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October 2001 Local polynomial estimation with a FARIMA-GARCH error process
Jan Beran, Yuanhua Feng
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Bernoulli 7(5): 733-750 (October 2001).


This paper considers estimation of the trend function $g$ as well as its $\nu$th derivative $g^{(\nu)}$ in a so-called semi-parametric FARIMA-GARCH model by local polynomial fits. The focus is on the derivation of the asymptotic normality of $\hat g^{(\nu)}$. A central limit theorem based on martingale theory is developed. Asymptotic normality of the sample mean of a FARIMA-GARCH process is proved. These results are then used to show the asymptotic normality of $\hat g^{(\nu)}$. As an auxiliary result, the weak consistency of a weighted sum is obtained for second-order stationary time series with short or long memory under very weak conditions. Formulae for the mean integrated square error and the asymptotically optimal bandwidth of $\hat g^{(\nu)}$ are also given..


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Jan Beran. Yuanhua Feng. "Local polynomial estimation with a FARIMA-GARCH error process." Bernoulli 7 (5) 733 - 750, October 2001.


Published: October 2001
First available in Project Euclid: 15 March 2004

zbMATH: 0985.62033
MathSciNet: MR2002H:62087

Keywords: asymptotic normality , FARIMA-GARCH process , Local polynomial estimation , long memory , Martingales , semi-parametric models

Rights: Copyright © 2001 Bernoulli Society for Mathematical Statistics and Probability

Vol.7 • No. 5 • October 2001
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