Xiaohong Chen, Timothy Christensen
arXiv 2 Nov 2013 · Mathematics — Statistics Theory · 14 citations (OpenAlex)
arXiv:1311.0412 · PDF · DOI · OpenAlex · Extracted main text
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 $L^2$-norm rates for severely ill-posed problems, and are power of $\log(n)$ slower than the optimal $L^2$-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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Blundell, R., X. Chen, and D. Kristensen (2007) Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves | 1.000 | 12 | 3 | 100% |
| 2 | Chen, X. and M. Reiss (2011) On Rate Optimality for Ill-Posed Inverse Problems in Econometrics self | 1.000 | 10 | 3 | 100% |
| 3 | Chen, X. and D. Pouzo (2012) Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals self | 1.000 | 9 | 4 | 100% |
| 4 | Hall, P. and J. L. Horowitz (2005) Nonparametric Methods for Inference in the Presence of Instrumental Variables | 1.000 | 6 | 3 | 100% |
| 5 | Newey, W. K (1997) Convergence Rates and Asymptotic Normality for Series Estimators | 1.000 | 5 | 3 | 100% |
| 6 | Huang, J. Z (1998) Projection Estimation in Multiple Regression with Application to Functional ANOVA Models | 0.941 | 6 | 3 | 83% |
| 7 | Horowitz, J. L (2011) Applied Nonparametric Instrumental Variables Estimation | 0.928 | 4 | 3 | 100% |
| 8 | Stone, C. J (1982) Optimal Global Rates of Convergence for Nonparametric Regression | 0.928 | 4 | 3 | 100% |
| 9 | Newey, W. K. and J. L. Powell (2003) Instrumental Variable Estimation of Nonparametric Models | 0.874 | 5 | 2 | 100% |
| 10 | Huang, J. Z (2003) Local Asymptotics for Polynomial Spline Regression | 0.776 | 15 | 6 | 47% |
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