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Multivariate Local Polynomial Estimators: Uniform Boundary Properties and Asymptotic Linear Representation

aDepartment of Economics, University of Washington, Seattle, WA, USA
bSchool of Economics and Finance, Queen Mary, University of London, London, UK

Essays in Honor of Aman Ullah

ISBN: 978-1-78560-787-5, eISBN: 978-1-78560-786-8

Publication date: 23 June 2016

Abstract

The asymptotic bias and variance of a general class of local polynomial estimators of M-regression functions are studied over the whole compact support of the multivariate covariate under a minimal assumption on the support. The support assumption ensures that the vicinity of the boundary of the support will be visited by the multivariate covariate. The results show that like in the univariate case, multivariate local polynomial estimators have good bias and variance properties near the boundary. For the local polynomial regression estimator, we establish its asymptotic normality near the boundary and the usual optimal uniform convergence rate over the whole support. For local polynomial quantile regression, we establish a uniform linearization result which allows us to obtain similar results to the local polynomial regression. We demonstrate both theoretically and numerically that with our uniform results, the common practice of trimming local polynomial regression or quantile estimators to avoid “the boundary effect” is not needed.

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Acknowledgements

Acknowledgements

We thank Ruixuan Liu for drawing our attention to Chen and Wu (2013) and Shih-Tang Hwu for pointing out an error in a preliminary draft. All remaining errors are ours. Yanqin Fan is grateful to Aman Ullah for introducing him to nonparametric econometrics and for his continued support throughout his career.

Citation

Fan, Y. and Guerre, E. (2016), "Multivariate Local Polynomial Estimators: Uniform Boundary Properties and Asymptotic Linear Representation", Essays in Honor of Aman Ullah (Advances in Econometrics, Vol. 36), Emerald Group Publishing Limited, Leeds, pp. 489-537. https://doi.org/10.1108/S0731-905320160000036023

Publisher

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Emerald Group Publishing Limited

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