Chapter 77 Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization

Abstract

In other words, while the solvability criterion of Kϕ = r imposes r ∈ Ψ for some γ ≥ 1, we will get a geometric rate of decay of the regularization bias as soon as r ∈ Ψ for some γ > 1. Note that the upper bound β = 2 of the rate of decay is just a matter of normalization of the parameter α of regularization. By (3.19), ϕ − ϕ © cannot go to zero faster than [1 − q (α, λ)] ©

Identification and estimation of nonclassical nonlinear errors-in-variables models with continuous distributions using instruments

“…Unfortunately, these results cannot trivially be extended to continuously distributed variables, because the number of nuisance parameters needed to describe the measurement error distribution (conditional on given values of the observable variables) becomes infinite. Identifying these parameters thus involves solving operator equations that exhibit potential ill-defined inverse problems (similar to those discussed in Carrasco, Florens, and Renault (2005), Darolles, Florens, and Renault (2002), and Newey and Powell (2003)). …”

“…An operator L is said to be injective if its inverse L −1 is defined over the range of the operator L (see Section 3.1 in Carrasco, Florens, and Renault (2005)). In a finite-dimensional space, the qualifier "injective" is synonymous with "invertible", but in an infinite-dimensional space the distinction is needed to account for the fact that inverses are often defined only over a restricted domain.…”

“…In a finite-dimensional space, the qualifier "injective" is synonymous with "invertible", but in an infinite-dimensional space the distinction is needed to account for the fact that inverses are often defined only over a restricted domain. As discussed in Carrasco, Florens, and Renault (2005), the weaker notion of injectivity is the concept needed to establish identification. In our setup, the inverses are guaranteed to be defined over a sufficiently large domain because the results of the inversions (such as L −1 x|x * L x|z = L x * |z , from Equation (4)) always yield a well-defined integral operator.…”

“…The condition (5) implies that L x|z is a Hilbert-Schmidt operator, which is necessarily compact (see Theorem 2.32 in Carrasco, Florens, and Renault (2005)). This in turn implies the existence of a singular value decomposition,…”

“…injectivity is sufficient, whether or not the inverse is continuous (Carrasco, Florens, and Renault (2005)). …”