Estimation of Hurst exponent revisited

Abstract

In order to estimate the Hurst exponent of long-range dependent time series numerous estimators such as based e.g. on rescaled range statistic ($R/S$) or detrended fluctuation analysis (DFA) are traditionally employed. Motivated by empirical behaviour of the bias of $R/S$ estimator, its bias-corrected version is proposed. It has smaller mean squared error than DFA and behaves comparably to wavelet estimator for traces of size as large as $2^{15}$ drawn from some commonly considered long-range dependent processes. It is also shown that several variants of $R/S$ and DFA estimators are possible depending on the way they are defined and that they differ greatly in their performance.

Introduction

Let ${\left(X_t\right)}_{t=1}^{\infty }$ be a real-valued stationary time series such that its covariance function $\gamma (k)≔\mathrm{Cov}\left(X_t,X_{i+k}\right)$ $\sim c_{\gamma }|k{|}^{-\gamma }$ for $\left|k\right|\to \infty$, where $0<\gamma <1$ and $\sim$ denotes asymptotic equivalence. This is the most important case of long-range dependence (LRD), which in a general situation is defined by the condition ${\sum }_{k=0}^{\infty }|\gamma (k)|=\infty$, and it encompasses two frequently studied processes having this property, namely a fractional Gaussian noise (fGn) and a fractional autoregressive integrated moving average (FARIMA). The phenomenon of long-range dependence is a topic of active research in statistics as well as in many areas of applied sciences e.g. in economics, geophysics and meteorology. We refer to Beran (1994) for a book-length treatment of this subject.

Assuming the above condition of the hyperbolic decay of covariance, it is easily seen that, with $Y_n={\sum }_{t=1}^n{X}_t$, we have

$$\mathrm{Var}\left(Y_n\right)\sim \frac{2c_{\gamma }}{(1-\gamma )(2-\gamma )}{n}^{2-\gamma }≕C_{\gamma }{n}^{2H}\text{,}$$

where $H≔1-\gamma /2$ is traditionally called the Hurst exponent (cf. Hurst (1951)). As the Hurst exponent describes the strength of dependence, its estimation is of a great interest. A process with a larger value of H is more regular and less erratic than a process with a smaller one. Most estimators are based on scaling properties similar to a property following from (1), namely that $\mathrm{Var}\left(Y_{\mathit{nk}}\right)\sim k^{2H}\mathrm{Var}\left(Y_n\right)$, for any $k\in \mathbb{N}$. Thus e.g. regressing a logarithm of some estimator of the variance of the partial sums against the logarithm of its size yields an estimator of $2H$. However, as in the case described above, this does not always yield a reliable estimator of H. For a discussion of such approach and a review of long-range dependence see Robinson (1994). A Monte Carlo experiment comparing performance of some such estimators is discussed in Taqqu et al. (1995).

We consider below three of the most popular estimators based on scaling property: $R/S$ estimator based on rescaled adjusted range statistic, DFA estimator pertaining to detrended fluctuation analysis and a wavelet estimator. Important competitors include, among others, the log periodogram estimator (Geweke and Porter-Hudak, 1983) based on an approximate scaling of spectral density and the Whittle (1953) estimator defined as the maximizer of an approximate version of loglikelihood. A semiparametric version of the last proposal called the local Whittle (LW) or Gaussian semiparametric estimator (Robinson, 1995) is proved to be more efficient than the log periodogram regression estimator. Modifications of the LW estimator intended to reduce its bias when the underlying LRD process is contaminated by noise have been proposed and studied by Andrews and Sun (2004), Arteche (2004) and Hurvich et al. (2005). They are shown to have a smaller mean squared error (MSE) than the previous proposals when the noise to signal ratio is considerable (see e.g. Hurvich and Ray, 2003). For recent contributions concerning estimation of Hurst exponent see also Lai (2004) and Stoev et al. (2006).

In Sections 2 and 3 we show that the performance of the studied estimators depends crucially on a way they are constructed. Moreover, as the main contribution of the paper, we indicate that $R/S$ estimator, which is immensely popular among practitioners, but is widely known to be suboptimal can be enhanced by a bias correction to the effect that it outperforms the DFA estimator with respect to the MSE and performs on par with the wavelet estimator. In Section 2 we describe possible variants of $R/S$ estimator, choose one of them and show how its performance can be improved by a bias correction. The bias correction method is based on approximate linearity of the bias of considered $R/S$ estimator and takes advantage of the exact value of the slope of the approximating line.

In Section 3 careful analysis of the DFA estimator is provided. In Section 4 we briefly describe the wavelet estimator and in Section 5 we compare the performance of all three semiparametric estimators and the Whittle estimator for FARIMA and fGn processes with provided sample sizes ranging from $2^9$ to $2^{15}$. An application to analysis of daily exchange rates listed by the National Bank of Poland is discussed. Section 6 concludes the paper.