures. In an empirical example dealing with the
effect of judicial eminent domain decisions on economic outcomes, the
Lasso-based IV estimator outperforms an intuitive benchmark.
  In developing the IV results, we establish a series of new results for Lasso
and Post-Lasso estimators of nonparametric conditional expectation functions
which are of independent theoretical and practical interest. We construct a
modification of Lasso designed to deal with non-Gaussian, heteroscedastic
disturbances which uses a data-weighted $\ell_1$-penalty function. Using
moderate deviation theory for self-normalized sums, we provide convergence
rates for the resulting Lasso and Post-Lasso estimators that are as sharp as
the corresponding rates in the homoscedastic Gaussian case under the condition
that $\log p = o(n^{1/3})$.
