Publication Date: November 2013
We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a diﬀicult ill-posed inverse problem with unknown operator in statistics. We ﬁrst establish a general upper bound on the sup-norm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal sup-norm convergence rates for spline and wavelet least squares regression estimators under weakly dependent data and heavy-tailed error terms. This upper bound also yields the sup-norm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal L2-norm rates for severely ill-posed problems, and are power of log(n) slower than the optimal L2-norm rates for mildly ill-posed problems. We then establish the minimax risk lower bound in sup-norm loss, which coincides with our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV estimators. This sup-norm rate optimality provides another justiﬁcation for the wide application of sieve NPIV estimators. Useful results on weakly-dependent random matrices are also provided.
Nonparametric instrumental variables, Statistical ill-posed inverse problems, Optimal uniform convergence rates, Weak dependence, Random matrices, Splines, Wavelets
JEL Classification Codes: C13, C14, C32