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 (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, horizontal ellipsis ), there exists a decomposition Y = mu(0)(X) + mu(1)(X)D + error with E(error|Z,X) = 0, where mu 1(X)equivalent to 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 mu(0)(X) and mu(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 mu(1)(X), which becomes E(Y1-Y0|complier) for randomized Z. Third, going further, a 'weighted IRE' can be done which is consistent for E{mu(1)(X)}. Empirical analyses as well as a simulation study are provided to illustrate our approaches.