Local Polynomial Estimation with a FARIMA-GARCH Error Process

This paper considers a class of semiparametric models being the sum of a nonparametric trend function g and a FARIMA-GARCH error process. Estimation of g( ), the th derivative of g, by local polynomial tting is investigated. The focus is on the derivation of the asymptotic normality of ĝ( ). At rst a central limit theorem based on martingale theory is developed and asymptotic normality of the sample mean of a FARIMA-GARCH process is proved. The central limit theorem is then extended from the case of an unweighted sum to a weighted sum in order to show the asymptotic normality of ĝ( ). As an auxiliary result, the weak consistency of a weighted sum is obtained for second order stationary time series with shortor long memory under very weak conditions. Asymptotic results on ĝ( ) in the presentation of long memory as well as antipersistence are also given for the current model.