  This chapter presents key concepts and theoretical results for analyzing
estimation and inference in high-dimensional models. High-dimensional models
are characterized by having a number of unknown parameters that is not
vanishingly small relative to the sample size. We first present results in a
framework where estimators of parameters of interest may be represented
directly as approximate means. Within this context, we review fundamental
results including high-dimensional central limit theorems, bootstrap
approximation of high-dimensional limit distributions, and moderate deviation
theory. We also review key concepts underlying inference when many parameters
are of interest such as multiple testing with family-wise error rate or false
discovery rate control. We then turn to a general high-dimensional minimum
distance framework with a special focus on generalized method of moments
problems where we present results for estimation and inference about model
parameters. The presented results cover a wide array of econometric
applications, and we discuss several leading special cases including
high-dimensional linear regression and linear instrumental variables models to
illustrate the general results.
