A maximum-likelihood-based approach to estimate the long memory parameter using fractional spline wavelets

Abstract

Fractional spline wavelets act as fractional differentiators for essentially low-pass signals, which is a useful property to analyze time series with fractal behavior. Moreover, under suitable scaling of wavelet coefficients, one can whiten a fractional Gaussian noise inputted in the wavelet channels. In this article, we demonstrate the two former statements and, by employing them, propose a method based on maximum likelihood to estimate the long-memory parameter using fractional spline wavelets. The proposed estimator exhibits competitive behavior in terms of mean squared error in the simulation studies we conducted, dominating all other methods as the sample size increases. In empirical applications, the proposed method outperformed other approaches in 10 of 12 evaluations for three different time series. Moreover, the proposed method allows for estimation in a continuous search grid, even when the Hurst coefficient lies outside of stationarity boundaries, which is a clear advantage over many algorithms: in such cases, the proposed method displays a stable behavior and an increasing outperformance margin. We also propose an identification procedure for the persistence or antipersistence of a given signal, based on the energy concentration of the coefficients of wavelet decomposition and its theoretical properties.