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LOCAL LINEAR FITTING UNDER NEAR EPOCH DEPENDENCE

Published online by Cambridge University Press:  06 December 2006

Zudi Lu  and
Oliver Linton
Zudi Lu
Affiliation:
London School of Economics, Chinese Academy of Sciences, Curtin University of Technology
Oliver Linton
Affiliation:
London School of Economics
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Abstract

Local linear fitting of nonlinear processes under strong (i.e., α-) mixing conditions has been investigated extensively. However, it is often a difficult step to establish the strong mixing of a nonlinear process composed of several parts such as the popular combination of autoregressive moving average (ARMA) and generalized autoregressive conditionally heteroskedastic (GARCH) models. In this paper we develop an asymptotic theory of local linear fitting for near epoch dependent (NED) processes. We establish the pointwise asymptotic normality of the local linear kernel estimators under some restrictions on the amount of dependence. Simulations and application examples illustrate that the proposed approach can work quite well for the medium size of economic time series.We thank Yuichi Kitamura and two referees for helpful comments. This research was partially supported by a Leverhulme Trust research grant, the National Natural Science Foundation of China, and the Economic and Social Science Research Council of the UK.

Information

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 1 , February 2007 , pp. 37 - 70
Copyright
© 2007 Cambridge University Press

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References

REFERENCES

Andrews, D.W.K. (1984) Non-strong mixing autoregressive processes. Journal of Applied Probability 21, 930–934.Google Scholar
Andrews, D.W.K. (1995) Nonparametric kernel estimation for semiparametric models. Econometric Theory 11, 560–596.Google Scholar
Bandi, F. (2004) On Persistence and Nonparametric Estimation (with an application to stock return predictability). Working paper, Graduate School of Business, University of Chicago.
Bierens, H. (1981) Robust Methods and Asymptotic Theory. Lecture Notes in Economics and Mathematical Systems 192. Springer.
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Carrasco, M. & X. Chen (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.Google Scholar
Cleveland, W.S. (1979) Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74, 829–836.Google Scholar
Cleveland, W.S. & C. Loader (1996) Smoothing by local regression: Principles and methods. In W. Härdle & M.G. Schimek (eds.), Statistical Theory and Computational Aspects of Smoothing, 10–49. Physica-Verlag.
Cline, D.B.H. & H.M.H. Pu (1999) Geometric ergodicity of nonlinear time series. Statistica Sinica 9, 1103–1118.Google Scholar
Craven, P. & G. Wahba (1979) Smoothing noise data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation. Numerical Mathematics 31, 377–403.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.
Deo, C.M. (1973) A note on empirical processes of strong mixing sequences. Annals of Probability 1, 870–875.Google Scholar
Devroye, L. & L. Györfi (1985) Nonparametric Density Estimation: The L1 View. Wiley.
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Fan, J. (1992) Design-adaptive nonparametric regression. Journal of the American Statistical Association 87, 998–1004.Google Scholar
Fan, J. & I. Gijbels (1992) Variable bandwidth and local linear regression smoothers. Annals of Statistics 20, 2008–2036.Google Scholar
Fan, J. & I. Gijbels (1995) Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaption. Journal of the Royal Statistical Society, Series B 57, 371–394.Google Scholar
Fan, J. & I. Gijbels (1996) Local Polynomial Modeling and Its Applications. Chapman and Hall.
Fan, J. & Q. Yao (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer-Verlag.
Gallant, A.R. (1987) Nonlinear Statistical Models. Wiley.
Gallant, A.R. & H. White (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Blackwell.
Goródetskii, V.V. (1977) On the strong mixing properties for linear sequences. Theory of Probability and Its Applications 22, 411–413.Google Scholar
Hastie, T. & C. Loader (1993) Local regression: Automatic kernel carpentry. Statistical Science 8, 120–143.Google Scholar
Ibragimov, I.A. (1962) Some limit theorems for stationary processes. Theory of Probability and Its Applications 7, 349–382.Google Scholar
Ibragimov, I.A. & Y.V. Linnik (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff.
Karlsen, H.A. & D. Tjøstheim (2001) Nonparametric estimation in null recurrent time series. Annals of Statistics 29, 372–416.Google Scholar
Ling, S. & W.K. Li (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 1184–1194.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.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. & Z. Jiang (2001) L1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Statistics and Probability Letters 51, 121–130.Google Scholar
Masry, E. (1986) Recursive probability density estimation from weakly dependent processes. IEEE Transactions on Information Theory IT-32, 254–267.Google Scholar
Masry, E. & J. Fan (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24, 165–179.Google Scholar
Masry, E. & D. Tjøstheim (1995) Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory 11, 258–289.Google 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 fur Wahrscheinlichskeittheorie 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
Moloche, G. (2000) Local Nonparametric Estimation of Scalar Diffusions. Working paper, Department of Economics, MIT.
Nakhapetyan, B.S. (1987) An approach to proving limit theorems for dependent random variables. Theory of Probability and Its Applications 32, 535–539.Google Scholar
Nze, P.A., P. Bühlmann, & P. Doukhan (2002) Weak dependence beyond mixing and asymptotics for nonparametric regression. Annals of Statistics 30, 397–430.Google Scholar
Nze, P.A. & P. Doukhan (2004) Weak dependence: Models and applications to econometrics. Econometric Theory 20, 995–1045.Google Scholar
Pham, T.D. (1986) The mixing properties of bilinear and generalized random coefficient autoregressive models. Stochastic Processes and Their Applications 23, 291–300.Google Scholar
Pham, D.T. & L.T. Tran (1985) Some strong mixing properties of time series models. Stochastic Processes and Their Applications 19, 297–303.Google Scholar
Phillips, P.C.B. & J.Y. Park (1998) Nonstationary Density Estimation and Kernel Autoregression. Cowles Foundation Discussion paper 1181.
Robinson, P.M. (1983) Nonparametric estimators for time series. Journal of Time Series Analysis 4, 185–207.Google Scholar
Ruppert, D. & M.P. Wand (1994) Multivariate locally weighted least squares regression. Annals of Statistics 22, 1346–1370.Google Scholar
Saikkonen, P. (2001) Stability Results for Nonlinear Vector Autoregressions with an Application to a Nonlinear Error Correction Model. Discussion papers of Interdisciplinary Research Project 373, http://sfb.wiwi.hu-berlin.de/.
Stone, C.J. (1977) Consistent nonparametric regression. Annals of Statistics 5, 595–620.Google 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.Google Scholar
Tong, H. (1990) Nonlinear Time Series: A Dynamical System Approach. Oxford University Press.
Wahba, G. (1977) A survey of some smoothing problems and the method of generalized cross-validation for solving them. In P.R. Krisnaiah (ed.), Applications of Statistics, pp. 507–523. North-Holland.
Weiss, A.A. (1984) ARMA model with ARCH errors. Journal of Time Series Analysis 5, 129–143.Google Scholar