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Local Polynomial Estimation with a FARIMA-GARCH Error Process

Jan Beran and Yuanhua Feng
Bernoulli
Vol. 7, No. 5 (Oct., 2001), pp. 733-750
DOI: 10.2307/3318539
Stable URL: http://www.jstor.org/stable/3318539
Page Count: 18
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Local Polynomial Estimation with a FARIMA-GARCH Error Process
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Abstract

This paper considers estimation of the trend function g as well as its vth derivative g(v) in a so-called semi-parametric FARIMA-GARCH model by local polynomial fits. The focus is on the derivation of the asymptotic normality of ĝ(v). 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 ĝ(v). 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 ĝ(v) are also given.

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