Honest confidence sets in nonparametric IV regression and other ill-posed models

摘要

This paper provides novel methods for inference in a very general class ofill-posed models in econometrics, encompassing the nonparametric instrumentalregression, different functional regressions, and the deconvolution. I focus onuniform confidence sets for the parameter of interest estimated with Tikhonovregularization, as in (Darolles, Fan, Florens, and Renault, 2011). I first showthat it is not possible to develop inferential methods directly based on theuniform central limit theorem. To circumvent this difficulty I develop twoapproaches that lead to valid confidence sets. I characterize expecteddiameters and coverage errors uniformly over a large class of models (i.e.constructed confidence sets are honest). Finally, I illustrate that introducedconfidence sets have reasonable width and coverage properties in samplescommonly used in applications with Monte Carlo simulations and consideringapplication to Engel curves.