Optimal Convergence Rates in Non‐Parametric Regression with Fractional Time Series Errors

Abstract

Consider the estimation of g^(v), the νth derivative of the mean function, in a fixed‐design non‐parametric regression model with stationary time series errors ξ_i. We assume that g ∈ C^k, ξ_i are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ξ_i has the form f(λ) ~ c_f|λ|^-α as λ→0 with constants c_f >0 and α ∈ (−1,1). Under regularity conditions, the optimal convergence rate of ^g^(v) is shown to be n^-r_v with r_v=(1−α)(k−ν)/(2k+1−α). This rate is achieved by local polynomial fitting. Moreover, in spite of including long memory and antipersistence, the required conditions on the innovation distribution turn out to be the same as in non‐parametric regression with iid errors.

Keywords: Optimal rate of convergence, non‐parametric regression, long memory, antipersistence