Instrument residual estimator for any response variable with endogenous binary treatment
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2021 · Volume 83 · Issue 3
Abstract
Given an endogenous/confounded binary treatmentD, a responseYwith its potential versions (Y0,Y1) and covariatesX, finding the treatment effect is difficult ifYis not continuous, even when a binary instrumental variable (IV)Zis available. We show that, for any form ofY(continuous, binary, mixed,…), there exists a decompositionY=μ0(X) +μ1(X)D+errorwithE(error|Z,X) = 0, whereand ‘compliers’ are those who get treated if and only ifZ= 1. First, using the decomposition, instrumental variable estimator (IVE) is applicable with polynomial approximations forμ0(X) andμ1(X) to obtain a linear model forY. Second, better yet, an ‘instrumental residual estimator (IRE)’ withZ−E(Z|X) as an IV forDcan be applied, and IRE is consistent for the ‘E(Z|X)-overlap’ weighted average ofμ1(X), which becomesfor randomizedZ. Third, going further, a ‘weighted IRE’ can be done which is consistent forE{μ1(X)}. Empirical analyses as well as a simulation study are provided to illustrate our approaches.
Keywords
effect on complier, endogenous treatment, heterogeneous effect, instrumental variable estimator, overlap weight
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