Simultaneous Confidence Bands in Nonlinear Regression Models with Nonstationarity

Degui Li; Weidong Liu; Qiying  Wang; Wei Biao Wu

doi:10.5705/ss.202015.0219

Simultaneous Confidence Bands in Nonlinear Regression Models with Nonstationarity

Degui Li, Weidong Liu, Qiying Wang, Wei Biao Wu

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider nonparametric estimation of the regression function g(·) in a nonlinear regression model Yt = g(Xt) + σ(Xt)et, where the regressor (Xt) is a nonstationary unit root process and the error (et) is a sequence of independent and identically distributed (i.i.d.) random variables. With proper centering and scaling, the maximum deviation of the local linear estimator of the regression function g is shown to be asymptotically Gumbel. Based on the latter result, we construct simultaneous confidence bands for g, which can be used to test patterns of the regression function. Our results substantially extend existing ones which typically require independent or stationary weakly dependent regressors. Furthermore, we examine the finite sample behavior of the proposed approach via the simulated and real data examples.

Original language	English
Pages (from-to)	1385-1400
Journal	Statistica Sinica
Volume	27
Issue number	3
DOIs	https://doi.org/10.5705/ss.202015.0219
Publication status	Published - 1 Jul 2017

Bibliographical note

Date of Acceptance: 07/07/2016. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

Access to Document

10.5705/ss.202015.0219

SS-2015-0219R1
This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details
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Cite this

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title = "Simultaneous Confidence Bands in Nonlinear Regression Models with Nonstationarity",

abstract = "We consider nonparametric estimation of the regression function g(·) in a nonlinear regression model Yt = g(Xt) + σ(Xt)et, where the regressor (Xt) is a nonstationary unit root process and the error (et) is a sequence of independent and identically distributed (i.i.d.) random variables. With proper centering and scaling, the maximum deviation of the local linear estimator of the regression function g is shown to be asymptotically Gumbel. Based on the latter result, we construct simultaneous confidence bands for g, which can be used to test patterns of the regression function. Our results substantially extend existing ones which typically require independent or stationary weakly dependent regressors. Furthermore, we examine the finite sample behavior of the proposed approach via the simulated and real data examples.",

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AU - Liu, Weidong

AU - Wang, Qiying

AU - Wu, Wei Biao

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N2 - We consider nonparametric estimation of the regression function g(·) in a nonlinear regression model Yt = g(Xt) + σ(Xt)et, where the regressor (Xt) is a nonstationary unit root process and the error (et) is a sequence of independent and identically distributed (i.i.d.) random variables. With proper centering and scaling, the maximum deviation of the local linear estimator of the regression function g is shown to be asymptotically Gumbel. Based on the latter result, we construct simultaneous confidence bands for g, which can be used to test patterns of the regression function. Our results substantially extend existing ones which typically require independent or stationary weakly dependent regressors. Furthermore, we examine the finite sample behavior of the proposed approach via the simulated and real data examples.

AB - We consider nonparametric estimation of the regression function g(·) in a nonlinear regression model Yt = g(Xt) + σ(Xt)et, where the regressor (Xt) is a nonstationary unit root process and the error (et) is a sequence of independent and identically distributed (i.i.d.) random variables. With proper centering and scaling, the maximum deviation of the local linear estimator of the regression function g is shown to be asymptotically Gumbel. Based on the latter result, we construct simultaneous confidence bands for g, which can be used to test patterns of the regression function. Our results substantially extend existing ones which typically require independent or stationary weakly dependent regressors. Furthermore, we examine the finite sample behavior of the proposed approach via the simulated and real data examples.

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