Instrument Residual Estimator for Any Response with Endogenous Binary Treatment

Given a binary treatment D, a response Y with its potential versions (Y⁰,Y¹) and covariates X, often D is endogenous. Unless Y is continuous, dealing with a binary endogenous D has been difficult even when an instrument δ is available. This paper shows that, for any form of Y (binary, ordinal, censored, continuous, ...), we always have a linear representation Y=μ₀(X)+μ₁(X)D+error with E(error|δ,X)=0, where μ₀(X) is an unknown 'X-conditional intercept' and μ₁(X)≡E(Y¹-Y⁰|complier,X) is the unknown 'X-conditional slope/effect'. We propose 'instrumental residual estimator (IRE)' using δ-E(δ|X) as an instrument for D. IRE is consistent for the Cov(δ,D|X)-weighted average of E(Y¹-Y⁰|complier,X), which screens out individuals with weak instruments in the sense of small |Cov(δ,D|X)|. IRE is easy to implement, and has an asymptotic variance estimator that works well in small samples as a simulation study demonstrates. If desired, a 'weighted IRE' can be used as well to estimate E{E(Y¹-Y⁰|complier,X)}.