Is completeness necessary? Penalized estimation in non-identified models

This paper studies non-identified ill-posed inverse models with estimated operator. Leading examples are the nonparametric IV regression and the functional linear IV regression. We argue that identification of infinite-dimensional parameters is less crucial than identification of finite-dimensional parameters. We show that in the case of identification failures, a very general family of continuously-regularized estimators is consistent for the best approximation of the parameter of interest and obtain $L_2$ and $L_\infty$ finite-sample risk bounds. This class includes Tikhonov, iterated Tikhonov, spectral cut-off, and Landweber-Fridman as special cases. We show that in many cases the best approximation coincides with the structural parameter and can be a useful and tractable object to infer relation between structural variables otherwise. Unlike in the identified case, estimation of the operator may have a non-negligible impact on the estimation accuracy and inference. We develop inferential methods for linear functionals in non-identified models as well as honest uniform confidence sets for the best approximation. Lastly, we demonstrate the discontinuity in the asymptotic distribution for extreme cases of identification failures where we observe a degenerate $U$-statistics asymptotics.