COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1626 On Rate Optimality for Ill-posed Inverse Problems in Econometrics Xiaohong Chen and Markus Reiss September 10, 2007 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. |
