## 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 - Jul 2021 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty