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..