Abstract. 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 wellapproximated 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.Optimal instruments are conditional expectations. 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 1-penalty function. By innovatively 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 ). We also provide a data-driven method for choosing the penalty level that must be specified in obtaining Lasso and Post-Lasso estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.

Jails and prisons are major sites of novel coronavirus (SARS-CoV-2) infection. Many jurisdictions in the United States have therefore accelerated the release of low-risk offenders. Early release, however, does not address how arrest and pretrial detention practices may be contributing to disease spread. Using data from Cook County Jail-one of the largest known nodes of SARS-CoV-2 spread in the United States-in Chicago, Illinois, we analyzed the relationship between jailing practices and community infections at the ZIP code level. We found that jailcommunity cycling was a significant predictor of cases of coronavirus disease 2019 (COVID-19), accounting for 55 percent of the variance in case rates across ZIP codes in Chicago and 37 percent of the variance in all of Illinois. Jail-community cycling far exceeds race, poverty, public transit use, and population density as a predictor of variance. The data suggest that cycling people through Cook County Jail alone is associated with 15.7 percent of all documented COVID-19 cases in Illinois and 15.9 percent of all documented cases in Chicago as of April 19, 2020. Our findings support arguments for reduced reliance on incarceration and for related justice reforms both as emergency measures during the present pandemic and as sustained structural changes vital for future pandemic preparedness and public health.