We develop results for the use of LASSO and Post-LASSO methods to form firststage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p, that apply even when p is much larger than the sample size, n.We rigorously develop asymptotic distribution and inference theory for the resulting IV estimators and provide conditions under which these estimators are asymptotically oracle-efficient.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 LASSObased IV estimator substantially reduces estimated standard errors allowing one to draw much more precise conclusions about the economic effects of these decisions.Optimal instruments are conditional expectations; and in developing the IV results, we also establish a series of new results for LASSO and Post-LASSO estimators of non-parametric conditional expectation functions which are of independent theoretical and practical interest.Specifically, we develop the asymptotic theory for these estimators that allows for non-Gaussian, heteroscedastic disturbances, which is important for econometric applications.By innovatively using moderate deviation theory for self-normalized sums, we provide convergence rates for these estimators that are as sharp as in the homoscedastic Gaussian case under the weak condition that log p = o(n 1/3 ).Moreover, as a practical innovation, we provide a fully data-driven method for choosing the user-specified penalty that must be provided in obtaining LASSO and Post-LASSO estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.

The accepted manuscript version (last revised 5 Jan 2018 (v8)) has 118 pages, 3 tables, 11 figures, and includes supplementary appendix. This version corrects some typos in Example 2 of the published version. This supplement contains 11 appendices with additional results and some omitted proofs. Appendices F-J include additional results for Sections 2-7, respectively. Appendix K gathers auxiliary results on algebra of covering entropies. Appendices L and M contain the proofs of Sections 4 and 5 omitted from the main text. Appendix N contains the proofs of Sections 6 omitted from the main text, together with the proofs of the additional results for Section 6 in Appendix I. Appendix O reports the results of a simulation experiment.