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LOCAL LINEAR FITTING UNDER NEAR EPOCH DEPENDENCE: UNIFORM CONSISTENCY WITH CONVERGENCE RATES

Published online by Cambridge University Press:  27 April 2012

Degui Li ,
Zudi Lu  and
Oliver Linton
Degui Li
Affiliation:
Monash University
Zudi Lu
Affiliation:
University of Adelaide
Oliver Linton*
Affiliation:
University of Cambridge
*
*Address correspondence to Oliver Linton, Faculty of Economics, Cambridge, CB3 9DD, United Kingdom; e-mail: [email protected].
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Abstract

Local linear fitting is a popular nonparametric method in statistical and econometric modeling. Lu and Linton (2007, Econometric Theory23, 37–70) established the pointwise asymptotic distribution for the local linear estimator of a nonparametric regression function under the condition of near epoch dependence. In this paper, we further investigate the uniform consistency of this estimator. The uniform strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results regarding uniform convergence rates for nonparametric kernel-based estimators are provided. The results of this paper will be of wide potential interest in time series semiparametric modeling.

Type
Articles
Information
Econometric Theory , Volume 28 , Issue 5 , October 2012 , pp. 935 - 958
Copyright
Copyright © Cambridge University Press 2012

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References

REFERENCES

Andrews, D.W.K. (1984) Non–strong mixing autoregressive processes. Journal of Applied Probability 21, 930–934.CrossRefGoogle Scholar
Andrews, D.W.K. (1995) Nonparametric kernel estimation for semiparametric models. Econometric Theory 11, 560–596.CrossRefGoogle Scholar
Bierens, H. (1981) Robust Methods and Asymptotic Theory. Lecture Notes in Economics and Mathematical Systems 192. Springer-Verlag.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, 2nd ed.Lecture Notes in Statistics 110. Springer-Verlag.Google Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Cline, D.B.H. & Pu, H.M.H. (1999) Geometric ergodicity of nonlinear time series. Statistica Sinica 9, 1103–1118.Google Scholar
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Annals of Statistics 25, 1–37.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. Chapman & Hall.Google Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer-Verlag.Google Scholar
Gallant, A.R. (1987) Nonlinear Statistical Models. Wiley.Google Scholar
Gallant, A.R. & White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Blackwell.Google Scholar
Hansen, B.E. (2008) Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726–748.Google Scholar
Hong, Y. (2000) Generalized spectral tests for serial dependence. Journal of the Royal Statistical Society, Series B 62, 557–574.Google Scholar
Ibragimov, I.A. (1962) Some limit theorems for stationary processes. Theory of Probability and Its Applications 7, 349–382.Google Scholar
Kanaya, S. (2010) Uniform Convergence Rates of Kernel-Based Nonparametric Estimators for Continuous Time Diffusion Processes: A Damping Function Approach. Manuscript, Department of Economics, University of Oxford.Google Scholar
Koo, B. & Linton, O. (2010) Semiparametric Estimation of Locally Stationary Diffusion Processes. Manuscript, LSE.Google Scholar
Kristensen, D. (2009) Uniform convergence rates of kernel estimators with heterogenous dependent data. Econometric Theory 25, 1433–1445.Google Scholar
Li, Q. & Racine, J.S. (2007) Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
Liebscher, E. (1996) Strong convergence of sums of α-mixing random variables with applications to density estimation. Stochastic Processes and Their Applications 65, 69–80.Google Scholar
Lin, Z. (2004) Strong near epoch dependence. Science in China (Series A) 47, 497–507.Google Scholar
Ling, S. (2007) Testing for change points in time series models and limiting theorems for NED sequences. Annals of Statistics 35, 1213–1237.CrossRefGoogle Scholar
Linton, O.B. & Mammen, E. (2005) Estimating semiparametric ARCH(∞) models by kernel smoothing methods. Econometrica 73, 771–836.CrossRefGoogle Scholar
Linton, O.B. & Sancetta, A. (2009) Consistent estimation of a general nonparametric regression function in time series. Journal of Econometrics 152, 70–78.Google Scholar
Lu, Z. (1996a) A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model. Statistics and Probability Letters 30, 305–311.Google Scholar
Lu, Z. (1996b) Geometric ergodicity of a general ARCH type model with applications to some typical models. Chinese Science Bulletin 41, 1630(digest).Google Scholar
Lu, Z. (1998) On geometric ergodicity of a non-linear autoregressive (AR) model with an autoregressive conditional heteroscedastic (ARCH) term. Statistica Sinica 8, 1205–1217.Google Scholar
Lu, Z. (2001) Asymptotic normality of kernel density estimators under dependence. Annals of the Institute of Statistical Mathematics 53, 447–468.Google Scholar
Lu, Z. & Cheng, P. (1997) Distribution-free strong consistency for nonparametric kernel regression involving nonlinear time series. Journal of Statistical Planning and Inference 65, 67–86.Google Scholar
Lu, Z. & Linton, O. (2007) Local linear fitting under near epoch dependence. Econometric Theory 23, 37–70.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.Google Scholar
Masry, E. & Tjøstheim, D. (1995) Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory 11, 258–289.CrossRefGoogle Scholar
McLeish, D.L. (1975a) A maximal inequality and dependent strong laws. Annals of Probability 3, 826–836.Google Scholar
McLeish, D.L. (1975b) Invariance principles for dependent variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 32, 165–178.Google Scholar
McLeish, D.L. (1977) On the invariance principle for nonstationary mixingales. Annals of Probability 5, 616–621.Google Scholar
Nze, P.A., Bühlmann, P., & Doukhan, P. (2002) Weak dependence beyond mixing and asymptotics for nonparametric regression. Annals of Statistics 30, 397–430.Google Scholar
Nze, P.A. & Doukhan, P. (2004) Weak dependence: Models and applications to econometrics. Econometric Theory 20, 995–1045.Google Scholar
Shao, Q. & Yu, H. (1996) Weak convergence for weighted empirical processes of dependent sequences. Annals of Probability 24, 2098–2127.CrossRefGoogle Scholar
Stone, C.J. (1980) Optimal rates of convergence for nonparametric estimators. Annals of Statistics 8, 1348–1360.Google Scholar
Tjøstheim, D. (1990) Nonlinear time series and Markov chains. Advances in Applied Probability 22, 587–611.CrossRefGoogle Scholar
Tong, H. (1990) Nonlinear Time Series: A Dynamical System Approach. Oxford University Press.Google Scholar
Wang, Q. & Phillips, P.C.B. (2009) Structural nonparametric cointegrating regression. Econometrica 77, 1901–1948.Google Scholar
Withers, C.S. (1981) Conditions for linear processes to be strong-mixing. Probability Theory and Related Fields 57, 477–480.Google Scholar