Multifractal regime detecting method for financial time series

Chaos, Solitons & Fractals 70 (2015) 117–129

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Multifractal regime detecting method for financial time series Hojin Lee, Woojin Chang ⇑ Department of Industrial Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, Republic of Korea

a r t i c l e

i n f o

Article history: Received 11 July 2014 Accepted 10 November 2014

a b s t r a c t We focus on time varying multifractality in time series and introduce a multifractal regime detecting method (MRDM) adopting a nonparametric statistical test for multifractality based on generalized Hurst exponent (GHE). MRDM is a practical method to discriminate multifractal regimes in a time series of any length using a moving time window approach with the adjustable time window size and the moving interval. MRDM is applied to simulations consisting of both multifractal and monofractal regimes, and the results confirm its validity. Using MRDM, we identify multifractal regimes in the time series of Korea composite stock price index (KOSPI) from 1990 through 2012 and observe the distinct stylized facts of the KOSPI return values in multifractal regimes such as the heavy tail distribution, high kurtosis, and the long memory in volatility. Surrogate tests based on improved amplitude adjusted Fourier transformation (IAAFT) algorithm, normal distribution, and generalized student t distribution are performed for the validation of MDRM, and the probable causes of multifractality in the KOSPI series are discussed. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Financial time series have various statistical stylized facts which are not expressed using Gaussian models. Some examples are as follow. Returns of financial asset have the fatter tail distribution than the Gaussian noise and show frequent extreme jumps. Volatility of returns is heteroscedastic, long-range dependent, and likely to be clustered. Multifractality is one of these stylized facts observed in many financial time series. The multifractality is rather a macroscopic concept, but there exist the time ranges showing strong multifractal properties in financial time series. This implies that a time series having multifractality as a whole can be segmented into the multifractal regimes and the non-identifiable regimes inside of the time series in microscopic view. Many financial time series reflecting economic cycles, growth and recession, exhibit the tendency of time-varying multifractality. ⇑ Corresponding author. Tel.: +82 2 880 8335. E-mail address: [email protected] (W. Chang). http://dx.doi.org/10.1016/j.chaos.2014.11.006 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.

Mandelbrot [23] developed a fractal theory measuring the complexity of a fractal structure by defining the fractal dimension distinct from the conventional Euclidean dimension. The fractal structure having multiple dimensions is called ‘‘multifractal’’, while a fractal structure with a single dimension is called ‘‘monofractal’’. The multifractality in a time series can be observed by the Hurst exponent estimation. Peng et al. [32] and Kantelhardt et al. [16] proposed a multifractal detrended fluctuation analysis (MF-DFA), which can reliably determines the multifractal scaling behavior of nonstationary time series, for the estimation of the exponent. Another method is called the generalized Hurst exponent (GHE) approach. The qth order moments of the distribution of increments of time series value are used to estimate the exponent [1,10]. Kantelhardt et al. [16] distinguished two sources of multifractality in time series: the properties of probability density function (PDF) for the values in time series, especially its heavy tail thickness (fat tail) and long range correlation of fluctuations in time series. Kumar and Deo [17] pointed out that both the properties of PDF and long

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range correlation give rise to the multifractality in Indian financial market. While Zunino et al. [43] and Barunik et al. [5] mentioned that heavy tail distribution is the main _ z_ et al. cause of multifractality, Kwapien´ et al. [18], Drozd [12], Oh et al. [28] and Zhou [41] claimed that multifractality in time series is mainly due to the long range correla_ z_ et al. [12] showed tion property of time series. Drozd that a uncorrelated time series of short length (less than 105 data points) can be misjudged as a multifractal series, _ z_ et al. [12] and Zhou [41] stated that the propand Drozd erties of PDF have an impact on the multifractality of time series only when the time series possesses long range correlation. Some overview of a series of research applying scaling property to model financial market is as follows. Bacry et al. [2,3] introduced multifractal random walks (MRW) with stationary increments and continuous dilation invariance. Górski et al. [13] studied the complicated multifractal nature of stock market dynamics. Calvet and Fisher [6–9] have done important work in multifractality in asset returns including volatility forecasting, Markov switching multifractal, and multifractal model of asset returns (MMAR). Lux [21] proposed a generalized method of moments approach for estimating multifractal parameters in Markov-switching multifractal model (MSM). Liu et al. [20] analyzed the multi-scaling properties of financial data using MSM. Os´wie˛cimka et al. [31] focused on the Lux extension to MMAR and applied the model to study the dynamics of the Polish stock market. Os´wie˛cimka et al. [29] applied MF-DFA to the high frequency stock market data. Kwapien´ et al. [18] pointed out that nonlinear temporal correlations weigh more than the fat-tailed probability distribution as a contributor to the multifractal dynamics of stock return. Os´wie˛cimka et al. [30] performed a comparative study for the detection of multifractality between MF-DFA and wavelet transform _ z_ et al. [12] modulus maxima (WTMM) method. Drozd applied both MF-DFA and WTMM method to detect multifractality in time series, and claimed that the genuine multifractality results from temporal correlation. Zunino et al. [42] applied MF-DFA to developed and emerging stock markets and introduced the multifractality degree to assess stages of stock market development. Zhou [41] decomposed the multifractality into three components caused by linear correlation, nonlinear correlation, and the fat-tailed probability density function (PDF), and maintained that the fat-tailed PDF have an impact on the multifractality with the presence of nonlinear correlation. _ z_ [19] provide a general overview of Kwapien´ and Drozd multifractality in complex systems. Suárez-García and Gómez-Ullate [36] applied MF-DFA to a multifractal and correlation analysis of the high-frequency returns of the IBEX 35 index of Madrid stock exchange. Among abundant studies on multifractality in financial time series, the research on a statistical multifractality test is relatively small. Wendt and Abry [38] and Wendt et al. [39] suggested bootstrap methods to discriminate multifractality for time series of large sample size (212 or 215 ). However, their methods may not be applicable to the time series of small sample size due to the inaccuracy of resampling. Jiang and Zhou [15] performed statistical tests upon

intraday minutely data within individual trading days to check whether the indexes possess multifractality. Morales et al. [25,26] suggested that financial time series have time varying multifractality. Morales et al. [25] computed weighted generalized Hurst exponent (wGHE) over a moving time window and monitored the dynamics of wGHE during the unstable periods in financial time series. Morales et al. [26] identified the time varying multifractal properties, comparing empirical observations of wGHE with the time series simulated via multifractal random walk by Bacry et al. [3]. Sensoy [33,34] studied the efficiency of stock markets (middle east and north african stock market and federation of Euro-Asian stock exchanges, respectively) using GHE over a moving time window. In this paper, we introduce a multifractal regime detecting method (MRDM) that identifies multifractal ranges in the time series through a moving time window. By applying MRDM to a time series, we can segment the time series into multifractal and non-identifiable regimes. The multifractality of a time series window is checked and a multifractal regime is detected by rolling the window forward with a regular interval at a time. MRDM adopts the GHE approach considering the time varying multifractal property and a simple nonparametric statistical test to select multifractal regimes. MRDM is appropriate to analyze the time varying multifractality of time series and performs well in the time series of small sample size. MRDM is applied to the simulation of multifractal model of asset returns (MMAR), which is a typical multifractal process, and the empirical data of Korea composite stock price index (KOSPI) ranging from 1990 to 2012. The remainder of this paper is as follows. In Section 2, stylized facts between the monofractal process and the multifractal process are compared and a multifractality test is introduced and validated using the simulation of monofractal and multifractal processes. In Section 3, MRDM is introduced and its type 1 and type 2 errors are measured based on simulation data consisting of fractional Brownian motion and MMAR. The empirical application of MRDM to the KOSPI series and the related surrogate test results are discussed in Section 4, and the summary and conclusion of this paper are in Section 5.

2. Discrimination of multifractality A time series, fXðtÞg, has the following scale property,

E½jDX s ðtÞjq   sfðqÞ

ð1Þ

where DX s ðtÞ ¼ Xðt þ sÞ  XðtÞ and fðqÞ is the scaling function. The scaling function of fractional Brownian motion (fBm) has a linear form of fðqÞ ¼ Hq, where H is the Hurst exponent, 0 < H < 1 [27]. Especially, fBm becomes an ordinary Brownian motion when H = 0.5. The fBm, BH ðtÞ, is a self similar process and has the following property,

BH ðctÞ  cH BH ðtÞ for c P 0

ð2Þ

When H is larger than 0.5, fBm is long range dependent.

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Table 1 Time series simulation properties (average of 1000 simulation samples) of fBm(H) and MMAR(H, k) process in two cases: Brownian motion (H = 0.5) without long range dependence and fBm(H = 0.7) with long range dependence. The p-value is for the null hypothesis of the comparison between MMAR and fBm.

H ¼ 0:5 DX

Y

fBmðHÞ

MMARðH; 1:1Þ

p-Value

h

1.00 0.00 3.00 0.37 0.37 0.18

0.82 0.00 4.22 0.54 0.54 0.23

0.00 0.58 0.00 0.00 0.00 0.00

h

0.18

0.23

0.50 0.50 0.50 0.00

0.60 0.57 0.55 0.03

h

0.99 0.00 2.99 0.37 0.37 0.18

h

Std. Skew. Kurt. MEF upper MEF lower kupper

X

H ¼ 0:7 DX

Y

klower Hð1Þ Hð2Þ Hð3Þ DHð1; 2Þ Std. Skew. Kurt. MEF upper MEF lower kupper

X

klower Hð1Þ Hð2Þ Hð3Þ DHð1; 2Þ

p-Value

MMARðH; 1:5Þ

p-Value

0.87 0.02 8.16 0.82 0.82 0.31

(0.00) 0.10 0.00 0.00 0.00 0.00

0.89 -0.02 17.13 1.13 1.13 0.39

0.00 0.42 0.00 0.00 0.00 0.00

0.00

0.30

0.00

0.39

0.00

0.00 0.00 0.00 0.00

0.63 0.55 0.51 0.08

0.00 0.00 0.00 0.00

0.65 0.53 0.48 0.13

0.00 0.00 0.00 0.00

0.92 0.00 4.55 0.59 0.59 0.24

0.00 0.76 0.00 0.00 0.00 0.00

1.05 -0.02 11.54 0.96 0.97 0.35

0.00 0.63 0.00 0.00 0.00 0.00

1.23 0.02 32.29 1.43 1.42 0.47

0.00 0.75 0.00 0.00 0.00 0.00

0.18

0.24

0.00

0.35

0.00

0.47

0.00

0.70 0.70 0.70 0.00

0.76 0.73 0.71 0.03

0.00 0.00 0.00 0.00

0.76 0.69 0.65 0.07

0.00 0.00 0.00 0.00

0.76 0.66 0.59 0.10

0.00 0.00 0.00 0.00

When fðqÞ is a nonlinear function, fXðtÞg is called a multifractal process. In this paper, we use the fBm and the multifractal model of asset returns (MMAR) as examples of monofractal and multifractal processes respectively for the validation of MRDM. 2.1. Multifractal model of asset return (MMAR) process

where hðtÞ is independent of BH ðtÞ, a time deformation process. The trading time, hðtÞ, is specified as cumulative distribution function of multifractal measure defined on ½0; T as follows.

k

Pk

ð4Þ

i

where Dt ¼ b , t ¼ i¼1 gi b , and the multiplier, M, is a positive, independent and identical random variable, M b ; b 2 f0; 1; . . . ; b  1g [9]. The multifractal measure is 1 canonical if E½M ¼ b , and the scaling property of multifractal measure is

E½h½t; t þ Dtq  ¼ E½hðDtÞq   cðqÞðDtÞfðqÞ q

as Dt ! 0

and f ðaÞ ¼ 1 

ða  kÞ2 4ðk  1Þ

By using the property of self similar process and Eqs. (3)–(5), we have the following as t goes to 0,

h i     E jXðtÞjq ¼ E hðtÞHq E jBH ð1Þjq  cX ðqÞt fX ðqÞ

ð6Þ

where cX ðqÞ ¼ ch ðHqÞE½jBH ð1Þj  and fX ðqÞ ¼ fh ðHqÞ.

ð3Þ

h½t; t þ Dt ¼ M g1 M g1 g2    M g1 gk

fðqÞ ¼ kq  ðk  1Þq2

q

Mandelbrot, Fisher, and Calvet [24] developed the multifractal model of asset returns (MMAR) as a compound process,

XðtÞ ¼ BH ½hðtÞ

MMARðH; 1:3Þ

ð5Þ

where fðqÞ ¼ logb E½M  and f  g means f ðxÞ=gðxÞ ! 1. Calvet and Fisher [9] derived MMARðH; kÞ model where M has the lognormal distribution, logb M  Nðk; r2 Þ with r2 ¼ 2ðk  1Þ= ln b under the canonical condition satisfied. In this case, the multifractal measure, h, has the scaling function, fðqÞ, and the multifractal spectrum function, f ðaÞ, as follows. For k > 1

2.2. Simulation of monofractal and multifractal processes Table 1 shows the simulation results where statistical stylized facts between a monofractal process, fBmðHÞ ¼ BH ðtÞ, and a multifractal process, MMARðH; kÞ ¼ BH ½hðtÞ, are compared. We generated 1000 simulation paths of length 1000 for both fBmðHÞ and MMARðH; kÞ, and performed analysis on stylized facts such as tail thickness and scaling properties. Table 1 describes two cases: one for the simulation with H ¼ 0:5, which is not long-range dependent, and the other for the simulation with H ¼ 0:7, which has long range dependence. The mean values of fBmðHÞ and MMARðH; kÞ processes for each stylized fact category are shown with the p-value of the two sample ttest for the null hypothesis that the mean of the MMARðH; kÞ process is equal to that of the fBmðHÞ process. The tail thickness for processes is measured using the mean excess function (MEF) and Hill estimator [14], h h k ; as follows. The larger values of MEF and k imply the heavier tail distribution.

MEFðuÞ ¼ E½YðtÞ  ujYðtÞ > u h

k ðqÞ ¼ E½ln YðtÞ  ln ujYðtÞ > u

ð7Þ ð8Þ

where YðtÞ is the normalized value of DXðtÞ ¼ DX 1 ðtÞ. We set u ¼ 2 throughout this paper.

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Table 2 Multifractality test result for MMARðH; kÞ processes. The parameters in Eq. (4) are set as b = 2 and k = 3. The average of difference between H(1) and H(2) for 2000 paths are denoted by DHð1; 2Þ. L

DHð1; 2Þ

pro0:05

DHð1; 2Þ

pro0:1

pro0:05

0.87 0.97 0.99 1.00

0.84 0.96 0.99 1.00

MMARð0:7; 1:1Þ 0.06 0.06 0.06 0.06

0.83 0.92 0.97 0.99

0.78 0.90 0.96 0.99

0.13 0.12 0.12 0.12

0.97 0.99 1.00 1.00

0.95 0.98 1.00 1.00

MMARð0:7; 1:3Þ 0.11 0.10 0.10 0.10

0.90 0.96 0.98 1.00

0.88 0.95 0.98 1.00

0.16 0.15 0.15 0.14

0.97 0.99 1.00 1.00

0.96 0.99 1.00 1.00

MMARð0:7; 1:5Þ 0.14 0.13 0.12 0.12

0.93 0.97 0.98 0.99

0.91 0.96 0.97 0.99

MMARð0:5; 1:1Þ 100 250 500 1000

0.07 0.07 0.07 0.07

MMARð0:5; 1:3Þ 100 250 500 1000 MMARð0:5; 1:5Þ 100 250 500 1000

pro0:1

Generalized Hurst exponent is denoted as HðqÞ ¼ fðqÞ=q, and DHðq; q0 Þ stands for the difference between two GHEs, DHðq; q0Þ ¼ HðqÞ  Hðq0 Þ [10]. For a monofractal process like fBm(H), HðqÞ is the constant value, H, for all q and Hðq; q0 Þ ¼ 0. Meanwhile, HðqÞ is distinct from Hðq0 Þ for q–q0 ðDHðq; q0 Þ – 0Þ due to the concaveness of the scaling function, fðqÞ, in multifractal process so that we can use this property to identify the multifractality of time series [5]. There are other methods to identify multifractality in time series including MF-DFA [16] mentioned in Section 1, but the GHE approach is used in this paper as GHE is more adaptable to the moving time window approach in MRDM and has the strength in the short sample size data with heavy tails [4]. Table 1 shows that MMARðH; kÞ has the larger kurtosis and the thicker tail distribution than fBmðHÞ and this tendency becomes stronger as k increases. We can observe that kurtosis, tail thickness indices (MEF and kh) and DHð1; 2Þ get larger as k increases, implying high correlation between k and intensification of multifractality. 2.3. Multifractality test method As mentioned in the previous section, the scaling function of Multifractal time series, fðqÞ, is nonlinear so that HðqÞ for multifractal process is dependent on q while HðqÞ for monofractal process is a constant value regardless of q. The generalized Hurst exponent HðqÞ is estimated as follows [10]. From Eq. (1) we have

K q ðsÞ  sfðqÞ

ð9Þ

ln K q ðsÞ ¼ qHðqÞ ln s þ c;

s ¼ f1; 2; . . . ; smax g

ð10Þ

The qth order moment of the distribution of increments for time series of length L (fXðkÞ : k ¼ 1; . . . ; Lg) is defined as, for s ¼ f1; 2; . . . ; smax g

    K q ðsÞ ¼ jXðLÞ  XðL  sÞjq = jXðL  sÞjq

ð11Þ

where



jXðLÞ  XðL  sÞj

q

s1 X 1 L jXðL  kÞ  XðL  s  kÞjq ¼ L  s k¼0



 jXðL  sÞjq ¼

s1 X 1 L jXðL  s  kÞjq L  s k¼0

The linear regression in Eq. (10) allows us to estimate HðqÞ. By setting smax 2 f5; 6; . . . ; 20g, we obtain 16 values (from 5 to 20) of H(q) using Eq. (10) and compute the mean value of those. We focused on the estimation of H(q) for q ¼ f1; 2; 3g as the scaling function, fðqÞ, in Eq. (9) is likely to diverge for q > 3 [22,40,5,26]. We claim that the multifractality can be observed when the null hypothesis, H0 : Hð1Þ ¼ Hð2Þ, is rejected with a significant level a ð0 < a < 1). The scaling behavior of the absolute values of increment is represented as H(1), and the scaling of autocorrelation function and the power spectrum of time series is associated with H(2) [11]. The null hypothesis for the multifractality test is rather simple, but the outcomes from simulation experiments and the empirical trials supported the efficiency of the single null hypothesis. As we cannot assume any condition of distribution, we perform Wilcoxon rank sum test for the null hypothesis, H0 . We simulated MMARðH; kÞ processes of length LðL ¼ 100; 250; 500; 1000Þ for each parameter pairs, fðk; HÞ : k ¼ 1:1; 1:3; 1:5; H ¼ 0:5; 0:7g. For each (L; k; HÞ block, we generated 2000 simulation paths and performed the multifractality test described above. Table 2 shows test results. The average value of H(1) less H(2) is denoted by DHð1; 2Þ. The proportion of sample paths that rejects H0 among 2000 paths in each cell is denoted as proa where a (a = 0.1, 0.05) is the corresponding significant level. For both the MMAR with H ¼ 0:5 and H ¼ 0:7, the value of proa becomes larger when the sample length, L, increases. The multifractality test introduced in this section is practical in that the statistical test is nonparametric and applicable to the time series of any length.

3. Detection of multifractal regimes In this section, we introduce the multifractal regime detecting method (MRDM), which is an algorithm detecting multifractal regimes in the time series using the multifractality test described in Section 2.3.

H. Lee, W. Chang / Chaos, Solitons & Fractals 70 (2015) 117–129

121

Fig. 1. The n sample windows containing a time range ½t; t þ D.

Fig. 2. Three periods in a simulation path consisting of monofractal and multifractal processes.

and set as 0.7 in this paper. For a time series of length T ¼ N D, we can compute IðtÞ at time points t ¼ ðn  1ÞD; nD; . . . ; T  nD ðN > nÞ and judge the existence of multifractality of the corresponding time ranges, ½t; t þ D for t ¼ ðn  1ÞD; nD; . . . ; T  nD (see Fig. 1). 3.2. Simulation test results

3.1. Multifractal regime detecting method (MRDM) The fractality in financial time series is time varying and the time ranges having multifractality can be identified throughout whole time series. To detect regimes where multifractality exists in the time series, we use the moving window method as follows. We set a sample window of length nD and let the window move forward by D unit time interval at a time. The multifractality test in Section 2.3 is performed in each sample window and the multifractality of the corresponding time series inside of the window is checked. There are n moving window samples that contain a time range, ½t; t þ D, as in Fig. 1. We check the existence of multifractality in ½t; t þ D by the following proportion index,

IðtÞ ¼

n 1X 1MðiÞ n i¼1

ð12Þ

where 1MðiÞ has value 1 when the multifractality is observed (reject the null hypothesis, H0 , in Section 2.3) in the ith sample window, ½t  ðn  iÞD; t þ iD, and 0 otherwise. The values of 1MðiÞ is sensitive to the window length, nD, which is relatively small in MRDM, comparing with the length of time series used for the estimation of GHE in previous research. The value of I ðtÞ is the proportion of sample windows rejecting the null hypotheses among the n sample windows that contain ½t; t þ D. The existence of multifractality in ½t; t þ D is determined if the value of IðtÞ is larger than the given threshold value, IM . The threshold value, IM , is critical to the selection of multifractal regime

The MRDM described in the previous section is applied to the simulated time series to confirm its validity. We generate 1000 simulation paths. Each simulation path consisting of 1500 time units (e.g. days) are divided into three periods. The first time series, fXðtÞ; t ¼ 1; . . . ; 500g, is monofractal process. The second one, fXðtÞ; t ¼ 501; . . . ; 1000g, is multifractal process. The third one, fXðtÞ; t ¼ 1001; . . . ; 1500g, is monofractal process. We use fractional Brownian motion and MMAR for the monofractal process and the multifractal process respectively. The moving interval, D, is 20, and the length of sample, nD, is 240 so that 12 sample windows (n ¼ 12Þ can cover a time range, ½t; t þ D. Fig. 2 illustrates three periods in a simulation path. Fig. 3 shows a simulation path of fXðtÞg and the corresponding I(t) values of in Eq. (11), which is 1 in the most part of given multifractal period (shaded middle range). Fig. 4 shows the autocorrelation function (ACF) of fjDXðtÞjg, which is regarded as the volatility of fDXðtÞg, in two categories: multifractal periods (MMAR) in the center, and monofractal (fBm) periods in both ends of Fig. 2. The long-range correlation of volatility is clearly shown in the multifractal range, implying that multifractality of the MMAR process accompanies the property of volatility clustering. Table 3 shows the type 1 and type 2 errors of MRDM for 24 cases of parameter combination, fðH; k; IM Þ : H ¼ 0:5; 0:7; k ¼ 1:1; 1:3; 1:5; IM ¼ 0:6; 0:7; 0:8; 0:9g. In each case, 1000 simulation paths are generated for the estimation of type 1 and type 2 errors. Type 1 error is the proportion that multifractal time ranges are misjudged the as

Fig. 3. Left panel: a simulation path of fXðtÞg generated using fBmð0:5Þ and MMARð0:5; 1:3Þ, Right panel: IðtÞ of a sample simulation path, for t ¼ 220 þ 20k; k ¼ 0; 1; . . . ; 52. The significant level (a) of null hypothesis for the multifractality test is 0.05.

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Fig. 4. ACF of fjDXðtÞjg for fBm(H, 1.3) ( ) and MMAR(H, 1.3) (o) (H = 0.5 for left panel and H = 0.7 for right panel).

Table 3 Type 1 and type 2 errors of MRDM. Results are obtained using 1000 simulation paths for each parameter combination belonging to the set, fðH; k; IM Þ : H ¼ 0:5; 0:7; k ¼ 1:1; 1:3; 1:5; IM ¼ 0:6; 0:7; 0:8; 0:9g. IM

H ¼ 0:5

k

0.6

H ¼ 0:7

capability of multifractality regime detection (type 1 error reduction) in spite of the possibility of sizable false regime detection (type 2 error increment). 4. Empirical study on KOSPI market

0.7

0.8

0.9

0.6

0.7

0.8

0.9

Type 1 error 1.1 0.07 1.3 0.02 1.5 0.02

0.12 0.04 0.03

0.19 0.07 0.06

0.28 0.12 0.11

0.10 0.06 0.05

0.15 0.09 0.08

0.23 0.14 0.13

0.34 0.22 0.20

Type 2 error 1.1 0.40 1.3 0.45 1.5 0.47

0.29 0.34 0.36

0.19 0.24 0.25

0.11 0.14 0.15

0.54 0.56 0.58

0.42 0.43 0.45

0.29 0.31 0.32

0.18 0.19 0.20

monofractal ones. Type 2 error is the proportion that the monofractal time ranges are misclassified as multifractal ones. Specifically, the misclassified time ranges from f½t; t þ D : t ¼ 220 þ 20k; k ¼ 14; . . . ; 38g are counted for type 1 error measurement, and the misclassified time ranges belonging to f½t; t þ D : t ¼ 220 þ 20k; k ¼ 0; 1; . . . ; 13 and k ¼ 39; . . . ; 52g are counted for type 2 error measurement. Type I error is positively correlated with the size of threshold value, IM , while type 2 error is negatively correlated with the value. In consideration of the trade-off between type 1 and type 2 errors in Table 3, we choose IM ¼ 0:7 for MRDM in this paper. Fig. 5 shows the proportion that a unit time range in f½t; t þ D : t ¼ 220 þ 20k; k ¼ 0; 1; . . . ; 52g is classified as multifractal one out of 1000 trials (simulations) when we set IM ¼ 0:7. As shown in Table 3 and Fig. 5, the threshold value IM ¼ 0:7 provides small type 1 errors at the cost of relatively large size of type 2 errors especially in the boundaries between multifractal and monofractal regimes. As one of the purpose of this research is to identify multifractal regimes in financial time, we tried to enlarge the

We applied MRDM to the empirical data, the log value of daily Korea composite stock price index (KOSPI) from 1990 through 2012, which amounts to 276 months. In the MRDM applied to KOSPI data, the sample window of length 12 months (nD ¼ 12 months or 240 days in general) moves forward by 1 month (D ¼ 1 month or 20 days in general), and the threshold value, IM , is set as 0.7. The multifractal regimes are detected using IðtÞ in Eq. (12). To check the credibility of GHE, H(q), used in MRDM, we compute the regression coefficient of determination, R2, which indicates the validity of the linear regression in Eq. (10). Each cell of Table 4 consists of the average of 254 R2 values and their minimum value in parenthesis. The high value of R2 in each category validates the estimation of H(q) for the KOSPI series. 4.1. Multifractal regime detection Fig. 6 shows (a) the log value of KOSPI time series (XðtÞ), (b) IðtÞ used for the detection of multifractal regimes based on IM . In the multifractality test for the determination of IðtÞ, the significant level (a) of the null hypothesis, H0 , is 0.05. Shaded ranges in the figure are the identified multifractal regimes. As our MRDM cannot detect all multifractal ranges due to the type I error described in Section 3.2, we are difficult to claim the pure monofractality of the non-shaded ranges which are not detected as multifractal ones in time series. For this reason, we categorize identified regimes and non-identified regimes as ‘‘multifractal’’ regimes and ‘‘non-identifiable’’ regimes, respectively.

Fig. 5. The series of proportion that a time range, ½t; t þ D, is classified as multifractal one out of 1000 MMAR(H, k) simulations. Left panel is for H = 0.5 and right panel is for H = 0.7. The thick line, the short dotted line, and the long dotted line are results for k ¼ 1:5, k ¼ 1:3, and k ¼ 1:1, respectively.

123

H. Lee, W. Chang / Chaos, Solitons & Fractals 70 (2015) 117–129 Table 4 The average value of R2 (minimum value in parenthesis) in each category of {(q, smax ): q = 1, 2, 3 and

smax

smax = 5, . . . , 20}.

5

6

7

8

9

10

11

12

0.997 (0.984) 0.998 (0.989) 0.997 (0.977)

0.996 (0.979) 0.998 (0.986) 0.997 (0.976)

0.996 (0.965) 0.998 (0.971) 0.997 (0.973)

0.996 (0.964) 0.997 (0.963) 0.996 (0.961)

0.996 (0.965) 0.997 (0.959) 0.996 (0.949)

0.996 (0.965) 0.997 (0.955) 0.996 (0.936)

0.995 (0.963) 0.997 (0.955) 0.995 (0.932)

0.995 (0.955) 0.997 (0.958) 0.995 (0.937)

13 0.995 (0.948) 0.996 (0.962) 0.995 (0.943)

14 0.995 (0.942) 0.996 (0.965) 0.995 (0.948)

15 0.995 (0.937) 0.996 (0.966) 0.995 (0.948)

16 0.995 (0.934) 0.996 (0.967) 0.994 (0.949)

17 0.995 (0.934) 0.996 (0.967) 0.994 (0.946)

18 0.995 (0.938) 0.996 (0.965) 0.994 (0.939)

19 0.995 (0.942) 0.996 (0.963) 0.994 (0.929)

20 0.995 (0.946) 0.995 (0.962) 0.993 (0.921)

q 1 2 3

1 2 3

There are 6 multifractal regimes and 6 non-identifiable regimes alternating sequentially in the KOSPI series. Table 5 describes the statistical stylized facts of the log return of KOSPI time series, fDXðtÞg, belong to each of multifractal and non-identifiable regimes. In general, multifractal regimes show higher Kurtosis and heavier tail indices (MFE, Hill estimator) than non-identifiable regimes. One exception is M3 regime. Generalized Hurst exponent, H(q), for q = 1, 2, 3 is estimated for each regime, and multifractal regimes show the clear difference in GHE values. In each regime, we performed the multifractality test in Section 2.3 for the confirmation of regime selection. In addition to the test of H0: DHð1; 2Þ ¼ 0, H0: DHð2; 3Þ ¼ 0 is also tested, and the results, mean and p-value for the

test, are shown for each regime. All multifractal regimes reject both null hypotheses clearly. Among non-identifiable regimes, only N5 regime accepts both null hypotheses, implying that most of non-identifiable regimes cannot be claimed as monofractal ones. In non-identifiable regimes, N4 does not accept H0: DHð1; 2Þ ¼ 0, and N1, N2, N3, and N6 do not accept H0: DHð2; 3Þ ¼ 0. Especially, N2 is somewhat questionable for its non-identifiable regime classification since the p-value for H0: DHð1; 2Þ ¼ 0 is 0.07. One possible reason for this is that the relatively long time interval of N2 regime, 931 days, might contain some short ranges (around 1995 in Fig. 6b) with multifractality. The values of DHð1; 2Þ and DHð2; 3Þ can be understood as the multifractality degree in Zunino et al. [42], who used

Fig. 6. Multifractal regime detection in KOSPI from December, 1990 to January, 2012.

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Table 5 Statistical stylized facts and GHE of the log return of KOSPI time series, fDXðtÞg, belong to each of multifractal regimes (a) and non-identifiable regimes (b).

(a) Multifractal regime Period Number Std. of DX Skewness of DX Kurtosis of DX (MEFupper , MEFlower ) h

h

(kupper , klower ) Hð1Þ Hð2Þ Hð3Þ DHð1; 2Þ (p-value) DHð2; 3Þ (p-value)

(b) Non-identifiable regime Period Number Std. of DX Skewness of DX Kurtosis of DX (MEFupper , MEFlower ) h

(kupper , MEFhlower ) Hð1Þ Hð2Þ Hð3Þ DHð1; 2Þ (p-value) DHð2; 3Þ (p-value)

M1

M2

M3

M4

M5

M6

1991.01–1991.06 144 0.01 0.48 6.48 (0.85, 0.88) (0.34, 0.36)

1993.03–1993.10 201 0.01 0.10 5.00 (0.92, 1.20) (0.38, 0.45)

1997.01–1999.07 728 0.03 0.04 3.91 (0.42, 0.48) (0.19, 0.20)

2000.12–2003.01 528 0.02 0.44 5.86 (0.37, 0.85) (0.15, 0.31)

2004.10–2009.02 1094 0.02 0.58 10.30 (0.97, 1.10) (0.35, 0.37)

2009.07–2011.01 401 0.01 0.71 4.83 (0.36, 0.77) (0.16, 0.30)

0.45 0.38 0.34 0.07 (0.00) 0.04 (0.00)

0.59 0.54 0.49 0.05 (0.00) 0.05 (0.00)

0.54 0.51 0.50 0.03 (0.00) 0.01 (0.00)

0.52 0.49 0.45 0.04 (0.00) 0.04 (0.00)

0.52 0.49 0.47 0.03 (0.00) 0.02 (0.00)

0.53 0.50 0.47 0.04 (0.00) 0.03 (0.00)

N1

N2

N3

N4

N5

N6

1991.07–1993.02 488 0.01 0.37 3.26 (0.44, 0.36) (0.19, 0.16)

1993.11–1996.12 931 0.01 0.23 3.56 (0.52, 0.33) (0.20, 0.14)

1999.08–2000.11 329 0.03 0.28 4.11 (0.27, 0.72) (0.12, 0.28)

2003.02–2004.09 410 0.02 0.30 4.19 (0.71, 0.63) (0.29, 0.26)

2009.03–2009.06 85 0.02 0.16 3.32 (0.39, 0.26) (0.18, 0.11)

2011.02–2011.12 227 0.02 0.34 4.31 (0.40, 0.59) (0.17, 0.23)

0.54 0.54 0.54 0.00 (0.75) 0.01 (0.00)

0.55 0.55 0.54 0.00 (0.07) 0.01 (0.00)

0.47 0.46 0.44 0.01 (0.25) 0.01 (0.02)

0.50 0.48 0.47 0.02 (0.03) 0.01 (0.15)

0.68 0.69 0.69 0.01 (0.98) 0.00 (0.98)

0.46 0.46 0.47 0.00 (0.81) 0.01 (0.00)

the degree to assess the market development stages. Morales et al. [25] reported that the (weighted) DH value increases in financial time series when the unstable period is reached. In the KOSPI series, multifractal regimes containing some financial crisis period tend to have the larger DH values than non-identifiable regimes. According to Zunino et al. [42] and Morales et al. [25], multifractal regimes identified in KOSPI series could be regarded as the periods with low market efficiency. In Fig. 6a, troughs of IðtÞ are related with four major stock market fall events in Korean stock market: Asian currency crisis in 1998, IT bubble shock in 2000, credit card crisis in 2003, and global financial crisis in 2008. Asian currency crisis in 1998 and global financial crisis in 2008 belong to multifractal regimes, M3 and M5, respectively. Credit card crisis in 2003 begins at the end of the multifractality regime, M4. However, IT bubble shock in 2000 is located in the non-identifiable regime, N3, which has a rather fat lower tail distribution. Fig. 7 shows the ACF of the volatility of fjDXðtÞjg in each regime. The long memory in volatility is observed in some multifractal regimes, M3, and M5. Other regimes including all non-identifiable regimes does not show any clear long range correlation of volatility. Based on the results in Table 5 and Fig. 7, we can classify multifractal regimes as follows. M1, M2, M4, and M6, have heavy tail distribution (especially in lower tail heaviness), but do not have long memory in volatility. M3 shows a strong long memory in volatility but no fat-tail property. M5, which include the recent global financial crisis period, accompanies both heavy tail distribution and long memory in volatility.

4.2. Surrogate test for multifractal regimes In this section, we perform some surrogate tests for the validation of multifractal regimes identified by MRDM in Section 4.1. MRDM was applied to multiple surrogate paths equivalent to the original KOSPI series. We performed surrogate tests to check whether MRDM can detect the same multifractal regimes (time intervals) in the surrogate series as those identified in the KOSPI series. The surrogate test procedure is as follows. We generated 200 surrogate sample series, and applied MRDM to them. Assuming that the classified regimes in the KOSPI series are infallible, we count the number of mislabeled time points in each series. Time points, which switch from multifractal (non-identifiable) regimes in the KOSPI series to non-identifiable (multifractal) regimes in a surrogate series, are categorized as type 1 (type 2). Based on these whole counts from 200 surrogate sample paths, we compute the proportion of time points misclassified as multifractal (non-identifiable) ones, which is type I error (type 2 error) in this section. There are many methods to generate surrogate series for the validation of MRDM, and each method has its own characteristics. From the result of a specific surrogate test, we can conjecture the main causes of multifractality in the KOSPI series. First, we applied MRDM to the surrogate series generated by Fourier phase randomization of KOSPI return series. Especially, we used improved amplitude adjusted Fourier transformation (IAAFT) algorithm [35] through which the surrogate series lose its nonlinear correlation but keeps its linear correlation and the properties due to

H. Lee, W. Chang / Chaos, Solitons & Fractals 70 (2015) 117–129

(a)

Multifractal regime

(b)

M1

N1

M2

N2

M3

N3

M4

N4

M5

N5

M6

N6

125

Non-identifiable regime

Fig. 7. ACF of the volatility, fjDXðtÞjg; in multifractal regime (M1, M2, M3, M4, M5, M6) and non-identifiable regime (N1, N2, N3, N4, N5, N6).

a probability density function including the tail property [37]. Each multifractal and non-identifiable regime (M1, . . . , M6, and N1, . . . , N6) in the KOSPI series is replaced with an equivalent surrogate regime generated by IAAFT so that each surrogate regime can retain the structure of the original regime except for its nonlinear correlation. Second, we made the surrogate series from a normal distribution. The parameters of normal distribution are estimated based on the original KOSPI return series. We generated a series of normal random values and reordered them to have the ordering structure identical to the original KOSPI return series. In other words, a surrogate path is obtained in the way that the kth largest value in the original KOSPI return series is replaced with the kth largest value generated from normal distribution. In this setting

the long range autocorrelation structure of the KOSPI series is preserved. Third, we used a generalized student t distribution to estimate the distribution of KOSPI return series. We tried to estimate the distribution of the KOSPI return series to obtain a surrogate series having the tail thickness equivalent to the original KOSPI series. We generated random values from a generalized student t distribution and reorder them to have the same ordering structure as that of the KOSPI return series. Fig. 8 shows the distribution fitted to the KOSPI return series (in left panel) and the comparison of ACF of volatility, fjDXðtÞjg, between the surrogate series generated from the distribution and the KOSPI return series (in right panel).

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(a) Fourier phase randomization (IAAFT) surrogate series vs. KOSPI series

(b) Normal surrogate series vs. KOSPI series

(c) Generalized student-t surrogate series vs. KOSPI series Fig. 8. The distribution fitted to the KOSPI return series and the comparison of ACF between the surrogate series generated from the distribution (dot mark) and the KOSPI return series (line).

The distribution of surrogate series in Fig. 8a is identical to that of KOSPI return series as the surrogate series generated from IAAFT can be regarded as the order shuffling of the KOSPI series. The ACF of volatility of IAAF surrogate shows persistency. We infer that the volatility clustering of the KOSPI return series still remains because the Fourier randomization is applied to each regime of the KOSPI return series separately. We observed that the long range ACF of volatility is disappeared when IAAFT is applied to the KOSPI return series as a whole unit. The normal distribution in Fig. 8b is not well fitted to that of the KOSPI return series, but its surrogate series have the ACF of volatility which is very close to that of the KOSPI series. The generalized student t distribution in Fig. 8c is best fitted to that of the KOSPI return series so that the surrogate ser-

ies generated from the distribution and the KOSPI return series can share many stylized facts in common. One example is the similarity of the volatility ACF structure between the surrogate series and the KOSPI return series. Table 6 shows the type 1 and type 2 errors of MRDM applied to the three kinds of surrogate series: IAAFT surrogates, normal surrogates, and generalized student t surrogates. For IAFFT surrogates, we obtained type 1 error of 0.18 and type 2 error of 0.52. For normal surrogates, the performance of MRDM detecting the given multifractal regimes is unsatisfactory, and the corresponding type 1 error is larger than that of IAAFT. The Kolmogorov–Smirnov (K–S) test shows that the normal distribution is not fit for the original KOSPI return series (p-value = 0.00). For the surrogates generated by generalized student t dis-

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Table 6 Three kinds of surrogate series generated from IAAFT, normal distribution and generalized student t distribution, and their K–S test statistics and type 1 and type 2 errors. Mean, standard deviation and degree of freedom of the distribution are denoted by l, r, and m. Surrogate

Parameters of distribution

K–S test stat (p-value)

Type 1 error

Type 2 error

IAAFT Normal Generalized student-t

– ðl; rÞ ¼ ð0; 0:02Þ ðt; l; rÞ ¼ ð3:14; 0:00; 0:01Þ

– 0.08 (0.00) 0.01 (0.79)

0.18 0.76 0.07

0.52 0.16 0.28

(a) IAAFT surrogate test

(b) Normal surrogate test

(c) Genralized student-t surrogate test Fig. 9. The series of rate that a unit time interval (month) is selected for having multifractality when MRDM is applied to 200 surrogate sample paths.

tribution, we obtained the smallest type 1 error of 0.07. We adjusted parameters of generalized student t distribution to make the distribution best fit for that of the KOSPI return series and K–S test statistics confirms this (pvalue = 0.79).

Fig. 9 shows the series of rate that a unit interval (month) is selected for having multifractality when MRDM is applied to 200 sample paths for each surrogate type: IAAFT, normal distribution, and generalized student t distribution.

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For the MRDM performance of IAAFT surrogate series in Fig. 9a, we can observe the increment of selection rate in the given multifractal regimes of the KOSPI series (shaded regions). The PDF properties of time series values including tail heaviness may play a role to cause multifractality in the KOSPI series. Also, the separate application of IAAFT to each regime might not eliminate the nonlinear correlation of the overall time series. For the MRDM performance of normal surrogate series in Fig. 9b, the high selection rate only appears in some parts of the multifractal regimes of the KOSPI series. This implies that the property of long range correlation needs to be accompanied by the properties of PDF such as tail heaviness to give rise to multifractality in the KOSPI series. MRDM shows the best performance in the reordered surrogate series generated from a generalized student t distribution. In Fig. 9c, the pattern of selection rate is relatively well matched with the multifractal regimes of the KOSPI series. This supports the claim that fat tail condition and long range correlation together bring synergies to the multifractality in the KOSPI series.

5. Conclusion In this paper, we introduced a multifractality test model and a multifractal regime detection method (MRDM) applicable to financial time series. The multifractality test model is a nonparametric method, and MRDM is quite effective for a time series of any length because the model allows the window size (nD) and moving interval ðDÞ introduced in Section 3 to be short. For this reason, the model is useful for the time varying multifractality analysis. Our method was applied to the simulation of MMAR processes and its validity was confirmed. We also identified multifractal regimes of the KOSPI time series data from 1990 to 2012 using MRDM, and determined 6 multifractal regimes and 6 non-identifiable regimes. We found some stylized facts for the multifractal regime distinct from the remaining non-identifiable regime. In general, multifractal regimes show higher Kurtosis and heavier tail indices (MFE, Hill estimator) than non-identifiable regimes. Based on the MMAR simulation in Section 3.2 and the empirical study on the KOSPI series, we claim that the ACF of volatility in multifractal regimes is more persistent than that in non-identifiable (monofractal) regimes. As the volatility clustering of time series is closely related to its long range correlation structure, we also claim that the time series having the long range ACF of volatility may also have long range correlation structure. In our empirical study of KOSPI series, the identified multifractal regime, M5 in which the global financial crisis occurred, can be an example supporting our argument. The further study on the relationship between long range ACF of volatility and multifractality is left as future work. The results of surrogate tests in Section 4.2 support that the causes of multifractality are due to a PDF and the long range correlation of time series values. Specifically, we observed the case that fat tail condition and long range correlation together bring synergies to the multifractality in the KOSPI series.

Although MRDM adopts a simple hypothesis test based on GHE, H(1) and H(2), MRDM can detect multifractal regimes well throughout the whole time series, in general. However, its capability to discriminate multifractality weakens around the regime boundaries (both ends of the regime) and a narrow multifractal regime located in the middle of a wide monofractal regime may not be detected. Our model needs to be revised to solve these problems. MRDM is meaningful in that it provides a practical method to discriminate multifractal regimes in a time series of any length. The difference of statistical properties observed in multifractal regimes from those in non-identifiable (monofractal possibly) regimes is a main reason why adaptive approaches considering multifractality are required for empirical time series studies. In the financial application, risk management for multifractal asset returns can be a typical example. The various application of MRDM in this regard is followed as future work. Acknowledgements The authors thank the three anonymous referees for their insightful and helpful suggestions that improved the quality of the paper. Financial support from the Institute for Research in Finance and Economics of Seoul National University (No. 0666A-2013007) is gratefully acknowledged. References [1] Barabasi AL, Vicsek T. Multifractality of self-affine fractals. Phys Rev A 1991;44:2730–3. [2] Bacry E, Delour J, Muzy JF. Modelling financial time series using multifractal random walks. Phys A Stat Mech Appl 2001;299:84–92. [3] Bacry E, Delour J, Muzy JF. Multifractal random walk. Phys Rev E 2001;64:026103. [4] Barunik J, Kristoufek L. On Hurst exponent estimation under heavytailed distributions. Phys A Stat Mech Appl 2010;389:3844–55. [5] Barunik J, Aste T, Di Matteo T, Liu R. Understanding the source of multifractality in financial markets. Phys A Stat Mech Appl 2012;391:4234–51. [6] Calvet L, Fisher A. Forecasting multifractal volatility. J Econ 2001;105:27–58. [7] Calvet L, Fisher A. Multifractality in asset returns: theory and evidence. Rev Econ Stat 2002;84:381–406. [8] Calvet L, Fisher A. How to forecast long-run volatility: regime switching and the estimation of multifractal processes. J Financial Econ 2004;2:49–83. [9] Calvet L, Fisher A. Multifractal volatility: theory, forecasting, and pricing. Academic Press; 2008. [10] Di Matteo T, Aste T, Dacorogna MM. Scaling behaviors in differently developed markets. Phys A Stat Mech Appl 2003;324:183–8. [11] Di Matteo T. Multi-scaling in finance. Quant Finance 2007;7:21–36. _ z_ S, Kwapien´ J, Os´wiecimka P, Rak R. Quantitative features of [12] Drozd multifractal subtleties in time series. Europhys Lett 2009;88:60003. _ z_ S, Speth J. Financial multifractality and its [13] Górski AZ, Drozd subtleties: an example of DAX. Phys A Stat Mech Appl 2002;316:496–510. [14] Hill Bruce M. A simple general approach to inference about the tail of a distribution. Ann Stat 1975;3:1163–74. [15] Jiang ZQ, Zhou WX. Multifractality in stock indexes: fact or fiction? Phys A Stat Mech Appl 2008;387:3605–14. [16] Kantelhardt JW, Zschiegner SA, Koscielny-Bunde E, Havlin S, Bunde A, Stanley HE. Multifractal detrended fluctuation analysis of nonstationary time series. Phys A Stat Mech Appl 2002;316:87–114. [17] Kumar S, Deo N. Multifractal properties of the Indian financial market. Phys A Stat Mech Appl 2009;388:1593–602. _ z_ S. Components of multifractality in [18] Kwapien´ J, Os´wie˛cimka P, Drozd high-frequency stock returns. Phys A Stat Mech Appl 2005;350:466–74.

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