Optimal uniform convergence rates for sieve nonparametric instrumental variables regression

Xiaohong Chen and Timothy M. Christensen

No 56/13, CeMMAP working papers from Institute for Fiscal Studies

Abstract: 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 matricies are also provided.

Date: 2013-11-04
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:azt:cemmap:56/13

DOI: 10.1920/wp.cem.2013.5613

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