Kernel Dependent Functions in Nonparametric Regression with Fractional Time Series Errors

This paper considers estimation of the regression function and its derivatives in nonparametric regression with fractional time series errors. We focus on investigating the properties of a kernel dependent function V (δ) in the asymptotic variance and finding closed form formula of it, where δ is the long-memory parameter. It is shown that V (δ) has a unified form for δ ∈ (−0.5, 0.5)\0 with V (0) := lim δ→0 V (δ) = R(K), the kernel constant for iid errors. General solution of V (δ) for polynomial kernels is given together with a few examples. It is also found, e.g. that the Uniform kernel is no longer the minimum variance one by strongly antipersistent errors and that, for a fourth order kernel, V (δ) at some δ > 0 is clearly smaller than R(K). The results are used to develop a general data-driven algorithm. Data examples illustrate the practical relevance of the approach and the performance of the algorithm.