Nonparametric Instrumental Variable Analysis Without Structural Equations: Debiased Inference on Functionals of Inverse Problems with No Solutions

Instrumental variable (IV) analyses generally start by posing a structural equation: Y = hstructural(X)+ϵ, (1) where hstructural represents the causal effect of X on Y, and X and ϵ may be endogenous (E[ϵ | X] = 0). Then given an exogenous instrument Z satisfying the exclusion restriction, the common statistical solution given joint observations of W = (X,Y,Z) P is to conduct inference on some continuous linear functional h 7 EP[m(W;h)] of a solution h H to the linear equation implied by exclusion: TPh = rP, (2) where TP: H G maps h 7 argming GEP(h(X) g(Z))2, rP = argminr GEP(Y r(Z))2, and H, G are closed linear subspaces of square-integrable functions of X and of Z, respectively. For example, if these are all square-integrable functions, then (TPh)(Z) = EP[h(X) | Z] is the conditional expectation.