TY - JOUR

T1 - Instrument residual estimator for any response variable with endogenous binary treatment

AU - Lee, Myoung jae

N1 - Funding Information:
The author is grateful to the Editor, the Associate Editor, Jin‐young Choi, and three reviewers for their helpful comments and for directing the author's attention to relevant studies in the literature. This research has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C1A01007786).
Publisher Copyright:
© 2021 Royal Statistical Society

PY - 2021/7

Y1 - 2021/7

N2 - Given an endogenous/confounded binary treatment D, a response Y with its potential versions (Y0, Y1) and covariates X, finding the treatment effect is difficult if Y is not continuous, even when a binary instrumental variable (IV) Z is available. We show that, for any form of Y (continuous, binary, mixed,…), there exists a decomposition Y = μ0(X) + μ1(X)D + error with E(error|Z,X) = 0, where (Formula presented.) and ‘compliers’ are those who get treated if and only if Z = 1. First, using the decomposition, instrumental variable estimator (IVE) is applicable with polynomial approximations for μ0(X) and μ1(X) to obtain a linear model for Y. Second, better yet, an ‘instrumental residual estimator (IRE)’ with Z−E(Z|X) as an IV for D can be applied, and IRE is consistent for the ‘E(Z|X)-overlap’ weighted average of μ1(X), which becomes (Formula presented.) for randomized Z. Third, going further, a ‘weighted IRE’ can be done which is consistent for E{μ1(X)}. Empirical analyses as well as a simulation study are provided to illustrate our approaches.

AB - Given an endogenous/confounded binary treatment D, a response Y with its potential versions (Y0, Y1) and covariates X, finding the treatment effect is difficult if Y is not continuous, even when a binary instrumental variable (IV) Z is available. We show that, for any form of Y (continuous, binary, mixed,…), there exists a decomposition Y = μ0(X) + μ1(X)D + error with E(error|Z,X) = 0, where (Formula presented.) and ‘compliers’ are those who get treated if and only if Z = 1. First, using the decomposition, instrumental variable estimator (IVE) is applicable with polynomial approximations for μ0(X) and μ1(X) to obtain a linear model for Y. Second, better yet, an ‘instrumental residual estimator (IRE)’ with Z−E(Z|X) as an IV for D can be applied, and IRE is consistent for the ‘E(Z|X)-overlap’ weighted average of μ1(X), which becomes (Formula presented.) for randomized Z. Third, going further, a ‘weighted IRE’ can be done which is consistent for E{μ1(X)}. Empirical analyses as well as a simulation study are provided to illustrate our approaches.

KW - effect on complier

KW - endogenous treatment

KW - heterogeneous effect

KW - instrumental variable estimator

KW - overlap weight

UR - http://www.scopus.com/inward/record.url?scp=85110973736&partnerID=8YFLogxK

U2 - 10.1111/rssb.12442

DO - 10.1111/rssb.12442

M3 - Article

AN - SCOPUS:85110973736

VL - 83

SP - 612

EP - 635

JO - Journal of the Royal Statistical Society. Series B: Statistical Methodology

JF - Journal of the Royal Statistical Society. Series B: Statistical Methodology

SN - 1369-7412

IS - 3

ER -