Is Completeness Necessary? Estimation and Inference in Non-identified Linear Models

Abstract

This paper studies non-identified linear ill-posed econometric models with estimated operator, including the nonparametric IV regression, the functional IV regression, and many other examples as special cases. We show that under identification failures, regularized estimators are consistent for the best approximation to the structural parameter of interest in a certain subspace of regular (or smooth) functions. We obtain $L_2$ and $L_\infty$ non-asymptotic risk bounds for regularized estimators, nesting Tikhonov, iterated Tikhonov, spectral cut-off, and Landweber-Fridman regularizations as special cases. Due to non-identification, the estimation of the operator can have a non-negligible impact on the estimation accuracy and inference. We provide tools for honest and uniform inference for the best approximation and its linear functionals under different levels of identification failures. We also illustrate that under extreme non-identification, the estimator has asymptotic distribution of the degenerate $U$-statistics.