## Multivariate local polynomial fitting for martingale

nonlinear regression models

### Zhan-Qian Lu

*Department of Mathematics, The Hong Kong University of Science and Technology, *

Clear Water Bay, Kowloon, Hong Kong, China
(Received November 20, 1997; revised May 22, 1998)

**Abstract.**
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.

*Key words and phrases*:
Partial derivative
estimation, nonlinearity in time series, confidence
intervals, nonparametric estimation, sunspots data.

**Source**
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