  We show that estimators based on spectral regularization converge to the best
approximation of a structural parameter in a class of nonidentified linear
ill-posed inverse models. Importantly, this convergence holds in the uniform
and Hilbert space norms. We describe several circumstances when the best
approximation coincides with a structural parameter, or at least reasonably
approximates it, and discuss how our results can be useful in the partial
identification setting. Lastly, we document that identification failures have
important implications for the asymptotic distribution of a linear functional
of regularized estimators, which can have a weighted chi-squared component. The
theory is illustrated for various high-dimensional and nonparametric IV
regressions.
