Summary: Since the seminal paper of C. W. J. Granger and R. Joyeux [J. Time Ser. Anal. 1, 15–29 (1980; Zbl 0503.62079)], the concept of a long memory has focused the attention of many statisticians and econometricians trying to model and measure the persistence of stationary processes. Many methods for estimating d, the long-range dependence parameter, have been suggested since the work of H. E. Hurst [Trans. Am. Soc. Civil Eng. 116, 770–799 (1951)]. They can be summarized in three classes: the heuristic methods, the semi-parametric methods and the maximum likelihood methods.
We try, by simulation, to verify the two main properties of \(\hat d\): the consistency and the asymptotic normality. Hence, it is very important for practitioners to compare the performance of the various classes of estimators. The results indicate that only the semi-parametric and the maximum likelihood methods can give good estimators. They also suggest that the AR component of the ARFIMA\((1,d,0)\) process has an important impact on the properties of the different estimators and that the P. Whittle [Hypothesis testing in time series analysis. Uppsala: Almquist and Wicksell (1951; Zbl 0045.41301)] method is the best one, since it has small mean squared error. We finally carry out an empirical application using the monthly seasonally adjusted US Inflation series, in order to illustrate the usefulness of the different estimation methods in the context of using real data.