We study the implications of including many covariates in a first-step estimate entering a two-step estimation procedure. We find that a first-order bias emerges when the number of included covariates is “large” relative to the square-root of sample size, rendering standard inference procedures invalid. We show that the jackknife is able to estimate this “many covariates” bias consistently, thereby delivering a new automatic bias-corrected two-step point estimator. The jackknife also consistently estimates the standard error of the original two-step point estimator. For inference, we develop a valid post-bias-correction bootstrap approximation that accounts for the additional variability introduced by the jackknife bias-correction. We find that the jackknife bias-corrected point estimator and the bootstrap post-bias-correction inference perform excellent in simulations, offering important improvements over conventional two-step point estimators and inference procedures, which are not robust to including many covariates. We apply our results to an array of distinct treatment effect, policy evaluation, and other applied microeconomics settings. In particular, we discuss production function and marginal treatment effect estimation in detail.

This paper illustrates how one can deduce preference from observed choices when attention is both limited and random. We introduce a random attention model where we abstain from any particular attention formation and instead consider a large class of nonparametric random attention rules. Our intuitive condition, monotonic attention, captures the idea that each consideration set competes for the decision maker’s attention. We then develop a revealed preference theory and obtain testable implications. We propose econometric methods for identification, estimation, and inference for the revealed preferences. Finally, we provide a general-purpose software implementation of our estimation and inference results and simulation evidence.