## Abstract

Given an endogenous/confounded binary treatment D, a response Y with its potential versions (Y^{0}, Y^{1}) 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 = μ_{0}(X) + μ_{1}(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 μ_{0}(X) and μ_{1}(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 μ_{1}(X), which becomes (Formula presented.) for randomized Z. Third, going further, a ‘weighted IRE’ can be done which is consistent for E{μ_{1}(X)}. Empirical analyses as well as a simulation study are provided to illustrate our approaches.

Original language | English |
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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 | |

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