Local Polynomial Estimation with a FARIMA-GARCH Error Process

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