Many Instruments and/or Regressors: A Friendly Guide

Abstract

This paper 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.Most recent theoretical literature tends to deal with all of the possible complications involved, and thus the degree of sophistication can become a stumbling block for applied researchers. The survey is structured so that complications are gradually scaffolded on an easily comprehensible skeleton model -a linear homoskedastic IVs model. In this section, we describe a road map for how the survey is organized.Section 3 presents the statistical foundations for many instrument/regressor asymptotics; an uninterested reader may skip this section during a first reading and jump to Section 4. First, we reiterate the idea of alternative asymptotic approximations, in particular, dimension asymptotics. Second, the historical evolution of limit theorems for bilinear forms -the main building block of many instrument/regressor MANY INSTRUMENTS AND/OR REGRESSORS: A FRIENDLY GUIDE 691 asymptotics -is shown, starting with an example of a simplest quadratic form, to show the differences between the approaches -the traditional asymptotics and dimension asymptotics -and how they interlock. Third, a certain important asymptotic property of the diagonal of the projection matrix is discussed.In Section 4, a linear homoskedastic IVs model is considered in detail. The reasons why 2SLS estimation fails and various remedies for such failures (bias correction, jackknifing, and LIML) are discussed. The alternative asymptotic distributions of consistent estimators are presented, initially for the case of normal errors. Then, we relax error normality to examine the arising complications and discuss how they can be handled. In addition to inferential procedures regarding parameter restrictions, the issues of asymptotic efficiency and specification testing ar...

Inference in Instrumental Variable Models With Heteroskedasticity and Many Instruments

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