Discrete Dynamics in Nature and Society

Volume 2018, Article ID 3051632, 7 pages

https://doi.org/10.1155/2018/3051632

## Memories of the Gold Foreign Exchange Market Based on a Moving -Statistic and Wavelet-Based Multiresolution Analysis

^{1}College of Management Science, Chengdu University of Technology, Chengdu, Sichuan Province 610059, China^{2}Geomathematics Key Laboratory of Sichuan Province, Chengdu, Sichuan Province 610059, China^{3}School of Economics and Management, University of Electronic Science and Technology, Chengdu, Sichuan Province 611731, China

Correspondence should be addressed to Bin Liu; moc.361@micnibuil

Received 19 January 2018; Revised 18 July 2018; Accepted 7 August 2018; Published 23 August 2018

Academic Editor: Francisco R. Villatoro

Copyright © 2018 Peng Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Memory in finance is the foundation of a well-established forecasting model, and new financial theory research shows that the stochastic memory model depends on different time windows. To accurately identify the multivariate long memory model in the financial market, this paper proposes the concept of a moving -statistic on the basis of a modified method to determine whether the time series has a long-range dependence and subsequently to apply wavelet-based multiresolution analysis to study the multifractality of the financial time series to determine the initial data windows. Finally, we check the moving -statistic estimation in wavelet analysis in the same condition; the paper selects the volatilities of the gold foreign exchange rates to evaluate the moving -statistic. According to the results, the method of testing memory established in this paper can identify the breakpoint of the memories effectively. Furthermore, this method can provide support for forecasting returns in the financial market.

#### 1. Introduction

The empirical analysis of long-term memory originated in the natural sciences. Since the 1980s, econometrists have introduced the long-term memory model in the financial field and considered that the cornerstones of the long-term memory of the finance market include the theories of noise trading [1], behavioral financial theory [2], and the fractal market hypothesis [3]. As the basis of an established forecasting model, many scholars have conducted extensive and in-depth research and have formed the Hurst index technology based on time domain analysis [4–6] and the fractional order difference parameter technique based on frequency domain analysis [7–9]. There is substantial literature improving the ability of the accurate judgment for the long-term memory by optimizing parameter algorithms [10–12].

With the continuous improvement of traditional methods of estimation and inspection, scholars have applied the long-term memory model in the field of gold foreign exchange market. Bentes analyzes the robustness and consistency of long memory volatility of gold price returns during different crisis periods by FIGARCH model [13]. Ali Habibnia establishes the model for the world gold price with Logistic Smooth Transition Autoregressive (STAR) model concerning long memory effect and compared it with other models [14]. Yang Na explores the long memory property on gold price volatility by calculating the Hurst index and establishes a family of long memory forecasting models based on fractal analysis. It shows the inherent volatility quality of gold price sequences and it has strong predictive capabilities [15]. Maurice Omane-Adjepong examines the presence of long-range dependence in the world’s gold market returns and volatility by using sampled historical daily gold market data to be less risky for hedging and portfolio diversification [16].

Mandelbrot (1997) introduced multifractal models to address the shortcomings of traditional models, which are not compatible with the stylized facts of time series, such as long-term memory and fat-tails in volatilities. Long-term memory models using wavelet-based multiresolution and Hurst index are widely researched from the perspective of multifractal in financial time series. As a comparatively new and powerful mathematical tool for time series analysis, multiresolution decomposition (MRD) is one of the basic tools of wavelet theory. Wavelet analysis is a time-frequency analysis method regarding signal in the time and frequency domain which has the ability of denoting partial signal characteristics. In 1989, Mallat and Meyer proposed the multiresolution analysis (MRA) theory and provided a numerical algorithm of discrete wavelet, namely, the Mallat tower algorithm (MTA) [17]. Wavelet-based multiresolution has a very wide range of applications in the financial sector, from descriptive analysis on different time scales to parameter estimation of multifractal properties and revelation of multifractality in cross-correlativity. For example, the correlation function of the wave logarithm in different time scales is analyzed to reveal causal information from low frequencies to high frequencies [18]. Schmitt shows the multifractal characteristics of foreign exchange earnings and estimates parameters expressing the small and medium strength fluctuation characteristics under the general multiface structure by means of multifractal analysis regarding the five daily foreign exchange rates [19]. The multifractal property is proved to exist in the cross-correlativity on the basis of an RMB/dollar exchange rate and daily price data of the Shanghai Composite Index [20].

The abovementioned literature studies the characteristics of memory by wavelet-based multiresolution analysis from the perspective of multifractal property depending on the fixed financial time series. Financial time series exhibit high degrees of nonlinear variability and multivariate long-term memory originates because of multiplicative interactions in different time windows; the multifractality of times series determines the multivariate long-term memory model.

To identify the breakpoint of memory in financial time series at a certain time scale and specific style of the multifractal form, this paper will estimate the dynamic value based on the specific branch level memory of the multifractal properties perspective. The remainder of the paper is organized as follows: after Section 1 outlines the development and application of long-term memory in financial time series, Section 2 introduces the concept of the moving -statistic after reviewing the modified theory and reviews wavelet-based multiresolution analysis, and Section 3 applies the model to evaluate the memory by selecting the high-frequency data of the gold price. The study’s conclusions are presented in Section 4.

#### 2. The Moving -Statistic and Wavelet-Based Multiresolution Analysis

##### 2.1. Modified Theory and -Statistic

In 1991, Lo put forward the modified theory [22] based on the classic theory to better distinguish between long- and short-range dependence. For the time series , given a sample of observation, , the definition of the modified rescaled range theory is as follows. is the square root of a consistent estimator of the partial sum’s variance.where and denotes the mean value of the time series. denotes the standard deviation of the time series after modification. This deviation involves not only sums of squared deviations of , but also its weight of autocovariances up to lag . where denotes the lag factor of the time series, according to Andrews’ (1991) data-dependent rule as in Lo (1991).where , denotes the greatest integer less than or equal to , and is the estimated first-order autocorrelation coefficient of the data.

The normalized classical Hurst-Mandelbrot rescaled range :

Compared with the classical analysis method, the main advantage of the modified analysis method is to avoid computing the Hurst index. The standard deviation is modified by introducing the lag factor to exclude the short-term memory of the time series for testing long-term memory, which makes long-term memory detection more robust.

##### 2.2. Definition of the Moving -Statistic

The characteristics of memory diversification are produced because the value of differs based on different time windows selected under the multifractal properties of the financial time series conditions. This eliminates interference of the memory test from the initial time window, which contributes to the multifractal property, to observe the dynamic change process of memory in the financial time series. This paper proposes the concept of a moving -statistic based on a nonfixed scale on the basis of the classical statistic.

*Definition 1. *Given time series , where k is the length of the time series:where denotes length of time series in the initial time windows, namely, in the time series , the initial data from the time series consist of . denotes the moving -statistic progress step (), with implied calculation accuracy.

In the process of testing the memory of the nonfixed scale by utilizing the moving -statistic, suppose that the starting data of time series for analysis will be written as , then the breakpoint of memory property of the time series , , ,…, should be identified. The initial window data should satisfy the following conditions:

(i) As to the initial time window data, ,…,, under the condition of multilevel fractal analysis, the fractal level of samples ,…, is at a lower fractal level than the time series data , , ,…, .

(ii) Under the condition of multilevel fractal analysis, the level of the fractal of the sample data , ,…, is the same as the level of the sample data ,…,.

The moving -statistic can eliminate all short-term memory within the target time windows, as well as the initial time window data, and would not affect memory independence in the target time window data.

##### 2.3. Wavelet-Based Multiresolution Analysis

A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most relevant to discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in the theory of differential equations (the ironing method) and the pyramid methods of image processing by Stephane Mallat and Yves Meyer in 1988/89. Multiresolution signal decomposition and the reconstruction algorithm (and the fast algorithm of orthogonal wavelet transform), generally called the Mallat algorithm, include two key steps: decomposition and reconstruction.

###### 2.3.1. Decomposition

###### 2.3.2. Reconstruction

Mallat’s algorithm is useful in representing the wavelet transform as a pyramid [23]. The base of the pyramid is the original data of high resolution, and the top is a low-resolution approximation, and the size and resolution will be reduced as the pyramid upper moves. Many studies establish a prediction model via multiresolution analysis method to forecast gold price volatility [24–26]. This paper will use wavelet-based multiresolution analysis to explore memory feature in gold price volatility in different time scales. The four criteria as followed are considered for selecting the mother wavelet adopted in this paper [27].

(i) Vanishing moments: the wavelet function should have a small enough vanishing moments to represent multifractality of the high-frequency data.

(ii) Cutoff frequencies: the wavelet should provide not sharp cutoff frequencies to magnify the adjacent resolution levels.

(iii) Orthonormal: the wavelet basis should be orthonormal.

(iv) The similarity of wavelet coefficients: for applications where the information lasts for a very short instant, wavelets with less number of coefficients are better choices.

There are several well-known families of orthogonal wavelets. An incomplete list includes Harr, Meyer family, Daubechies family, Coiflet family, and Symmlet family [28]. Prior studies [29, 30] show that gold has nonlinear multiresolution characteristic in different time scales. Daubechies wavelets are selected in this paper due to their outstanding performance in detecting waveform discontinuities for evaluating the memorial breakpoint [31].

The hour returns of exchange rate between gold and the US dollar are chosen as the target data for detecting the memorial breakpoint of the high-frequency data from the perspective of multifractality, rather than not smoothing signal, while the larger the vanishing moment of wavelet filter is, the shaper its cutoff frequency is. So filter banks of Daubechies 3 (db3) are selected for determining the initial data window and evaluation in comparison with Daubechies 5 (db5).

##### 2.4. Analysis Process of the Moving -Statistic

To resolve multifractality of the time series, this paper utilizes wavelet analysis to recognize and reconstruct financial time series and later calculate the moving -statistic to determine the breakpoint of the memory. The results are then subjected to wavelet analysis for evaluation and this process is shown in Figure 1.