On rate optimality for ill-posed inverse problems in econometrics

In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.

💡 Research Summary

The paper provides a unified theoretical treatment of two central non‑parametric econometric problems—non‑parametric indirect regression (NPIR) and non‑parametric instrumental variables (NPIV) regression—by framing them as ill‑posed inverse problems. After a concise motivation, the authors introduce a compact, self‑adjoint operator (T) that links the latent structural function (f) to observable data. They impose only two basic regularity conditions: (i) a source condition (f = T^{s}g) with smoothness index (s>0) and (g) square‑integrable, and (ii) an eigenvalue decay condition on the spectrum of (T). The decay can be polynomial ((\lambda_j \asymp j^{-\alpha})) for mildly ill‑posed problems or exponential ((\lambda_j \asymp \exp(-c j^{\beta}))) for severely ill‑posed problems. These two conditions simultaneously cover the full range of ill‑posedness encountered in econometric applications.

Under this framework the authors derive minimax lower bounds for the mean integrated squared error (MISE) risk of any estimator of (f). The bounds are expressed in closed form: for mildly ill‑posed cases the optimal rate is (n^{-2s/(2s+\alpha)}); for severely ill‑posed cases it is ((\log n)^{-2s/\beta}). The derivation relies on a careful decomposition of risk into bias and variance components using the eigenbasis of (T) and on the concept of effective dimension, which quantifies how many singular directions can be reliably estimated given the sample size.

The second major contribution is to show that very simple estimators attain these lower bounds. For NPIR the authors consider a projection estimator onto a finite‑dimensional subspace (V_m) spanned by the first (m) eigenfunctions of (T). By choosing the projection dimension as (m \asymp n^{1/(2s+\alpha)}) (polynomial decay) or (m \asymp (\log n)^{1/\beta}) (exponential decay), the estimator’s MISE matches the minimax rate. For NPIV they propose a sieve minimum‑distance estimator: one selects a sieve space (\mathcal{S}K) generated by a basis ({\psi_k}{k=1}^K) and solves a least‑squares problem that enforces the conditional moment restriction (E