Long memory analysis is one of the most active areas in econometrics and time series where various methods have been introduced to identify and estimate the long memory parameter in partially integrated time series. One of the most common models used to represent time series that have a long memory is the ARFIMA (Auto Regressive Fractional Integration Moving Average Model) which diffs are a fractional number called the fractional parameter. To analyze and determine the ARFIMA model, the fractal parameter must be estimated. There are many methods for fractional parameter estimation. In this research, the estimation methods were divided into indirect methods, where the Hurst parameter is estimated first, and then the fractional integration parameter is estimated from it by a relation between them. As for direct methods, the fractional integration parameter is estimated directly without relying on Hurst's parameter, and most of them are semi parametric methods. In this paper, some of the most common direct methods were used to estimate the fraction modulus namely (Geweke-Porter-Hudak, Smoothed Geweke-Porter-Hudak, Local Whittle, Wavelet and weighted wavelet), using simulation method with different value of (d) and different size of time series. The comparison between the methods was done using the mean squared error (MSE). It turns out that the best methods to estimate the fractional parameter is (Local Whittle).
The ARFIMA model was generated by a function programmed by the MATLAB statistical program
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