Multifractal detrended fluctuation analysis on high-frequency SZSE in Chinese stock market

Physica A 521 (2019) 225–235

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Physica A journal homepage: www.elsevier.com/locate/physa

Multifractal detrended fluctuation analysis on high-frequency SZSE in Chinese stock market Danlei Gu, Jingjing Huang

∗

School of Science, Beijing Information Science & Technology University, Beijing 100192, PR China

highlights • • • •

Use MF-DFA to investigate the multifractal behavior of high-frequency SZSE. Determine generalized Hurst exponent and singularity spectrum of SZSE. All data are divided into 6 units and their main parameters are obtained. Compare the MF-DFA results for the original SZSE with shuffled series.

article

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Article history: Received 13 July 2018 Received in revised form 15 October 2018 Available online 23 January 2019 Keywords: Multifractal detrended fluctuation analysis (MF-DFA) Generalized Hurst exponent Multifractal spectrum High-frequency stock data

a b s t r a c t We use multifractal detrended fluctuation analysis (MF-DFA) method to investigate the multifractal behavior of Shenzhen Component Index (SZSE) 5-minute high-frequency stock data from 2017.6.15 - 2018.4.11. We determine generalized Hurst exponent and singularity spectrum and find that these fluctuations have multifractal nature. In order to maintain the long-term memory of the stock , all 9696 data are divided into 6 units. According to the multifractal spectrum, the main parameters of the six units are obtained. Comparing the MF-DFA results for the original SZSE high-frequency time series with those for shuffled series, we conclude that the origin of multifractality is due to both the broadness of probability density function and long-range correlation. The generalized Hurst exponent obviously depend on the order of fluctuation function and change with it. The curve of scaling function clearly departs from a straight line, i.e. it shows clearly nonlinear property, and the multifractal spectrum displays the commonly observed bell shape. This will provide an important and theoretical foundation for researching the forecasting of finance markets. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In the past few years, the analysis of financial time series has become the focus of research [1–7]. People often think that the change of stock price is a random process. Until recently, people discovered that the change of stock price was not strictly random but correlated and predictable even in the most competitive market. With the development of science, the application of fractal theory has provided new technical support for stock enthusiasts [8,9]. It can solve the problems that the effective market theory cannot solve more effectively, and analyze the nonlinear behavioral characteristics of the stock market as detailed as possible. Single fractals are obviously not able to accurately describe the real situation of the stock market, and a powerful way to characterize these features is the new development of fractal theory — Multifractal theory. Multifractal theory can obtain different fluctuation information of financial asset prices on different time scales, and provide more analysis for unpredictable financial markets [10,11]. ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Huang). https://doi.org/10.1016/j.physa.2019.01.040 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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The oldest multifractal analysis method is multifractal box counting (MF-BOX) [12], which, however, fails in the presence of non-stationary series such as trends. This deficiency led to the development of wavelet transform modulus maxima (WTMM) method- a generalized box counting approach based on wavelet transform [13–15]. Another approach to study multifractality in time series is the multifractal detrended fluctuation analysis (MF-DFA) [16], which is a less complicated and computational involved procedure compared with the WTMM algorithm, and thus became a popular alternative in the applied sciences. Recently, some scholars began to use the MF-DFA method to study the characteristics of the stock market [17–23]. Wei and Huang study high-frequency (per 5 min) data of Shanghai Stock Exchange Composite index (SSEC) from January 1999 to July 2001 with multifractal method [24]. They use box counting method to calculate the multifractal spectra of SSEC, and propose a new market risk measurement, Rf , which contains the integrated information of the two parameters of multifractal spectra, ∆f and ∆α . Lin (2001) finds the fractal features of the Shanghai and Shenzhen stock index indices by comparing the distribution of weekly returns and daily returns [25]. Hao, Yu and Chen use the multifractal method of box counting to demonstrate the multifractal characteristics of the Chinese stock market, selects different highfrequency stock price data for the same period of time. A preliminary solution to the issue of frequency selection of stock data was made [26]. For the stock market, many authors have concluded that stock indices of various markets (American, Indian, Polish, Brazilian among others) exhibit long-range correlations and multifractal characteristics [27–40]. However, these analysis have not been performed on the Shenzhen Component Index (SZSE) high-frequency stock data in Chinese stock market. In order to maintain the long-term memory of stocks, the Shenzhen Component Index (SZSE) is divided into six units, which are studied by multifractal detrended fluctuation analysis method. The use of shuffled series indicates that the origin of multifractality is due to both the broadness of probability density function and long-range correlation. The reminder of the paper is organized as follows: Section 2 mainly focuses on the description of MF-DFA method. Section 3 describes the data of SZSE high-frequency stock sequence in Chinese stock market and using the Jarque–Bera method to test the normal distribution of the data. Section 4 provides the empirical results, which compares the original high-frequency stock sequence with shuffled sequences. Section 5 is the conclusions. 2. Methodology In recent years, PENG et al. proposed detrended fluctuation analysis (DFA) [41]. DFA is a scaling exponent calculation method based on DNA mechanism. It is used to analyze the long-range correlation of time series. One of the advantages of DFA is that it can effectively filter out the trend components of the sequence, detect the long-range correlation of noisy and polynomial trend signals, and is suitable for the long-range power law correlation analysis of non-stationary time series. Building on DFA, Kantelhardt et al. [16] gave an improved version, MF-DFA, which is capable of studying the multipoint correlation of the non-stationary series. This method can accurately quantify the long-range correlation of non-stationary time series, which is based on random walk theory and can avoid artificially induced time series instability. And it can be used for a global detection of multifractal behavior and does not involve more effort in programming than the conventional DFA. MF-DFA method can be described as follows. Let us suppose that xk (k = 1, . . . , N) is a series of length N, and this series is of compactly support, i.e. xk = 0 for an insignificant fraction of the values only. The generalized MF-DFA procedure consists of five steps. Step 1: The cumulative deviation of the sequence for the mean Y (i) ≡

i ∑

|xk − ⟨x⟩|, i = 1, . . . , N .

(1)

k=1

⟨x ⟩ =

N 1 ∑

N

xk

(2)

k=1

Subtraction of the mean ⟨x⟩ is not compulsory, since it would be eliminated by the later detrending in the third step. Step 2: Divide the profile Y(i) into Ns ≡ int(N /s) non-lapping segments of equal length s. Since the length N of the series is often not a multiple of the considered time scale s, a short part at the end of the profile may remain. In order not to disregard this part of the series, the same procedure is repeated starting from the opposite end. Thereby, 2Ns segments are obtained altogether. Step 3: Calculate the local trend for each of the 2Ns segments by a least-square fit of the series. Then determine the variance F 2 (s, ν ) ≡

s 1∑

s

{Y [(ν − 1)s + i] − yν (i)}2 ,

(3)

i=1

for each segment ν , ν = 1, . . . , Ns and F 2 (s, ν ) ≡

s 1∑

s

{Y [N − (ν − Ns )s + i] − yν (i)}2 ,

i=1

(4)

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for each segment ν = Ns + 1, . . . , 2Ns . Here, yν (i) is the fitting polynomial in segment ν . Linear, quadratic, cubic, or higher order polynomials can be used in the fitting procedure. Generally denoted as MF-DFAm , m is the order of the fitting polynomial. Since the detrending of the time series is done by the subtraction of the polynomial fits from the profile, different order MF-DFA differs in their capability of eliminating trends in the series. In MF-DFAm (mth order MF-DFA) trends of order m in the profile (or, equivalently, of order m-1 in the original series) are eliminated. Thus a comparison of the results for different orders of MF-DFA allows one to estimate the type of the polynomial trend in the series. Step 4: For 2Ns intervals average over all segments to obtain the qth order fluctuation function Fq (s) ≡ {

2Ns 1 ∑

2Ns

q

1

[F 2 (s, ν )] 2 } q ,

(5)

ν=1

When q = 0, the fluctuation function can be determined by the following form F0 (s) = exp{

2Ns 1 ∑

4Ns

ln[F 2 (s, ν )]}

(6)

ν=1

where, in general, the index variable q can take any real value except zero. For q = 2, MF-DFA degenerates into DFA. When q < 0, the size of Fq (s) mainly depends on the size of the small fluctuation deviation F 2 (s, ν ), and when q>0, the size of Fq (s) mainly depends on the size of the large fluctuation deviation F 2 (s, ν ). In this way, different q describes the effect of different degrees of fluctuation on Fq (s). We are interested in how the generalized q dependent fluctuation function Fq (s) depend on the time scale s for different values of q. Hence, we must repeat steps 2–4 for several time scale s. It is apparent that Fq (s) will increase with increasing s. Step 5: Determine the scaling behavior of the fluctuation functions by analyzing log–log plots Fq (s) versus s for each value of q. If the series {xk } is long-range power law correlated, Fq (s) increases for large values of s as a power law: Fq (s) ∼ sH(q) .

(7)

In general, the exponent H(q) may depend on q. For stationary time series, H(2) is identical to the well-known Hurst exponent H. Thus ,we will call the function H(q) generalized Hurst exponent. Hurst exponent is the exponent to judge whether time series data follow random walk or biased random walk process. There are three forms of the Hurst exponent: (1) If H(q) = 0.5 indicates that time series can be described by random walk. (2) If 0.5 < H(q) < 1, there is a long-term memory in time series. (3) If 0 is less than H(q) < 0.5, it means that the pink noise (anti persistent) is the mean recovery process. That is to say, as long as H ̸ = 0.5, the time series data can be described by biased Brownian movement (fractal Brownian movement). In addition, H(q) = 0.5 indicates that the sequence is random and unpredictable; the larger the Hurst exponent, the stronger the correlation between the data before and after can be used to predict. The family of generalized Hurst exponent H(q) can be obtained by observing the slope of log–log plot of Fq (s) versus s through the method of least squares. In order to make Fq (s) have a higher degree of stability, usually the value of s does not exceed Ns /4. When the sequence {xk } is monofractal, the scale of the deviation F 2 (s, ν ) in all intervals is consistent, so that H(q) is independent of q as a constant. In particular, when the sequence {xk } is uncorrelated or short-range correlated, H(q) = 21 ; when H(q) depends on q as a function of q, the sequence {xk } is a multifractal. For monofractal time series with compact support, H(q) is independent of q, since the scaling behavior of the variances F 2 (s, ν ) is identical for all segments ν , and the averaging produce will give just this identical scaling behavior for all values of q. Only if small and large fluctuations scale differently, there will be a significant dependence of H(q) on q. If we consider positive values of q, the segments ν with large variance F 2 (s, ν ) (i.e. , large deviations from the corresponding fit) will dominate the average Fq (s). Thus, for positive values of q, H(q) describes the scaling behavior of the segments with large fluctuations. On the contrary, for negative values of q, H(q) describes the scaling behavior of the segments ν with small variance F 2 (s, ν ) will dominate the average Fq (s). Hence, for negative values of q, H(q) describes the scaling behavior of the segments with small fluctuations, which are usually characterized by a larger scaling exponent. In the traditional multifractal scaling approach used in multifractal box counting and WTMM analysis one studies the partition function Zq (s) ∼ sτ (q) . However, the MF-DFA method can only determine the signal properties of the positive generalized Hurst exponent H(q). When H(q) approaches 0, the signal exhibits a strong inverse correlation. At this time, the MF-DFA method cannot accurately determine the nature of the signal. This requires an amendment to the MF-DFA process. The simplest method is to find the cumulative dispersion in the first step twice before the MF-DFA method, as follows: Y˜ (i) =

i ∑ [Y (k) − ⟨Y ⟩]

(8)

k=1

The following steps are the same as the MF-DFA method, which gives a generalized fluctuation function: ˜

F˜q (s) ∝ sH(q) = sH(q)+1

(9)

˜ For large scales, there are H(q) = H(q) + 1. Generalized Hurst exponent H(q) and τ (q) have the following relationship: τ (q) = qH(q) − 1

(10)

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Fig. 1. SZSE high-frequency closing price fluctuation diagram.

In this way, the relationship between the generalized Hurst exponent H(q) defined in the MF-DFA method and the scale exponent τ (q) in the conventional method is obtained by Legendre transformation [12,42]:

α = τ ′ (q) f (α ) = qα − τ (q)

(11) (12)

If the time series of the study is monofractal, the function f (α ) is a certain value; if the time series is multifractal, the function f (α ) is a unimodal bell image. Let the interval where f (α ) ≥ 0 is greater than 0 be recorded as [αmin , αmax ] dτ (q) dτ (q) (αmin = dq |q→+∞ , αmax = dq |q→−∞ , and f (αmin ) = f (αmax ) = 0). Several parameters can be used to describe the degree of multifractality of the time series. They are fmax = f (α0 )(α0 ∈ [αmin , αmax ]), and the width W = αmin − αmax of interval [αmin , αmax ]. Through the description of the above three parameters of the time series, the larger the maximum value of f (α ) and the wider the width W of the interval [αmin , αmax ], the better the symmetry of the spectrum curve f (α ) and the stronger the multifractality of the time series. 3. Data description The Shenzhen Component Index, or SZSE, is the major stock index of the Shenzhen Component Index. High-frequency time series is a new time scale refinement of the stock market price, which can reflect the non-linear characteristics of the stock market in detail. The higher the frequency of the stock, the richer the information contained in the stock price, and the easier it is to provide more accurate and reliable basis for predicting the future trend of the stock price [24,26]. So we randomly downloaded the high-frequency data of SZSE from June 15th, 2017 to April 11th, 2018. The time interval is 5 min. Excluding weekends and holidays, the Shenzhen Component Index has a total of 9696 data for research analysis. The actual fluctuations are shown in Fig. 1, where the abscissa represents the time and the ordinate represents the closing price. We amplify the data points for a certain period of time during this period of time (selected here from 4673 data from September 2017 to January 2018) to obtain Fig. 2. The enlarged part presents new details again. Because of the different coordinate scales, the difference is inevitable. These details are somewhat self-similar to the overall 9696 data. We continue to magnify some of the data in Fig. 2 to obtain new images. We find that the new image still exhibits selfsimilarity to a certain extent to the whole, and it initially shows the characteristics of fractals. It is like a tortuous coastline or a stretch of peaks. After each part is enlarged, it can show similarity with the whole. This is the characteristic of fractals. Although an important feature of fractals is to maintain the invariance of the scale, the fact that the stock market has multifractal features based on the above figures is obviously one-sided. Afterwards, we will analyze it from a mathematical perspective and further demonstrate the multifractal characteristics of the stock market. The basic statistical analysis of the high-frequency data of SZSE and Jarque–Bera test results are shown in Table 1, and the results of the QQ diagram are shown in Fig. 3. By calculating the kurtosis, skewness and JB value of each sample, we can see that: (1) The distribution of SZSE shows skewness. The skewness of the high-frequency closing price distribution of SZSE is less than 0.

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Fig. 2. SZSE high-frequency closing price fluctuation diagram during 2017.09–2018.01.

Table 1 SZSE high-frequency closing price basic statistics and JB test results. Value

Statistics

Mean Median value Variance Standard deviation Minimum maximum Skewness Kurtosis Jarque–Bera Critical value

10919.4553 10978.9150 152992.965 391.14315 9881.37 11712.18 −0.181 −0.956 422.0885 5.9815

Fig. 3. SZSE high-frequency closing price QQ chart.

(2) The distribution of high-frequency closing prices for five minutes shows the characteristic of partial kurtosis. Compared with normal distribution, its distribution is light-tailed, and its tail length is larger than normal distribution. The observational measurement is less concentrated. (3) The JB test significantly rejects the assumption of a normal distribution of closing prices and the QQ diagram of the normal distribution is not on a straight line, which is consistent with the above described skewness and kurtosis characteristics. (4) The high-frequency closing price sequence of the SZSE showed some degree of autocorrelation.

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Fig. 4. ShenZhen Composite Index log–log figure.

Fig. 5. Generalized Hurst exponents of Shenzhen Composite Index.

4. Empirical results 4.1. SZSE high-frequency closing price analysis of experimental results This section provides the empirical results by the MF-DFA for the SZSE. We utilize Shenzhen Component Index highfrequency stock data from 2017.6.15–2018.4.11. Calculate the value of the fluctuation function after eliminating the trend from the accumulated deviation segment of the mean value of SZSE series. Analyze the relationship between Fq (s) and s in a double logarithmic graph in Fig. 4, where q ranges from −5 to 5 with a step of 0.5 (s = 10, 20, 30, . . . , 300). The slopes of the regression curves give the generalized Hurst exponent, H(q). We can see from Fig. 4 that the double logarithmic image of the high-frequency fluctuation function Fq (s) and s of SZSE is a straight line, so it conforms to the power law relationship. Fig. 5 shows the generalized Hurst exponent H(q) for different values of q ranging from −5 to 5 for the high-frequency closing price sequence of SZSE. From Fig. 5, we can see that H(q) decreases with the q varying from −5 to 5, implying that the high-frequency closing price sequence of SZSE has a multifractal nature. As already mentioned before, we can study multifractality in a time series in a better way by converting q to α and f (α ) by a Legendre transform and plot the multifractal singularity spectrum, f (α ). The shape of the multifractal singularity spectrum, f (α ) is like an inverted parabola with its maximum at α0 , which corresponds to the moment q= 0. The left part of the spectrum (α < α0 ) corresponds to the positive values of the moment q and the right part of the spectrum (α > α0 ) corresponds to the negative values of the moment q. The spectrum function graph is analyzed as below:

∆f = f (αmin ) − f (αmax ) Through the size of ∆f , the ratio of the highest and lowest stock prices can be calculated, that is, the ratio of the price to the peak (highest) and to the valley (lowest) positions. When ∆f > 0, it indicates that the price is more at the crest, and the top of the spectrum is relatively mellow, and the stock price has a rising trend; When ∆f < 0, it indicates that the price is more at the bottom, and the top of the spectrum is relatively sharp at this time, and the stock price has a downward trend. In order to maintain the long-term memory of the stock, all 9696 data are divided into 6 units. According to the calculation method of main parameters of multifractal spectrum, we can obtain the main parameters of the multifractal spectrum of the six units corresponding to SZSE from June 2017 to April 2018, as is shown in Table 2 (see Figs. 6–11). We can clearly recognize that the spectrum curve presents the typical single-humped shape that characterizes multifractal signals. Observing the multifractal spectrum of SZSE, we can see that ∆f > 0 during 2017.7.19–2017.9.18, a slightly larger part of the index is at the crest, while the range of α > α0 is slightly greater than the range of α < α0 , and the index has

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Fig. 6. Shenzhen Component Index 2017.6.15–2017.7.18 high-frequency multifractal spectrum.

Fig. 7. Shenzhen Component Index 2017.7.19–2017.9.18 high-frequency multifractal spectrum.

Fig. 8. Shenzhen Component Index 2017.9.19–2017.11.9 high-frequency multifractal spectrum.

Table 2 The main parameters of the multifractal spectrum of each of the six unit groups in Shenzhen Component Index. Unit⧹ parameters

αmin

αmax

∆α

f (αmin )

f (αmax )

∆f

1 unit 2 unit 3 unit 4 unit 5 unit 6 unit

0.97045 1.5096 1.1403 1.5434 1.0901 1.5039

1.6319 1.802 1.8757 1.8334 1.5595 1.7696

0.66144 0.29243 0.73536 0.28997 0.46944 0.26567

0.006076 0.66894 0.044651 0.74186 0.30256 0.53739

0.52185 0.35778 0.44856 0.4766 0.76515 0.6078

<0 >0 <0 >0 <0 <0

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Fig. 9. Shenzhen Component Index 2017.11.10–2017.12.27 high-frequency multifractal spectrum.

Fig. 10. Shenzhen Component Index 2017.12.28–2018.2.13 high-frequency multifractal spectrum.

Fig. 11. Shenzhen Component Index 2018.2.14–2018.4.11 high-frequency multifractal spectrum.

a decreasing trend. During 2017.6.15–2017.7.18, 2017.9.19–2017.11.9 and 2017.12.28–2018.4.11, ∆f < 0, a slightly larger portion of the index is at the bottom, and the interval of α > α0 is slightly smaller than the interval of α < α0 . The index has an upward trend. 4.2. Comparison of the multifractality for original and shuffled series For analyzing the source of multifractality in SZSE high-frequency stock data, we have analyzed the modified times series including shuffled times series. The main rationale behind this approach is that generally two different types of sources for multifractality in time series are identified: (i) multifractality due to different long-range temporal correlations for small and large fluctuations (ii) multifractality related to the fat-tailed probability distributions (PDF) of variations. Shuffling and phase randomization are the main procedures to find the contributions of two sources of multifractality. Shuffling preserves the distribution of the variations but destroys any temporal correlations. In fact, one can destroy the temporal correlations by randomly shuffling the corresponding time series of variations. What then remains are data with exactly the same fluctuation distributions but without any correlation.

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Fig. 12. Generalized Hurst exponents of SZSE original time series.

Fig. 13. Generalized Hurst exponents of SZSE shuffled time series.

To check the multifractal nature, we compare the fluctuation function Fq (s) for the original series with the results of shuf the corresponding shuffled, Fq (s). Differences between shuffled fluctuation functions with the original series indicate the presence of long-range correlation or broadness of probability density function in the original series. These differences can shuf be observed in a plot of ratio Fq (s)/Fq (s) versus s [16]. The scaling behavior of this ratio is Fq (s)/Fqshuf (s) ∼ sH(q)−Hshuf (q) = sHcor (q)

(13)

If only fatness of the PDF is responsible for the multifractality, one should find H(q) = Hshuf (q) and Hcor (q) = 0. If only correlation multifractality is present, one should find Hshuf (q) = 0.5. If both distribution and correlation multifractality are present, Hshuf (q) depends on q. The q-independence of the exponent H(q) for original, shuffled time series are shown in Figs. 12 and 13. We can see that Hshuf (q) depends on q, which shows that the multifractality nature of the SZSE time series is due to both broadness of the PDF and long-range correlation. The multifractal spectrum gives information about the relative importance of various fractal exponents present in the series. In order to better analyze the multifractal strength for original and shuffled time series, the singularity spectra are shown in Figs. 14 and 15. There are obvious difference between spectra of the original and shuffled time series. We can see that the width of the original series is larger than that of the shuffled series. The shuffled series exhibits weaker multifractal scaling than the original series, which shows that both kinds of multifractality are present. 5. Conclusion This paper mainly uses MF-DFA method to analyze the time series of the high-frequency closing prices of SZSE during 2017.6.15–2018.4.11. The results show that there exist multifractal characters in Shenzhen Component Index. By analyzing the generalized Hurst exponent H(q) and the multifractal spectrum f (α ), it is concluded that MF-DFA method can accurately determine the multifractal nature of the time series. The multifractal spectrums of SZSE are all bell-shaped and have good symmetry. The maximum value of f (α ) and the width of interval ∆α are all greater than 0, which means that the highfrequency closing price sequence of SZSE has a certain multifractal nature. During 2017.7.19–2017.9.18, the index has a decreasing trend. During 2017.6.15–2017.7.18, 2017.9.19–2017.11.9 and 2017.12.28–2018.4.11, the index has an upward trend. By comparing the generalized Hurst exponent of the original time series with the shuffled time series, we have found that the multifractality of SZSE is due to both broad probability density function and long-range correlation. Applying the multifractal principle to scientifically analyze high-frequency stock data, it is of great significance to the theoretical modeling , short-term forecasting and customization of management and control strategies of stock markets. The innovation

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Fig. 14. SZSE original time series high-frequency multifractal spectrums.

Fig. 15. SZSE shuffled time series high-frequency multifractal spectrums.

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