## Bernoulli

• Bernoulli
• Volume 7, Number 5 (2001), 733-750.

### Local polynomial estimation with a FARIMA-GARCH error process

#### Abstract

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..

#### Article information

Source
Bernoulli, Volume 7, Number 5 (2001), 733-750.

Dates
First available in Project Euclid: 15 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1079399539

Mathematical Reviews number (MathSciNet)
MR2002h:62087

Zentralblatt MATH identifier
0985.62033

#### Citation

Beran, Jan; Feng, Yuanhua. Local polynomial estimation with a FARIMA-GARCH error process. Bernoulli 7 (2001), no. 5, 733--750. https://projecteuclid.org/euclid.bj/1079399539

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