Instrument residual estimator for any response variable with endogenous binary treatment

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4 Citations (Scopus)


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 languageEnglish
Pages (from-to)612-635
Number of pages24
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Issue number3
Publication statusPublished - 2021 Jul


  • 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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