Jean-Pierre Florens, Université Toulouse
Is completeness necessary? Penalized estimation in non identifed linear models
Identification is an important issue in many econometric models. This paper studies potentially non-identified and/or weakly identified ill-posed inverse models. The leading examples are the nonparametric IV regression and the functional linear IV regression. 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. We obtain L2 and L1 convergence rates for this general class of regularization schemes, including Tikhonov, iterated Tikhonov, spectral cut-off, and Landweber-Fridman. Unlike in the identified case, estimation of the operator has non-negligible impact on convergence rates and inference. We develop inferential methods for linear functionals in such models. Lastly, we demonstrate the discontinuity in the asymptotic distribution in case of weak identification. In particular, the estimator has a degenerate U-statistics type behavior, in the extreme case of weak instrument.