Instrument residual estimator for any response variable with endogenous binary treatment

Myoung jae Lee

doi:10.1111/rssb.12442

Instrument residual estimator for any response variable with endogenous binary treatment

Myoung jae Lee

Department of Economics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	612-635
Number of pages	24
Journal	Journal of the Royal Statistical Society. Series B: Statistical Methodology
Volume	83
Issue number	3
DOIs	https://doi.org/10.1111/rssb.12442
Publication status	Published - 2021 Jul

Bibliographical note

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

Keywords

effect on complier
endogenous treatment
heterogeneous effect
instrumental variable estimator
overlap weight

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1111/rssb.12442

Cite this

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title = "Instrument residual estimator for any response variable with endogenous binary treatment",

abstract = "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 {\textquoteleft}compliers{\textquoteright} 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 {\textquoteleft}instrumental residual estimator (IRE){\textquoteright} with Z−E(Z|X) as an IV for D can be applied, and IRE is consistent for the {\textquoteleft}E(Z|X)-overlap{\textquoteright} weighted average of μ1(X), which becomes (Formula presented.) for randomized Z. Third, going further, a {\textquoteleft}weighted IRE{\textquoteright} can be done which is consistent for E{μ1(X)}. Empirical analyses as well as a simulation study are provided to illustrate our approaches.",

keywords = "effect on complier, endogenous treatment, heterogeneous effect, instrumental variable estimator, overlap weight",

author = "Lee, {Myoung jae}",

note = "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: {\textcopyright} 2021 Royal Statistical Society",

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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

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

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ER -

Instrument residual estimator for any response variable with endogenous binary treatment

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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