Instrument residual estimator for any response variable with endogenous binary treatment

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 μ1(X)≡E(Y1‐Y0|complier,X) 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 E(Y1‐Y0|complier) 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.