Many Instruments and/or Regressors: A Friendly Guide

Abstract

This article surveys the state of the art in the econometrics of regression models with many instruments or many regressors based on alternative – namely, dimension – asymptotics. We list critical results of dimension asymptotics that lead to better approximations of properties of familiar and alternative estimators and tests when the instruments and/or regressors are numerous. Then we consider the problem of estimation and inference in the basic linear instrumental variables regression setup with many strong instruments. We describe the failures of conventional estimation and inference, as well as alternative tools that restore consistency and validity. We then add various other features to the basic model such as heteroskedasticity, instrument weakness, etc., in each case providing a review of the existing tools for proper estimation and inference. Subsequently, we consider a related but different problem of estimation and testing in a linear mean regression with many regressors. We also describe various extensions and connections to other settings, such as panel data models, spatial models, time series models, and so on. Finally, we provide practical guidance regarding which tools are most suitable to use in various situations when many instruments and/or regressors turn out to be an issue.

Keywords: linear regression, instrumental variables, alternative asymptotic theory, dimension asymptotics, many instruments, weak instruments, many regressors

JEL Classification: C12, C13, C18, C21, C26, C55