FRACTAL DETRENDED FLUCTUATION ANALYSIS ... - World Scientific
December 15, 2010
16:14
WSPC/S0218-1274
02808
International Journal of Bifurcation and Chaos, Vol. 20, No. 11 (2010) 3753–3768 c World Scientific Publishing Company DOI: 10.1142/S0218127410028082
FRACTAL DETRENDED FLUCTUATION ANALYSIS OF CHINESE ENERGY MARKETS JUNHUAN ZHANG and JUN WANG∗ College of Science, Beijing Jiaotong University, Beijing 100044, P. R. China ∗[email protected] Received January 21, 2010; Revised April 3, 2010 In this paper, we analyze and compare long-range power-law correlations of returns, absolute returns, squared returns, cubed returns and square waved returns for sixteen individual stocks from the block of energy sources of Chinese stock market and five stock indices (Shanghai Composite Index, Shenzhen Component Index, Dow Jones Industrial Average index, Nasdaq Composite Index, the Standard and Poor’s 500 Index) by using a detrended fluctuation analysis approach. The empirical evidence suggests that Shanghai Composite Index is very close to Shenzhen Component Index and Nasdaq, DJIA is very close to S&P 500 in all cases. And the exponent trends of the returns are close to that of square waved returns. Also, five indices deviate from other sixteen individual energy stocks in all cases except square waved returns. Further, there are long-range correlations and persistence in volatility series of absolute returns and squared returns. Moreover, we investigate the long-term memory of these returns by applying Lo’s modified rescaled range statistic. We find that the China energy market exhibits fractal and persistence properties. Keywords: Energy market; return; detrended fluctuation analysis; Lo’s modified rescaled range statistic.
1. Introduction In recent decades, China has been becoming an important part in global economic matters and energy markets. China is now the world’s second largest energy consumer behind the United States and the world’s third largest oil import nation, and it is driven to expand its energy needs with its economic trajectory. The increasing energy demand makes the country much dependent on the oil import. The objective of this paper is to investigate and compare long-range power-law correlations and fractal detrended fluctuation of returns, absolute returns, squared returns, cubed returns and square waved returns for sixteen individual energy stocks and five stock indices by using a detrended ∗
fluctuation analysis (DFA) approach and Lo’s modified rescaled range statistic. The databases are from Shanghai Stock Exchange and Shenzhen Stock Exchange, which are sixteen individual stocks from energy sources block of Chinese stock market, Shanghai Stock Exchange (SSE) Composite Index, Shenzhen Stock Exchange (SZSE) Component Index. Also, we consider Dow Jones Industrial Average index (DJIA), Nasdaq Composite Index (IXIC), and the Standard and Poor’s 500 Index (S&P 500). The details of the databases are given in Sec. 2, the corresponding statistical analysis may be useful for understanding the fluctuation of Chinese energy market. Recently, some research work has been done to study the behavior of oil price series and stock
Author for correspondence 3753
December 15, 2010
3754
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang
prices in energy market, see [Alvarez-Ramirez et al., 2008; Ayadi et al., 2009; Brooks, 1998; Ghouri, 2006; Grau-Carles, 2001; Hauser, 1997; Hurst, 1951; Liao & Wang, 2010; Lo, 1991; Oberndorfer, 2009; Papapetrou, 2001; Sadique & Silvapulle, 2001; Wang, 2007; Wang & Wang, 2009]. The fluctuations of crude oil markets and stock markets are analyzed by the fractal analysis [Calvet & Fisher, 2008; Mandelbrot, 1982; Mandelbrot et al., 1997; Peters, 1994] and other statistical analysis [Gaylord & Wellin, 1995; Ilinski, 2001; Lamberton & Lapeyre, 2000; Mills, 1999]. For example, Ghouri [2006], Oberndorfer [2009] and Papapetrou [2001] studied the relationships between oil price shocks and stock markets, Brooks [1998], Liao and Wang [2010], Wang [2007], Wang and Wang [2009] investigated and predicted stock indices and returns volatility. Further, some work has been done in the investigation of long-term memory in stock market returns, see [Hauser, 1997; Hurst, 1951; Lo, 1991; Peters, 1994; Sadique & Silvapulle, 2001]. This paper proceeds as follows: Sec. 2 gives the databases which is analyzed in the present paper. Section 3 presents the statistical analysis of Chinese energy stocks and five stock indices, and we analyze and compare the corresponding long-range power-law correlations of these returns.
2. Databases of the Empirical Research In the present paper, sixteen individual stocks from the energy sources block of Chinese stock market are analyzed. Among them, eight stocks come from Shanghai Stock Exchange and eight stocks come from Shenzhen Stock Exchange, and these energy stocks can well reflect Chinese energy market to some extent. They are Rongfeng Holding Group (000668), Sinopec ShanDong TaiShan Pectroleum (000554), Taian Lurun (600157), Yueyang Xingchang Petro-Chemical (000819), Shenzhen Guangju Energy (000096), Maoming Petro-Chemical Shihua (000637), Zhejiang Haiyue (600387), Shenyang Chemical Industry (000698), China Union Holdings (000036), China-Kinwa High Technology (600110), Guangzhou Development Industry Holdings (600098), Shenergy Company Limited (600642), Offshore Oil Engeneering (600583), Changjiang Securities Company Limited (000783), Shanghai Petrochemical (600688) and China Petroleum & Chemical Corporation (600028), where the numbers in brackets are
the corresponding registered stock tickers. The databases are from Jan 4th 2000 to Nov 27th 2009 for the stocks of 000036, 000554, 000637, 000668, 000698, 000783, 000819, 600098, 600110, 600157, 600642 and 600688. In addition, the selected data of stock 000096 is from Jul 24th 2000 to Nov 27th 2009; the data of 600028 is from Aug 8th 2001 to Nov 27th 2009; the data of 600387 is from Feb 18th 2004 to Nov 27th 2009; and the observed data of 600583 is from Feb 5th 2002 to Nov 27th 2009. Furthermore, we select the daily data of five market indices from all over the world. SSE Composite Index and SZSE Component Index are two of the most important security indices in Chinese stock markets. The database is from the index of SSE in the 10-year period from Jan 4th 2000 to Nov 27th 2009, the total number of observed data is 2391. The daily data of SZSE is from Jan 4th 2000 to Nov 27th 2009, the total number of observed data is about 2390. Another database is the Standard and Poor’s 500 index, the daily data of S&P 500 is observed in 10-year period from Jan 3rd 2000 to Nov 27th 2009, and the total number of observed data is 2492. And we also study Dow Jones Industrial Average index, the sample daily data of DJIA is from Jan 3rd 2000 to Nov 27th 2009, the total data number is about 2492. At last, the daily data of Nasdaq Composite Index (IXIC) is from Jan 3rd 2000 to Nov 27th 2009, the data size is 2492 in total. In the present paper, we analyze the statistical behaviors of market returns by using the daily closing prices for five indices and sixteen individual stocks (where individual stocks are from the energy source block in Chinese stock markets). The Chinese stock markets are actually developing the financial markets. Especially from 2000, with the reformation and development of Chinese economic systems, the Chinese financial markets have developed rapidly. In order to study the actual behaviors of the current Chinese stock markets, the daily empirical research data is selected after 2000.
3. Statistical Analysis of Chinese Energy Stocks and Five Market Indices We introduce the concepts of return, absolute return, squared return, cubed return for the stock market. According to the theory of mathematical finance [Lamberton & Lapeyre, 2000; Wang, 2007], we have the formula of stock logarithmic return r(t) = ln P (t + 1) − ln P (t)
December 15, 2010
16:14
WSPC/S0218-1274
02808
Fractal Detrended Fluctuation Analysis of Chinese Energy Markets
where t = 1, 2, . . . , N , P (t + 1) and P (t) denote the daily closing stock prices at (t + 1)th and tth trade days, respectively. The corresponding absolute return is defined as r1 (t) = |ln P (t + 1) − ln P (t)| = |r(t)|. And the formula of squared return is defined as r2 (t) = [ln P (t + 1) − ln P (t)]2 = [r(t)]2 . And finally, the cubed return is given by r3 (t) = [ln P (t + 1) − ln P (t)]3 = [r(t)]3 . Furthermore, the definition of square waved return is introduced in Sec. 3.1. In this section, we investigate and compare the long-range power-law correlations and the fractal detrended fluctuations of return, absolute return, squared return, cubed return and square waved return for sixteen individual energy stocks and five market indices by using a detrended fluctuation analysis (DFA) approach and the Lo’s modified rescaled range statistic.
3.1. The detrended fluctuation analysis of returns In this section, we investigate the long range powerlaw correlations in the stock returns time series by applying a detrended fluctuation analysis (DFA) method, see [Grau-Carles, 2001]. DFA is a scaling technique applied to estimate a scaling exponent from the behavior of the average fluctuation of a random variable around its local trend. We consider a time series x(t) (t = 1, . . . , N ), and the integrated time series y(k) is defined as k N 1 x(t) . x(t) − y(k) = N t=l
k=1
The integrated time series is divided into intervals of equal size, m. In each box of length m, a least squares line is fit to the data (representing the trend in that box). The y coordinate of the straight line segments is denoted by ym (k). In each interval, the integrated time series, y(k), is detrended by subtracting the local trend, ym (k). For a given interval size m, the root mean square fluctuation is calculated by 12 N 1 [y(t) − ym (t)]2 . F (m) = N t=1
The above definition is repeated for all the divided intervals. There is a power-law relation
3755
between F (m) and m : F (m) ∝ mα (where F (m) is calculated by log F (m) = β + α log m). The parameter α is the scaling exponent or the correlation exponent, which exhibits the long range powerlaw correlations of the time series. For α = 0.5, it indicates that the time series is uncorrelated (white noise); for the value 0 < α < 0.5, it indicates the power-law anticorrelations; for 0.5 < α < 1, the time series has the persistent long-range power-law correlations; for α > 1, it indicates that the correlations exist but not in a power-law form. In the following, according to the method of DFA, we study and compare the statistical behaviors of five kinds of returns. Firstly, we analyze the behavior of returns for sixteen individual energy stocks and five market indices in Fig. 1 and Table 1. The empirical result reveals that the smallest exponent corresponds to DJIA index and the largest one is for Sinopec ShanDong TaiShan Pectroleum (000554). In Table 1, we also find that the exponents for four individual stocks (600583, 000668, 600157 and 000783) and two indices (S&P 500 and IXIC) are smaller than 0.5, which indicates the power-law anticorrelations. And the exponents for other twelve individual stocks and three indices are bigger than 0.5, this indicates the long-range power-law correlations. In addition, the empirical result shows that five indices deviate from other sixteen individual energy stocks in Fig. 1. The SSE Composite Index is close to IXIC while DJIA is close to S&P 500. Further, the front part of SZSE Component Index is close to those of IXIC and SSE Composite Index, this implies that these three time series have similar power-law correlation (or anticorrelation) behaviors at short time scales. The absolute returns for sixteen individual energy stocks and five market indices are investigated in Fig. 2 and Table 2. The empirical research shows that the smallest exponent corresponds to Offshore Oil Engeneering (600583) and the largest one is for IXIC index. In Table 2, the results show that the values of the scaling exponents are bigger than 0.5 in all cases, so we can conclude that there are the long-range correlations and persistence in volatility series. Furthermore, we also find that the front parts of five indices deviate from other sixteen individual energy stocks in Fig. 2, this may imply that the five indices have similar power-law correlation (or anticorrelation) behaviors at short time scales, similarly to the group of sixteen individual energy stocks. And the values of F (m) for five indices are smaller to those for other individual
December 15, 2010
3756
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang 0
10
0
10
−1
10
−2
10
−1
10
−3
10
1
2
10
3
10
10
−1
−2
10
10
−2
10
−3
1
2
10
10
1
3
10
10
2
10
3
10
Fig. 1.
10
The plot of DFA test statistics of returns.
Table 1.
DFA test statistics of returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−2.3943 −2.3045 −2.6292 −2.4538 −2.5076 −2.6288 −2.6217
0.50121 0.4356 0.57505 0.45303 0.51879 0.56213 0.6186
000819 600110 600387 600157 000637 600098 000783
−2.4682 −2.3731 −2.4503 −2.3764 −2.5313 −2.6231 −2.2863
0.52544 0.51142 0.55736 0.49803 0.52202 0.56057 0.48924
600028 000698 SSE SZSE S&P500 IXIC DJIA
−2.5722 −2.4846 −2.8587 −2.8156 −2.6989 −2.637 −2.7139
stocks. The SSE Composite Index and SZSE Component Index are close to IXIC, and DJIA is very close to S&P 500. DFA test statistics of squared returns for sixteen individual energy stocks and five stock indices are investigated in Fig. 3 and Table 3.
α 0.52385 0.51559 0.59305 0.59246 0.40154 0.46495 0.39933
We find that the smallest exponent corresponds to Changjiang Securities Company Limited (000783) and the largest one is for IXIC. In Table 3, there are two exponents for individual stocks (600583 and 000783) smaller than 0.5. The rest are bigger than 0.5, so there is a indication of the long-range
0
10
0
10
−1
10
−2
10
−1
10
−3
10
1
2
10
10
3
10
0
10
−2
10
−1
10
−2
10
−3
10
1
10
−3
10
1
2
10
2
10
10
Fig. 2.
The plot of DFA test statistics of absolute returns.
3
10
3
10
December 15, 2010
16:14
WSPC/S0218-1274
02808
Fractal Detrended Fluctuation Analysis of Chinese Energy Markets Table 2.
DFA test statistics of absolute returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−2.7681 −2.6521 −3.0087 −3.0554 −2.8465 −3.0121 −3.0005
0.69086 0.64452 0.75985 0.79742 0.68192 0.71773 0.81961
000819 600110 600387 600157 000637 600098 000783
−2.9054 −2.8249 −2.7524 −2.866 −3.0525 −3.0076 −2.6368
0.73284 0.74238 0.68495 0.73748 0.82091 0.77787 0.67452
600028 000698 SSE SZSE S&P500 IXIC DJIA
−3.0624 −2.9594 −3.2582 −3.2376 −3.4932 −3.4108 −3.5275
power-law correlations in the squared returns of the stocks and indices except 600583 and 000783. Furthermore, we also find that the front parts of five indices deviate from other sixteen individual energy stocks in Fig. 3, which may also imply that the five indices have similar power-law correlation (or anticorrelation) behaviors at short time scales, similarly to the group of sixteen individual energy stocks. And the values of F (m) for five indices are smaller than those for other individual stocks. The SSE Composite Index and SZSE Component Index are close to IXIC, and DJIA is very close to S&P 500.
α 0.76416 0.73472 0.778 0.78791 0.87331 0.89042 0.87927
We analyze DFA test statistics of cubed returns for sixteen individual energy stocks and five stock indices in Fig. 4 and Table 4. The empirical research exhibits that the smallest exponent corresponds to Changjiang Securities Company Limited (000783) and the largest one is for IXIC. Table 4 shows that the exponents for seven individual stocks and five indices are smaller than 0.5, which indicates the power-law anticorrelations. The rest are bigger than 0.5. Hence, this indicates the long-range power-law correlations. Furthermore, we also find that the five indices deviate from other sixteen individual energy stocks in Fig. 4. And the values of F (m) for them
0
10
−2
10
1
2
10
−2
3
10
10
10
−2
10
−4
10
−4
10
−6
10
1
2
10
1
3
10
10
2
10
3
10
Fig. 3.
10
The plot of DFA test statistics of squared returns.
Table 3.
3757
DFA test statistics of squared returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−3.3369 −3.0284 −3.6747 −4.013 −3.7859 −4.402 −3.9634
0.56504 0.44791 0.53679 0.61489 0.56775 0.63179 0.65068
000819 600110 600387 600157 000637 600098 000783
−4.2318 −3.4668 −3.8247 −3.4472 −3.6335 −4.178 −2.6264
0.68467 0.56158 0.65183 0.55501 0.53523 0.73285 0.43393
600028 000698 SSE SZSE S&P500 IXIC DJIA
−4.2664 −4.0355 −4.7343 −4.7218 −4.909 −4.8463 −4.9647
α 0.65853 0.59002 0.70399 0.72181 0.76965 0.8461 0.77537
December 15, 2010
3758
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang
−2
10
−5
10
0
10
−4
10
−6
−6
10
1
2
10
10
3
10
1
2
10
10
3
10
10
−2
10
−4
10
−6
10
1
2
10
Fig. 4.
10
The plot of DFA test statistics of cubed returns.
where x is the floor function, sgn(x) is the sign function, and tanh−1 x is the inverse hyperbolic tangent. Let the period of the square wave be 2L. Since the square wave function is odd, the Fourier series has a0 = an = 0 and
are smaller than those for other individual stocks. The SSE Composite Index is very close to SZSE Component Index. At the same time, IXIC is close to SZSE Component Index, and DJIA is close to S&P 500. In the next part, we introduce the square wave, which is a periodic waveform consisting of instantaneous transitions between two levels, and sometimes also called the Rademacher function [Thompson et al., 1986]. The square wave has period 2 and levels −1/2 and 1/2. Other common levels for square waves include (−1, 1) and (0, 1) (digital signals). Analytic formulas for the square wave S(x) with half-amplitude A, period T and offset x0 are given by
bn =
L
sin
nπx
0
dx
2 [1 − (−1)n ] nπ for n even 0, = . 4 , for n odd nπ =
Then the Fourier series for the square wave with period 2L, phase offset 0 and half-amplitude 1 can be concluded as
2i [tanh−1 (e−iπ(x−x0 )/T ) π
f (x) =
− tanh−1 (eiπ(x−x0 )/T )] Table 4.
2 L
L 4 2 1 sin nπ = nπ 2
S(x) = A(−1)2(x−x0 )/T 2π(x − x0 ) = A sgn sin T =A
3
10
4 π
∞ n=1,3,5...
nπx 1 sin . n L
DFA test statistics of cubed returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−3.8589 −3.4588 −4.3674 −4.4685 −4.6252 −5.7207 −4.7813
0.56692 0.38525 0.50749 0.27961 0.5756 0.52607 0.58901
000819 600110 600387 600157 000637 600098 000783
−5.2753 −4.0605 −4.5687 −4.0182 −4.3509 −5.2337 −2.6499
0.46281 0.51662 0.56633 0.5217 0.48292 0.78273 0.294
600028 000698 SSE SZSE S&P500 IXIC DJIA
−5.1459 −4.505 −5.8173 −5.823 −5.5303 −5.7669 −5.7174
α 0.48742 0.2133 0.42324 0.43139 0.14444 0.46602 0.22261
December 15, 2010
16:14
WSPC/S0218-1274
02808
Fractal Detrended Fluctuation Analysis of Chinese Energy Markets
3759
2
10
2
10
1
10
1
0
10
10
−1
10
1
2
10
3
10
10
0
10
0
10
1
−1
10
10
1
10
3
10
3
10
Fig. 5.
∞
1 sin n n=1,3,5...
10
The plot of DFA test statistics of square waved returns.
Thus the square waved return is defined as 4 f (r(t)) = π
2
10
2
The advantage of the square wave method is that the latter parts of DFA test statistics can be easily analyzed than those of other methods. But there is also a disadvantage, that is, it becomes more difficult to study the front parts for DFA test statistics, since the corresponding front parts gather together in a line. In their latter parts, the SSE Composite Index is very close to SZSE Component Index. And the front parts of IXIC are close to SZSE Component Index, and DJIA is close to S&P 500 index. Finally, we compare the scaling exponents α for the returns, absolute returns, squared returns, cubed returns and square waved returns in Fig. 6. The result shows that the SSE Composite Index is very close to SZSE Component Index and IXIC. DJIA is close to S&P 500 in all cases. And the exponent trend of the returns are close to that of square waved returns. Moreover, we can find that all the exponents α for absolute returns are larger than 0.5, and most exponents α for squared returns are larger than 0.5, see Table 3 and Fig. 6. This implies that
nπr(t) . L
Then, we study DFA test statistics of the square waved returns for sixteen individual energy stocks and five market indices in Fig. 5 and Table 5. In Table 5, the smallest exponent corresponds to Changjiang Securities Company Limited (000783) and the largest value is for SSE Composite Index. Table 5 shows that the exponents for three individual stocks and two indices are smaller than 0.5, which indicates the power-law anticorrelations. The rest are bigger than 0.5, this indicates the longrange power-law correlations. Also, we find that the front parts of all the financial series gather together in a line and their latter parts depart from each other in Fig. 5, this denotes that the method of square waved return results in deviations in DFA test statistics at longer time scales. Table 5.
DFA test statistics of square waved returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−0.61529 −0.66068 −0.71813 −0.59654 −0.69202 −0.74015 −0.76065
0.51042 0.53689 0.56849 0.48891 0.54259 0.59551 0.60193
000819 600110 600387 600157 000637 600098 000783
−0.62929 −0.76325 −0.83942 −0.58993 −0.69351 −0.67448 −0.525
0.52047 0.59799 0.64186 0.49551 0.55791 0.53174 0.43888
600028 000698 SSE SZSE S&P500 IXIC DJIA
−0.61488 −0.71984 −0.88057 −0.85617 −0.56455 −0.6365 −0.61224
α 0.50171 0.56845 0.67945 0.67477 0.45667 0.52711 0.49592
December 15, 2010
3760
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang 1 Absolute returns
Squared returns
0.8
0.6
0.4
0.2 Cubed returns
60
36 00
60
00
05
0
Square waved returns
83 06 42 00 06 68 00 00 96 60 06 88 00 05 54 00 08 19 60 01 10 60 03 87 60 01 57 00 06 37 60 00 98 00 07 83 60 00 28 00 06 98 SS E SZ SE S& P5 00 IXI C DJ IA
Returns
Fig. 6. The plot of the scaling exponents α for the returns, absolute returns, squared returns, cubed returns and square waved returns.
there are long-range correlations and persistence in volatility series for absolute returns and for most squared returns.
where σ ˆn2 (q)
3.2. The fractal detrended fluctuation analysis
×
In this subsection, we analyze the long-term memory of five kinds of returns by using Lo’s modified rescaled range statistic [Lo, 1991]. The long memory is measured by the Hurst exponent H, calculated by Lo’s modified rescaled range statistic. For 1/2 < H < 1, the series exhibits the long term persistence, with the maintenance of tendency; for 0 < H < 0.5, the series is the antipersistent, presenting reversion to the mean; and for H = 0.5, the series corresponds to a random walk. By estimating the Hurst exponent H, we wish to find whether the Chinese energy market exhibits the fractal and the persistence properties. We con. . , Xn and let X n sider a sample of series X1 , X2 , . denote the sample mean (1/n) j Xj . Then the modified rescaled range statistic, denoted by Qn , is defined by k 1 max (Xj − X n ) Qn = σ ˆn (q) 1≤k≤n j=1
− min
1≤k≤n
k j=1
(Xj − X n )
n
q
j=1
j=1
1 2 = (Xj − X n )2 + ωj (q) n n n
(Xi − X n )(Xi−j
i=j+1
=σ ˆx2 + 2
q
− X n)
ωj (q)ˆ γj
j=1
and ωj (q) = 1 −
j , q+1
q
16:14
WSPC/S0218-1274
02808
International Journal of Bifurcation and Chaos, Vol. 20, No. 11 (2010) 3753–3768 c World Scientific Publishing Company DOI: 10.1142/S0218127410028082
FRACTAL DETRENDED FLUCTUATION ANALYSIS OF CHINESE ENERGY MARKETS JUNHUAN ZHANG and JUN WANG∗ College of Science, Beijing Jiaotong University, Beijing 100044, P. R. China ∗[email protected] Received January 21, 2010; Revised April 3, 2010 In this paper, we analyze and compare long-range power-law correlations of returns, absolute returns, squared returns, cubed returns and square waved returns for sixteen individual stocks from the block of energy sources of Chinese stock market and five stock indices (Shanghai Composite Index, Shenzhen Component Index, Dow Jones Industrial Average index, Nasdaq Composite Index, the Standard and Poor’s 500 Index) by using a detrended fluctuation analysis approach. The empirical evidence suggests that Shanghai Composite Index is very close to Shenzhen Component Index and Nasdaq, DJIA is very close to S&P 500 in all cases. And the exponent trends of the returns are close to that of square waved returns. Also, five indices deviate from other sixteen individual energy stocks in all cases except square waved returns. Further, there are long-range correlations and persistence in volatility series of absolute returns and squared returns. Moreover, we investigate the long-term memory of these returns by applying Lo’s modified rescaled range statistic. We find that the China energy market exhibits fractal and persistence properties. Keywords: Energy market; return; detrended fluctuation analysis; Lo’s modified rescaled range statistic.
1. Introduction In recent decades, China has been becoming an important part in global economic matters and energy markets. China is now the world’s second largest energy consumer behind the United States and the world’s third largest oil import nation, and it is driven to expand its energy needs with its economic trajectory. The increasing energy demand makes the country much dependent on the oil import. The objective of this paper is to investigate and compare long-range power-law correlations and fractal detrended fluctuation of returns, absolute returns, squared returns, cubed returns and square waved returns for sixteen individual energy stocks and five stock indices by using a detrended ∗
fluctuation analysis (DFA) approach and Lo’s modified rescaled range statistic. The databases are from Shanghai Stock Exchange and Shenzhen Stock Exchange, which are sixteen individual stocks from energy sources block of Chinese stock market, Shanghai Stock Exchange (SSE) Composite Index, Shenzhen Stock Exchange (SZSE) Component Index. Also, we consider Dow Jones Industrial Average index (DJIA), Nasdaq Composite Index (IXIC), and the Standard and Poor’s 500 Index (S&P 500). The details of the databases are given in Sec. 2, the corresponding statistical analysis may be useful for understanding the fluctuation of Chinese energy market. Recently, some research work has been done to study the behavior of oil price series and stock
Author for correspondence 3753
December 15, 2010
3754
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang
prices in energy market, see [Alvarez-Ramirez et al., 2008; Ayadi et al., 2009; Brooks, 1998; Ghouri, 2006; Grau-Carles, 2001; Hauser, 1997; Hurst, 1951; Liao & Wang, 2010; Lo, 1991; Oberndorfer, 2009; Papapetrou, 2001; Sadique & Silvapulle, 2001; Wang, 2007; Wang & Wang, 2009]. The fluctuations of crude oil markets and stock markets are analyzed by the fractal analysis [Calvet & Fisher, 2008; Mandelbrot, 1982; Mandelbrot et al., 1997; Peters, 1994] and other statistical analysis [Gaylord & Wellin, 1995; Ilinski, 2001; Lamberton & Lapeyre, 2000; Mills, 1999]. For example, Ghouri [2006], Oberndorfer [2009] and Papapetrou [2001] studied the relationships between oil price shocks and stock markets, Brooks [1998], Liao and Wang [2010], Wang [2007], Wang and Wang [2009] investigated and predicted stock indices and returns volatility. Further, some work has been done in the investigation of long-term memory in stock market returns, see [Hauser, 1997; Hurst, 1951; Lo, 1991; Peters, 1994; Sadique & Silvapulle, 2001]. This paper proceeds as follows: Sec. 2 gives the databases which is analyzed in the present paper. Section 3 presents the statistical analysis of Chinese energy stocks and five stock indices, and we analyze and compare the corresponding long-range power-law correlations of these returns.
2. Databases of the Empirical Research In the present paper, sixteen individual stocks from the energy sources block of Chinese stock market are analyzed. Among them, eight stocks come from Shanghai Stock Exchange and eight stocks come from Shenzhen Stock Exchange, and these energy stocks can well reflect Chinese energy market to some extent. They are Rongfeng Holding Group (000668), Sinopec ShanDong TaiShan Pectroleum (000554), Taian Lurun (600157), Yueyang Xingchang Petro-Chemical (000819), Shenzhen Guangju Energy (000096), Maoming Petro-Chemical Shihua (000637), Zhejiang Haiyue (600387), Shenyang Chemical Industry (000698), China Union Holdings (000036), China-Kinwa High Technology (600110), Guangzhou Development Industry Holdings (600098), Shenergy Company Limited (600642), Offshore Oil Engeneering (600583), Changjiang Securities Company Limited (000783), Shanghai Petrochemical (600688) and China Petroleum & Chemical Corporation (600028), where the numbers in brackets are
the corresponding registered stock tickers. The databases are from Jan 4th 2000 to Nov 27th 2009 for the stocks of 000036, 000554, 000637, 000668, 000698, 000783, 000819, 600098, 600110, 600157, 600642 and 600688. In addition, the selected data of stock 000096 is from Jul 24th 2000 to Nov 27th 2009; the data of 600028 is from Aug 8th 2001 to Nov 27th 2009; the data of 600387 is from Feb 18th 2004 to Nov 27th 2009; and the observed data of 600583 is from Feb 5th 2002 to Nov 27th 2009. Furthermore, we select the daily data of five market indices from all over the world. SSE Composite Index and SZSE Component Index are two of the most important security indices in Chinese stock markets. The database is from the index of SSE in the 10-year period from Jan 4th 2000 to Nov 27th 2009, the total number of observed data is 2391. The daily data of SZSE is from Jan 4th 2000 to Nov 27th 2009, the total number of observed data is about 2390. Another database is the Standard and Poor’s 500 index, the daily data of S&P 500 is observed in 10-year period from Jan 3rd 2000 to Nov 27th 2009, and the total number of observed data is 2492. And we also study Dow Jones Industrial Average index, the sample daily data of DJIA is from Jan 3rd 2000 to Nov 27th 2009, the total data number is about 2492. At last, the daily data of Nasdaq Composite Index (IXIC) is from Jan 3rd 2000 to Nov 27th 2009, the data size is 2492 in total. In the present paper, we analyze the statistical behaviors of market returns by using the daily closing prices for five indices and sixteen individual stocks (where individual stocks are from the energy source block in Chinese stock markets). The Chinese stock markets are actually developing the financial markets. Especially from 2000, with the reformation and development of Chinese economic systems, the Chinese financial markets have developed rapidly. In order to study the actual behaviors of the current Chinese stock markets, the daily empirical research data is selected after 2000.
3. Statistical Analysis of Chinese Energy Stocks and Five Market Indices We introduce the concepts of return, absolute return, squared return, cubed return for the stock market. According to the theory of mathematical finance [Lamberton & Lapeyre, 2000; Wang, 2007], we have the formula of stock logarithmic return r(t) = ln P (t + 1) − ln P (t)
December 15, 2010
16:14
WSPC/S0218-1274
02808
Fractal Detrended Fluctuation Analysis of Chinese Energy Markets
where t = 1, 2, . . . , N , P (t + 1) and P (t) denote the daily closing stock prices at (t + 1)th and tth trade days, respectively. The corresponding absolute return is defined as r1 (t) = |ln P (t + 1) − ln P (t)| = |r(t)|. And the formula of squared return is defined as r2 (t) = [ln P (t + 1) − ln P (t)]2 = [r(t)]2 . And finally, the cubed return is given by r3 (t) = [ln P (t + 1) − ln P (t)]3 = [r(t)]3 . Furthermore, the definition of square waved return is introduced in Sec. 3.1. In this section, we investigate and compare the long-range power-law correlations and the fractal detrended fluctuations of return, absolute return, squared return, cubed return and square waved return for sixteen individual energy stocks and five market indices by using a detrended fluctuation analysis (DFA) approach and the Lo’s modified rescaled range statistic.
3.1. The detrended fluctuation analysis of returns In this section, we investigate the long range powerlaw correlations in the stock returns time series by applying a detrended fluctuation analysis (DFA) method, see [Grau-Carles, 2001]. DFA is a scaling technique applied to estimate a scaling exponent from the behavior of the average fluctuation of a random variable around its local trend. We consider a time series x(t) (t = 1, . . . , N ), and the integrated time series y(k) is defined as k N 1 x(t) . x(t) − y(k) = N t=l
k=1
The integrated time series is divided into intervals of equal size, m. In each box of length m, a least squares line is fit to the data (representing the trend in that box). The y coordinate of the straight line segments is denoted by ym (k). In each interval, the integrated time series, y(k), is detrended by subtracting the local trend, ym (k). For a given interval size m, the root mean square fluctuation is calculated by 12 N 1 [y(t) − ym (t)]2 . F (m) = N t=1
The above definition is repeated for all the divided intervals. There is a power-law relation
3755
between F (m) and m : F (m) ∝ mα (where F (m) is calculated by log F (m) = β + α log m). The parameter α is the scaling exponent or the correlation exponent, which exhibits the long range powerlaw correlations of the time series. For α = 0.5, it indicates that the time series is uncorrelated (white noise); for the value 0 < α < 0.5, it indicates the power-law anticorrelations; for 0.5 < α < 1, the time series has the persistent long-range power-law correlations; for α > 1, it indicates that the correlations exist but not in a power-law form. In the following, according to the method of DFA, we study and compare the statistical behaviors of five kinds of returns. Firstly, we analyze the behavior of returns for sixteen individual energy stocks and five market indices in Fig. 1 and Table 1. The empirical result reveals that the smallest exponent corresponds to DJIA index and the largest one is for Sinopec ShanDong TaiShan Pectroleum (000554). In Table 1, we also find that the exponents for four individual stocks (600583, 000668, 600157 and 000783) and two indices (S&P 500 and IXIC) are smaller than 0.5, which indicates the power-law anticorrelations. And the exponents for other twelve individual stocks and three indices are bigger than 0.5, this indicates the long-range power-law correlations. In addition, the empirical result shows that five indices deviate from other sixteen individual energy stocks in Fig. 1. The SSE Composite Index is close to IXIC while DJIA is close to S&P 500. Further, the front part of SZSE Component Index is close to those of IXIC and SSE Composite Index, this implies that these three time series have similar power-law correlation (or anticorrelation) behaviors at short time scales. The absolute returns for sixteen individual energy stocks and five market indices are investigated in Fig. 2 and Table 2. The empirical research shows that the smallest exponent corresponds to Offshore Oil Engeneering (600583) and the largest one is for IXIC index. In Table 2, the results show that the values of the scaling exponents are bigger than 0.5 in all cases, so we can conclude that there are the long-range correlations and persistence in volatility series. Furthermore, we also find that the front parts of five indices deviate from other sixteen individual energy stocks in Fig. 2, this may imply that the five indices have similar power-law correlation (or anticorrelation) behaviors at short time scales, similarly to the group of sixteen individual energy stocks. And the values of F (m) for five indices are smaller to those for other individual
December 15, 2010
3756
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang 0
10
0
10
−1
10
−2
10
−1
10
−3
10
1
2
10
3
10
10
−1
−2
10
10
−2
10
−3
1
2
10
10
1
3
10
10
2
10
3
10
Fig. 1.
10
The plot of DFA test statistics of returns.
Table 1.
DFA test statistics of returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−2.3943 −2.3045 −2.6292 −2.4538 −2.5076 −2.6288 −2.6217
0.50121 0.4356 0.57505 0.45303 0.51879 0.56213 0.6186
000819 600110 600387 600157 000637 600098 000783
−2.4682 −2.3731 −2.4503 −2.3764 −2.5313 −2.6231 −2.2863
0.52544 0.51142 0.55736 0.49803 0.52202 0.56057 0.48924
600028 000698 SSE SZSE S&P500 IXIC DJIA
−2.5722 −2.4846 −2.8587 −2.8156 −2.6989 −2.637 −2.7139
stocks. The SSE Composite Index and SZSE Component Index are close to IXIC, and DJIA is very close to S&P 500. DFA test statistics of squared returns for sixteen individual energy stocks and five stock indices are investigated in Fig. 3 and Table 3.
α 0.52385 0.51559 0.59305 0.59246 0.40154 0.46495 0.39933
We find that the smallest exponent corresponds to Changjiang Securities Company Limited (000783) and the largest one is for IXIC. In Table 3, there are two exponents for individual stocks (600583 and 000783) smaller than 0.5. The rest are bigger than 0.5, so there is a indication of the long-range
0
10
0
10
−1
10
−2
10
−1
10
−3
10
1
2
10
10
3
10
0
10
−2
10
−1
10
−2
10
−3
10
1
10
−3
10
1
2
10
2
10
10
Fig. 2.
The plot of DFA test statistics of absolute returns.
3
10
3
10
December 15, 2010
16:14
WSPC/S0218-1274
02808
Fractal Detrended Fluctuation Analysis of Chinese Energy Markets Table 2.
DFA test statistics of absolute returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−2.7681 −2.6521 −3.0087 −3.0554 −2.8465 −3.0121 −3.0005
0.69086 0.64452 0.75985 0.79742 0.68192 0.71773 0.81961
000819 600110 600387 600157 000637 600098 000783
−2.9054 −2.8249 −2.7524 −2.866 −3.0525 −3.0076 −2.6368
0.73284 0.74238 0.68495 0.73748 0.82091 0.77787 0.67452
600028 000698 SSE SZSE S&P500 IXIC DJIA
−3.0624 −2.9594 −3.2582 −3.2376 −3.4932 −3.4108 −3.5275
power-law correlations in the squared returns of the stocks and indices except 600583 and 000783. Furthermore, we also find that the front parts of five indices deviate from other sixteen individual energy stocks in Fig. 3, which may also imply that the five indices have similar power-law correlation (or anticorrelation) behaviors at short time scales, similarly to the group of sixteen individual energy stocks. And the values of F (m) for five indices are smaller than those for other individual stocks. The SSE Composite Index and SZSE Component Index are close to IXIC, and DJIA is very close to S&P 500.
α 0.76416 0.73472 0.778 0.78791 0.87331 0.89042 0.87927
We analyze DFA test statistics of cubed returns for sixteen individual energy stocks and five stock indices in Fig. 4 and Table 4. The empirical research exhibits that the smallest exponent corresponds to Changjiang Securities Company Limited (000783) and the largest one is for IXIC. Table 4 shows that the exponents for seven individual stocks and five indices are smaller than 0.5, which indicates the power-law anticorrelations. The rest are bigger than 0.5. Hence, this indicates the long-range power-law correlations. Furthermore, we also find that the five indices deviate from other sixteen individual energy stocks in Fig. 4. And the values of F (m) for them
0
10
−2
10
1
2
10
−2
3
10
10
10
−2
10
−4
10
−4
10
−6
10
1
2
10
1
3
10
10
2
10
3
10
Fig. 3.
10
The plot of DFA test statistics of squared returns.
Table 3.
3757
DFA test statistics of squared returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−3.3369 −3.0284 −3.6747 −4.013 −3.7859 −4.402 −3.9634
0.56504 0.44791 0.53679 0.61489 0.56775 0.63179 0.65068
000819 600110 600387 600157 000637 600098 000783
−4.2318 −3.4668 −3.8247 −3.4472 −3.6335 −4.178 −2.6264
0.68467 0.56158 0.65183 0.55501 0.53523 0.73285 0.43393
600028 000698 SSE SZSE S&P500 IXIC DJIA
−4.2664 −4.0355 −4.7343 −4.7218 −4.909 −4.8463 −4.9647
α 0.65853 0.59002 0.70399 0.72181 0.76965 0.8461 0.77537
December 15, 2010
3758
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang
−2
10
−5
10
0
10
−4
10
−6
−6
10
1
2
10
10
3
10
1
2
10
10
3
10
10
−2
10
−4
10
−6
10
1
2
10
Fig. 4.
10
The plot of DFA test statistics of cubed returns.
where x is the floor function, sgn(x) is the sign function, and tanh−1 x is the inverse hyperbolic tangent. Let the period of the square wave be 2L. Since the square wave function is odd, the Fourier series has a0 = an = 0 and
are smaller than those for other individual stocks. The SSE Composite Index is very close to SZSE Component Index. At the same time, IXIC is close to SZSE Component Index, and DJIA is close to S&P 500. In the next part, we introduce the square wave, which is a periodic waveform consisting of instantaneous transitions between two levels, and sometimes also called the Rademacher function [Thompson et al., 1986]. The square wave has period 2 and levels −1/2 and 1/2. Other common levels for square waves include (−1, 1) and (0, 1) (digital signals). Analytic formulas for the square wave S(x) with half-amplitude A, period T and offset x0 are given by
bn =
L
sin
nπx
0
dx
2 [1 − (−1)n ] nπ for n even 0, = . 4 , for n odd nπ =
Then the Fourier series for the square wave with period 2L, phase offset 0 and half-amplitude 1 can be concluded as
2i [tanh−1 (e−iπ(x−x0 )/T ) π
f (x) =
− tanh−1 (eiπ(x−x0 )/T )] Table 4.
2 L
L 4 2 1 sin nπ = nπ 2
S(x) = A(−1)2(x−x0 )/T 2π(x − x0 ) = A sgn sin T =A
3
10
4 π
∞ n=1,3,5...
nπx 1 sin . n L
DFA test statistics of cubed returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−3.8589 −3.4588 −4.3674 −4.4685 −4.6252 −5.7207 −4.7813
0.56692 0.38525 0.50749 0.27961 0.5756 0.52607 0.58901
000819 600110 600387 600157 000637 600098 000783
−5.2753 −4.0605 −4.5687 −4.0182 −4.3509 −5.2337 −2.6499
0.46281 0.51662 0.56633 0.5217 0.48292 0.78273 0.294
600028 000698 SSE SZSE S&P500 IXIC DJIA
−5.1459 −4.505 −5.8173 −5.823 −5.5303 −5.7669 −5.7174
α 0.48742 0.2133 0.42324 0.43139 0.14444 0.46602 0.22261
December 15, 2010
16:14
WSPC/S0218-1274
02808
Fractal Detrended Fluctuation Analysis of Chinese Energy Markets
3759
2
10
2
10
1
10
1
0
10
10
−1
10
1
2
10
3
10
10
0
10
0
10
1
−1
10
10
1
10
3
10
3
10
Fig. 5.
∞
1 sin n n=1,3,5...
10
The plot of DFA test statistics of square waved returns.
Thus the square waved return is defined as 4 f (r(t)) = π
2
10
2
The advantage of the square wave method is that the latter parts of DFA test statistics can be easily analyzed than those of other methods. But there is also a disadvantage, that is, it becomes more difficult to study the front parts for DFA test statistics, since the corresponding front parts gather together in a line. In their latter parts, the SSE Composite Index is very close to SZSE Component Index. And the front parts of IXIC are close to SZSE Component Index, and DJIA is close to S&P 500 index. Finally, we compare the scaling exponents α for the returns, absolute returns, squared returns, cubed returns and square waved returns in Fig. 6. The result shows that the SSE Composite Index is very close to SZSE Component Index and IXIC. DJIA is close to S&P 500 in all cases. And the exponent trend of the returns are close to that of square waved returns. Moreover, we can find that all the exponents α for absolute returns are larger than 0.5, and most exponents α for squared returns are larger than 0.5, see Table 3 and Fig. 6. This implies that
nπr(t) . L
Then, we study DFA test statistics of the square waved returns for sixteen individual energy stocks and five market indices in Fig. 5 and Table 5. In Table 5, the smallest exponent corresponds to Changjiang Securities Company Limited (000783) and the largest value is for SSE Composite Index. Table 5 shows that the exponents for three individual stocks and two indices are smaller than 0.5, which indicates the power-law anticorrelations. The rest are bigger than 0.5, this indicates the longrange power-law correlations. Also, we find that the front parts of all the financial series gather together in a line and their latter parts depart from each other in Fig. 5, this denotes that the method of square waved return results in deviations in DFA test statistics at longer time scales. Table 5.
DFA test statistics of square waved returns.
Symbol
β
α
Symbol
β
α
Symbol
β
000036 600583 600642 000668 000096 600688 000554
−0.61529 −0.66068 −0.71813 −0.59654 −0.69202 −0.74015 −0.76065
0.51042 0.53689 0.56849 0.48891 0.54259 0.59551 0.60193
000819 600110 600387 600157 000637 600098 000783
−0.62929 −0.76325 −0.83942 −0.58993 −0.69351 −0.67448 −0.525
0.52047 0.59799 0.64186 0.49551 0.55791 0.53174 0.43888
600028 000698 SSE SZSE S&P500 IXIC DJIA
−0.61488 −0.71984 −0.88057 −0.85617 −0.56455 −0.6365 −0.61224
α 0.50171 0.56845 0.67945 0.67477 0.45667 0.52711 0.49592
December 15, 2010
3760
16:14
WSPC/S0218-1274
02808
J. Zhang & J. Wang 1 Absolute returns
Squared returns
0.8
0.6
0.4
0.2 Cubed returns
60
36 00
60
00
05
0
Square waved returns
83 06 42 00 06 68 00 00 96 60 06 88 00 05 54 00 08 19 60 01 10 60 03 87 60 01 57 00 06 37 60 00 98 00 07 83 60 00 28 00 06 98 SS E SZ SE S& P5 00 IXI C DJ IA
Returns
Fig. 6. The plot of the scaling exponents α for the returns, absolute returns, squared returns, cubed returns and square waved returns.
there are long-range correlations and persistence in volatility series for absolute returns and for most squared returns.
where σ ˆn2 (q)
3.2. The fractal detrended fluctuation analysis
×
In this subsection, we analyze the long-term memory of five kinds of returns by using Lo’s modified rescaled range statistic [Lo, 1991]. The long memory is measured by the Hurst exponent H, calculated by Lo’s modified rescaled range statistic. For 1/2 < H < 1, the series exhibits the long term persistence, with the maintenance of tendency; for 0 < H < 0.5, the series is the antipersistent, presenting reversion to the mean; and for H = 0.5, the series corresponds to a random walk. By estimating the Hurst exponent H, we wish to find whether the Chinese energy market exhibits the fractal and the persistence properties. We con. . , Xn and let X n sider a sample of series X1 , X2 , . denote the sample mean (1/n) j Xj . Then the modified rescaled range statistic, denoted by Qn , is defined by k 1 max (Xj − X n ) Qn = σ ˆn (q) 1≤k≤n j=1
− min
1≤k≤n
k j=1
(Xj − X n )
n
q
j=1
j=1
1 2 = (Xj − X n )2 + ωj (q) n n n
(Xi − X n )(Xi−j
i=j+1
=σ ˆx2 + 2
q
− X n)
ωj (q)ˆ γj
j=1
and ωj (q) = 1 −
j , q+1
q
