December 1999, Volume 51, Issue 4, pp 691-706
Multivariate Local Polynomial Fitting for Martingale Nonlinear Regression Models
- Zhan-Qian Lu
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
Local polynomial modelling is a useful tool for nonlinear time series analysis. For nonlinear regression models with martingale difference errors, this paper presents a simple proof of local linear and local quadratic fittings under apparently minimal short-range dependence condition. Explicit formulae for the asymptotic bias and asymptotic variance are given, which facilitate numerical evaluations of these important quantities. The general theory is applied to nonparametric partial derivative estimation in nonlinear time series. A bias-adjusted method for constructing confidence intervals for first-order partial derivatives is described. Two examples, including the sunspots data, are used to demonstrate the use of local quadratic fitting for modelling and characterizing nonlinearity in time series data.
Castellana, J. V. and Leadbetter, M. R. (1986). On smoothed probability density estimation for stationary processes, Stochastic. Process. Appl., 21, 179–193.
Fan, J. and Gijbel, I. (1996). Local Polynomial Modelling and Its Applications, Chapman and Hall, London.
Härdle, W. and Tsybakov, A. (1997). Local polynomial estimators of the volatility function in nonparametric autoregression, J. Econometrics, 81, 223–242.
Härdle, W., Tsybakov, A. and Yang, L. (1996). Nonparametric vector autoregression, Working Papers, No. 61, Humboldt University, Berlin.
Lu, Z. Q. (1996). Multivariate locally weighted polynomial fitting and partial derivative estimation, J. Multivariate Anal., 59, 187–205.
Masry, E. (1996). Multivariate regression estimation: local polynomial fitting for time series, Stochastic. Process. Appl., 65(1), 81–101.
Masyr, E. and Fan, J. (1997). Local polynomial estimation of regression functions for mixing processes, Scand. J. Statist., 24(2), 165–179.
Mills, T. C. (1993). The Econometric Modelling of Financial Time Series, Cambridge University Press, Cambridge.
Nychka, D., Ellner, S., McCaffrey, D., and Gallant, A. R. (1992). Finding chaos in noisy systems, J. Roy. Statist. Soc. Ser. B, 54 (2), 399–426.
Priestley, M. B. (1988). Non-linear and Non-stationary Time Series Analysis, Academic Press, London.
Ruppert, D. and Wand, M. P. (1994). Multivariate locally weighted least squares regression, Ann. Statist., 22(3), 1346–1370.
Shiryayev, A. N. (1984). Probability, Springer, New York.
Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach, Oxford University Press, Oxford.
Welsh, A. H. (1996). Robust estimation of smooth regression and spread functions and their derivatives, Statist. Sinica, 6, 347–366.
- Multivariate Local Polynomial Fitting for Martingale Nonlinear Regression Models
Annals of the Institute of Statistical Mathematics
Volume 51, Issue 4 , pp 691-706
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Partial derivative estimation
- nonlinearity in time series
- confidence intervals
- nonparametric estimation
- sunspots data
- Industry Sectors
- Zhan-Qian Lu (1)
- Author Affiliations
- 1. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China