Using wavelets to obtain a consistent ordinary least squares estimator of the long-memory parameter

Abstract

We develop an ordinary least squares estimator of the long-memory parameter from a fractionally integrated process that is an alternative to the Geweke and Porter-Hudak (1983) estimator. Using the wavelet transform from a fractionally integrated process, we establish a log-linear relationship between the wavelet coecients' variance and the scaling parameter equal to the log-memory parameter. This log-linear relationship yields a consistent ordinary least squares estimator of the long-memory parameter when the wavelet coecients' population variance is replaced by their sample variance. We derive the small sample bias and variance of the ordinary least squares estimator and test it against the GPH estimator and the McCoy±Walden maximum likelihood wavelet estimator by conducting a number of Monte Carlo experiments. Based upon the criterion of choosing the estimator which minimizes the mean squared error, the wavelet OLS approach was superior to the GPH estimator, but inferior to the McCoy±Walden wavelet estimator for the processes simulated. However, given the simplicity of programming and running the wavelet OLS estimator and its statistical inference of the long-memory parameter we feel the general practitioner will be attracted to the wavelet OLS estimator.

Commodity futures returns: more memory than you might think!†

“…Although a little less supportive than the wavelet analysis, these results still 11 The computations are implemented using OX 4.1 and Matlab 7.8. contrast with Crato and Ray (2000) who found no evidence of long memory. In any case and as noted earlier, Jensen (1999) demonstrates that the wavelet estimator has a significantly smaller MSE than the GPH estimator. Jin et al (2006) argue that this may imply considerably lower sampling variability which may translate into greater power against the null of no fractional integration.…”

Fractional Differencing in Discrete Time

“…For gold, the GP H estimator for the fractional differencing parameter is negative and insignificant, while the GSP , W OLS and BW M LE estimators are negative and significant. The W OLS estimator is negative and largest in magnitude, consistent with its large negative bias in Monte-Carlo studies (Jensen (1999), Elder and Jubinski (2010)).…”

“…We use four methods to estimate the fractional differencing parameter. Two methods are based on frequency domain analysis, as developed in Geweke and Porter-Hudak (1983) (GP H) and Robinson (1995a) (GSP ), and two methods are based on wavelet analysis, as developed by Wornell and Oppenheim (1992), Jensen (1999) and Jensen (2000). Our empirical results suggest some evidence of antipersistence in gold and copper returns, and strong evidence of fractional differencing in each of the volatility series, as measured by absolute returns.…”

Estimators for the long-memory parameter in LARCH models, and fractional Brownian motion

“…This remark is also supported by numerical evidence, since our explicit formula (24) for e k allows us to compute the estimation error empirically.…”