Elsevier

Economics Letters

Volume 165, April 2018, Pages 10-12
Economics Letters

Identification and estimation using heteroscedasticity without instruments: The binary endogenous regressor case

https://doi.org/10.1016/j.econlet.2018.01.003Get rights and content

Abstract

Lewbel (2012) provides an estimator for linear regression models containing an endogenous regressor, when no outside instruments or other such information is available. The method works by exploiting model heteroscedasticity to construct instruments using the available regressors. Some authors have considered the method in empirical applications where an endogenous regressor is binary (e.g., endogenous Diff-in-Diff or endogenous binary treatment models). The present paper shows that the assumptions required for Lewbel’s estimator can indeed be satisfied when an endogenous regressor is binary. Caveats regarding application of the estimator are discussed.

Introduction

Linear regression models containing endogenous regressors are generally identified using outside information such as exogenous instruments, or by parametric distribution assumptions. Some papers obtain identification without outside instruments by exploiting heteroscedasticity, including Rigobon (2003), Klein and Vella (2010), Lewbel (2012), and Prono (2014). See also Lewbel (2017).

Some authors, including include Emran et al. (2014) and Hoang et al. (2014), have questioned whether the Lewbel (2012) estimator can be used when the endogenous regressor is binary. Others, including Le Moglie et al. (2015), have applied the Lewbel (2012) estimator with a binary endogenous variable, though without verifying if the assumptions hold.

Examples of such applications would include Diff-in-Diff models with endogenous fixed effects, or models with binary endogenous treatment indicators. Binary endogenous regressors are a natural case to consider in part because they imply that the instrument equation will automatically have heteroscedastic errors, which is one of the requirements of the estimator.

This paper shows validity of the Lewbel (2012) estimator when an endogenous regressor is binary. So, e.g., the estimator might be applied to estimate a (homogeneous) treatment effect when the binary treatment is not randomly assigned and when exogenous instruments are not available. However, the sufficient conditions given here do impose strong restrictions on the error term of the model.

Section snippets

The model and estimator

Assume a sample of observations of endogenous variables Y1 and Y2, and a vector of exogenous covariates X. We wish to estimate γ and the vector β in the model Y1=Xβ+Y2γ+ε1Y2=Xα+ε2where the errors ε1 and ε2 may be correlated. As in Lewbel (2012), we also consider the more general case where Y2=gX+ε2 for some nonlinear, possibly unknown function g.

Standard instrumental variables estimation depends on having an element of X that appears in the Y2 equation but not in the Y1 equation, and uses

A binary endogenous regressor

Suppose that Y2 is binary. Then Y2=Xα+ε2 is a linear probability model. But we also wish to allow for more general models, so let Y2=gX+ε2 where gX=EY2X. Here gX is possibly nonlinear and possibly unknown. For example, if Y2 satisfies a probit or logit model, then gX=FXα where F is the cumulative normal or logistic distribution function. Also included are nonparametric models, where gX is estimated by a nonparametric regression of Y2 on X.

If the Y2 equation is a linear probability model,

Conclusions and caveats

Theorem 1 shows that the assumptions required to apply the Lewbel (2012) can be satisfied when Y2 is binary, and a supplemental appendix to this paper provides a different way to satisfy these assumptions when both Y1 and Y2 are binary. So, e.g., the STATA module IVREG2H by Baum and Schaffer (2012) can be used without change when just Y2 is binary, or when both Y1 and Y2 are binary.

A drawback of these results is that there are no obvious behavioral models that directly imply Assumption A2. This

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