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# Inference on Strongly Identified Functionals of Weakly Identified Functions

Aug 2022

In a variety of applications, including nonparametric instrumental variable(NPIV) analysis, proximal causal inference under unmeasured confounding, andmissing-not-at-random data with shadow variables, we are interested ininference on a continuous linear functional (e.g., average causal effects) ofnuisance function (e.g., NPIV regression) defined by conditional momentrestrictions. These nuisance functions are generally weakly identified, in thatthe conditional moment restrictions can be severely ill-posed as well as admitmultiple solutions. This is sometimes resolved by imposing strong conditionsthat imply the function can be estimated at rates that make inference on thefunctional possible. In this paper, we study a novel condition for thefunctional to be strongly identified even when the nuisance function is not;that is, the functional is amenable to asymptotically-normal estimation at$\sqrt{n}$-rates. The condition implies the existence of debiasing nuisancefunctions, and we propose penalized minimax estimators for both the primary anddebiasing nuisance functions. The proposed nuisance estimators can accommodateflexible function classes, and importantly they can converge to fixed limitsdetermined by the penalization regardless of the identifiability of thenuisances. We use the penalized nuisance estimators to form a debiasedestimator for the functional of interest and prove its asymptotic normalityunder generic high-level conditions, which provide for asymptotically validconfidence intervals. We also illustrate our method in a novel partially linearproximal causal inference problem and a partially linear instrumental variableregression problem.

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