Is completeness necessary? Estimation in nonidentified linear models

This paper documents the consequences of the identification failures in a class of linear ill-posed inverse models. The Tikhonov-regularized estimator converges to a well-defined limit equal to the best approximation of the structural parameter in the orthogonal complement to the null space of the operator. We illustrate that in many instances the best approximation may coincide with the structural parameter or at least may reasonably approximate it. We obtain new nonasymptotic risk bounds in the uniform and the Hilbert space norms for the best approximation. Nonidentification has important implications for the large sample distribution of the Tikhonov-regularized estimator, and we document the transition between the Gaussian and the weighted chi-squared limits. The theoretical results are illustrated for the nonparametric IV and the functional linear IV regressions and are further supported by the Monte Carlo experiments.