An Introduction to Nonparametric Long Memory Time Series

This chapter introduces nonparametric methods for analyzing long memory time series, offering flexible alternatives to parametric models when the underlying functional form is unknown or complex nonlinear relationships are present. The general framework for nonparametric time series models is established, where the conditional mean function m(·) relating current observations to past values is estimated without imposing rigid structural assumptions. Kernel-based estimation methods are developed as the primary nonparametric tool, with the Nadaraya-Watson estimator presented for both first-order and higher-order autoregressive structures. A comprehensive treatment of kernel functions is provided, including Gaussian, Epanechnikov, uniform, bi-weight, and tri-cube kernels, with visual illustrations of their weighting properties. The critical issue of bandwidth selection is addressed through cross-validation methodology, with R code implementation demonstrating the bias-variance tradeoff. The role of smoothing parameters in estimation quality is illustrated through synthetic examples showing how different bandwidth choices affect the smoothness and accuracy of estimated functions. Simulation studies compare nonparametric kernel estimators with parametric ARFIMA models for long memory time series forecasting. Practical exercises guide readers in applying these methods to real datasets including airline passenger data and US macroeconomic indicators.