股票市场的复杂性与有效性(英文

摇 摇 doi:10. 3969 / j. issn. 1000-2162. 2013. 01. 001 The complexity and efficiency of stock markets GU Rong鄄bao (School of Finance, Nanjing University of Finance & Economics, Nanjing摇 210046, China) Abstract: In the paper, we proposed the efficiency index and the complexity index for financial markets, and investigated the relationships between the indices of Shanghai and Shenzhen stock markets employing the technique of rolling window. Empirical results showed that there was two-way Granger causality relation between the complexity and efficiency for Shanghai stock market and so does for Shenzhen market, there were two -way Granger causality relations between not only the efficiencies but also the complexities for the two markets. The complexity had an increasing influence on efficiency, while the efficiency had a decreasing influence on complexity. The interaction between the complexity and efficiency of Shanghai market was stronger than that of Shenzhen market, and the interaction between the efficiencies was stronger than that between complexities for the two stock markets. Those suggested that the complexity was the essential characteristic of the stock market and the efficiency was the external performance of the complexity, and so the change of complexity was intrinsic factor of the change of efficiency. Those empirical results not only supported Liu蒺s theoretical result that the financial complexity could improve the efficiency of financial market, but also further illustrated some distinct differences in the degree of interaction between the indices of the two stock markets. Key words: stock market; complexity index; efficiency index; multifractal analysis CLC number: F224摇 摇 摇 摇 Document code: A摇 摇 摇 摇 Article ID:1000-2162(2013)01-0001-07 股票市场的复杂性与有效性 顾荣宝 (南京财经大学摇 金融学院,江苏 南京摇 210046) 摘摇 要:对金融市场提出有效性指标和复杂性指标,利用“滑动窗技术冶研究了上海和深圳股票市场有效性与 复杂性之间的关联.结果显示,上海股票市场的复杂性和有效性之间以及复杂性之间存在双向的 Granger 因 果关系,深圳股票市场亦是如此;两个市场的有效性之间存在双向的 Granger 因果关系,两个市场的复杂性之 间亦是如此;上海股票市场的有效性之间以及复杂性之间相互影响强于深圳股票市场,两个市场有效性之间 的相互影响强于复杂性之间的相互影响.论文的实证结果支持了刘维奇关于金融复杂性可以改进金融市场 效率的理论研究的论断. 关键词:股票市场;复杂性;有效性;多重分形分析 Received date:2012-07-19 Foundation item: Supported by the National Natural Science Foundation of China (70871058, 71071071), the Humanities and Social Science Project of Ministry of Education of China (12YJAZH020, 09YJA7909199), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions ( PAPD), the Project Funded by Jiangsu Modern Service Institute (PMS) and the Science Foundation of Nanjing University of Finance & Economics (A2010017) Author蒺s brief:GU Rong鄄bao(1956—),male, born in Mingguang of Anhui Province,professor of Nanjing University of Finance & Economics, master supervisor, Ph. D. 2013 年 1 月 第 37 卷 第 1 期 安徽大学学报(自然科学版) Journal of Anhui University (Natural Science Edition) January 2013 Vol. 37 No. 1 0摇 Introduction Since the efficiency market hypothesis has been introduced by Fama[1], the theoretical research and empirical test of various financial markets, especially the stock markets, have drawn the concern of numerous experts and scholars. By applying a wide range of nonlinear analytical techniques, much significant progresses has been achieved not only in terms of efficiency of stock markets but also in terms of complex characterization of stock markets, see [2-10]. Recently, Liu[11] has given the qualitative analysis theoretically about the relationship between efficiency and complexity of financial market in China. His research shows that the complexity of the finance market has created the innovative room for the market. This has guaranteed the stability of the finance system, but also improved the efficiency of finance market. In this paper, empirical method is used to investigate Shanghai and Shenzhen stock markets to understand the relationship between efficiency and complexity. In order to make effective analysis, the efficiency index and complexity index will be introduced, and some econometric methods will be used. 1摇 Choice of indices To establish the efficiency index and the complexity index for financial market, we introduce Rescaled Range Analysis and Multifractal Detrended Fluctuation Analysis, and we also give their meanings on finance. 1. 1摇 Efficiency index of market According to Efficiency Market Hypothesis which is introduced by Fama[12], the efficient markets have three levels, including weak鄄form efficiency, semi鄄strong鄄form efficiency, and strong鄄form efficiency. A market is deemed as weak鄄form efficiency if the asset prices can reflect all historical information. A market is semi鄄strong鄄form efficiency if the asset prices can reflect not only all historical information but also all public information. A market is strong鄄form efficiency if the asset prices can reflect not only all historical information and all pubic information but all insider information. If a market is weak鄄form efficiency, all historical information will be included in the current prices and the prices will follow a random walk. Rescaled Range Analysis (R / S) which is proposed by Hurst[13] can be used to test whether a time series follows a random walk. Let {x1,x2,…,xn} be a time series and x - n = 1 / n移 n t = 1 xt denotes the sample mean where n is the time span considered. Then the R / S statistic is given by (R / S) n 以 1 sn [ max 1臆t臆n 移 t k = 1 (xk - x - n) - min1臆t臆n移 t k = 1 (xk - x - n)], (1) where sn is the usual standard deviation estimator sn 以 [ 1 n移 n t = 1 (xt - x - n) 2] 1 / 2 . (2) 摇 摇 Hurst found that the rescaled range, R / S, for many records in time is very well described by the following empirical relation (R / S) n = (an) H, (3) where H is called the Hurst exponent. By performing a linear least鄄squares regression, one finds the slope of the regression which is the estimate of the Hurst exponent H . If H >0. 5, the time series is persistent or long鄄 run memory. If H <0. 5, the time series is anti鄄persistent or mean recurrence. If H = 0. 5, the time series displays random walk behavior. For a financial market, if the Hurst exponent of the asset prices or returns is closer to 0. 5, the market 2 安徽大学学报(自然科学版) 第 37 卷 will be closer to weak鄄form efficiency. This indicates that the Hurst exponent is an appropriate indicator measuring the efficiency of a financial market ( also see Eom et al. [9] ). Thus, we can define the market efficiency as EFF =| H - 0. 5 | . (4) 摇 摇 This means that, the smaller the EFF value is, the higher the efficiency of the market could be. 1. 2摇 Complexity index of market Kantelhardt et al. [14] proposed the Multifractal Detrended Fluctuation Analysis (MF-DFA) which can be used for a global detection of multifractal behavior of a non鄄stationary time series. The MF-DFA procedure consists of five steps as follows: Let {xt,t = 1,…,N} be a time series, where N is the length of the series. Step 1. Determine the profile yk =移 k t = 1 (xt - x - ),k = 1,2,…,N, (5) where x - denotes the averaging over the whole time series. Step 2. Divide the profile {yk} k = 1,…,N into Ns 以 int (N / s) non鄄overlapping segments of equal length s . Since the length N of the series is often not a multiple of the considered time scale s , a short part at the end of the profile may remain. In order not to disregard this part of the series, the same procedure is repeated starting from the opposite end. Thereby, 2Ns segments are obtained altogether. Introduced by Peng et al. [15] we get 10< s < Ns / 5 . Step 3. Calculate the local trend for each of the 2Ns segments by a least鄄square fit of the series. Then determine the variance F2( s,姿) 以 1s 移 s j = 1 [y(姿-1) s+j - P姿( j)] 2, (6) for 姿 = 1,2,…,Ns and F2( s,姿) 以 1s 移 s j = 1 [yN-(姿-Ns) s+j - P姿( j)] 2, (7) for 姿 = Ns + 1,Ns + 2,…,2Ns . Here, P姿( j) is the fitting polynomial with order m in segment 姿 . Step 4. Average over all segments to obtain the q鄄th order fluctuation function Fq( s) = { 1 2Ns 移 2Ns 姿 = 1 [F2( s,姿)]q / 2}1 / q, (8) for any real value q 屹0 and F0( s) = exp { 1 4Ns 移 2Ns 姿 = 1 ln [F2( s,姿)]} . (9) 摇 摇 We repeat steps 2 to 4 several time scales s . It is apparent that Fq( s) will increase with increasing s . Of course, Fq( s) depends on the DFA order m . By construction, Fq( s) is only defined for s 逸 m + 2. Step 5. Determine the scaling behavior of the fluctuation functions by analyzing log鄄log plots Fq( s) versus s for each value of q . For large values of s, as a power鄄law, Fq( s) ~ sh(q), where h ( q) is called the generalized Hurst exponent, which can be obtained by observing the slope of log鄄log plot of Fq( s) versus s through the method of least square. The time series is called to be multifractal if h ( q ) depends on q and monofractal if h ( q) is independent of q . A mutifractal series has a structure that is relatively more complicated than that of a monofractal series. Shi and Ai[16] suggested that generalized Hurst exponent h(q) decreases as q increases. To our regret that this is not true in general ( see, although many stock markets possess this property ( see [7])). So, the multifractality degree proposed by Zunino et al. [7] should be 3第 1 期 顾荣宝:股票市场的复杂性与有效性(英文) modified as 驻h = max q h(q) - min q h(q) . For a financial market, the higher multifractality degree of the asset prices or returns is, the more complicated the market is. This indicates that the multifractality degree is an appropriate indicator measuring the complexity of a financial market. Thus, we can define the complexity index of the market as COM = max q h(q) - min q h(q) . This means that, the larger the COM value is, the stronger the complexity of the market could be. 2摇 Static analysis on efficiency and complexity of stock markets 2. 1摇 Data The data of the study consist of the daily spot prices for Shanghai and Shenzhen stock markets. We analyze the Shanghai stock exchange com posite index for Shanghai and Shenzhen composite index for Shenzhen. All the data used in this paper was taken from the Wind financial database. The data span from Apr. 3, 1990 to Mar. 31,2010, namely 4 656 observations for both Shanghai and Shenzhen markets. Let pt be the price of a stock on day t . The daily price returns, rt is calculated as its logarithmic difference rt = log (pt +1 / pt) . (10) 2. 2摇 Basic characteristic of stock markets Tab. 1 reports the generalized Hurst exponents of Shanghai and Shenzhen stock markets which show that the two stock markets exhibit the multifractal characterization. Tab. 1摇 The generalized Hurst exponents of stock markets q -10 -8 -6 -4 -2 2 4 6 8 10 Hsh(q) 0. 857 0 0. 835 8 0. 804 1 0. 754 2 0. 672 3 0. 535 0 0. 417 7 0. 354 1 0. 319 9 0. 298 7 Hsz(q) 0. 848 3 0. 827 9 0. 797 0 0. 747 1 0. 684 8 0. 620 4 0. 553 3 0. 502 0 0. 468 3 0. 445 8 摇 摇 From Tab. 2, we find that Shanghai stock market is more weak-form efficient than Shenzhen market and the complexity of Shanghai stock market is greater than that of Shenzhen market. Tab. 2摇 The efficiency indices and complexity indices of stock markets Stock market Efficiency index Complexity index Shanghai 0. 035 0 0. 558 3 Shenzhen 0. 120 4 0. 402 5 3摇 Dynamic analysis on complexity and efficiency of stock markets To study the dynamics of efficiency and complexity of Shanghai and Shenzhen stock markets, considering the approach proposed in Cajueiro and Tabak[3], we estimate the time鄄varying generalized Hurst exponents (using MF-DFA) for moving windows of fixed length of 1 008 observations ( approximately 4 years) at a time. The data in the x鄄axis stand for the beginning of the sample used in the estimation of the Hurst exponents. Therefore, for a data Jan鄄90 the Hurst exponents were evaluated for the sample beginning in Jan鄄90 and ending 1 year later (Jan-91) and so forth. We denote by EFFsh, EFFsz, COMsh and COMsz the efficiency index of Shanghai stock market, efficiency index of Shenzhen stock market, complexity index of Shanghai stock market and complexity index of Shenzhen stock market, respectively. 4 安徽大学学报(自然科学版) 第 37 卷 3. 1摇 Correlation test for efficiency and complexity indices Tab. 3 presents the correlations between these variables. As we can see, there is remarkably positive correlation between efficiency and complexity indices of Shanghai stock market, and so is for Shenzhen stock market. While the correlation of Shanghai stock market is stronger than that of Shenzhen, the efficiency of Shanghai stock market is positively correlated remarkably with that of Shenzhen and the complexity of Shanghai stock market is also positively correlated with that of Shenzhen, but the correlation between efficiencies is stronger than that between complexities for the two markets. Tab. 3摇 Correlation matrix EFFsh EFFsz COMsh COMsz EFFsh 1. 000 0 EFFsz 0. 630 2 1. 000 0 COMsh 0. 540 0 0. 676 1 1. 000 0 COMsz 0. 140 9 0. 264 5 0. 400 0 1. 000 0 3. 2摇 Unit root test for series of complexity and efficiency indices Before establishing the VAR model (see Sims[18]), a unit root test is initially completed to examine the properties of variables. Dickey and Fuller[19] (ADF), and Philips and Perron[20] ( PP) tests are the most commonly used methods to test for unit roots. The results are presented in Tab. 4. The common suggestion of the two tests is that all variables are stationary, thus the VAR model will be used to do the following analysis. Tab. 4摇 Unit root test results Intercept EFFsh EFFsz COMsh COMsz Intercept and trend EFFsh EFFsz COMsh COMsz ADF -2. 998 0b -4. 232 8a -5. 553 3a -6. 841 7a -3. 456 24b -4. 353 6a -5. 456 6a -7. 161 9a PP -10. 038a -10. 715a -4. 999 6a -9. 622 1a -18. 275 7a -13. 163a -4. 847 4a -9. 479 5a 摇 摇 Note: the letters a and b represent significance at 1% and 5% levels, respectively. Lag lengths are determined via AIC. 3. 3摇 Granger causality test Granger causality test proposed by Granger[21] is a basic method to analyze the causal relationship between economic variables. Tab. 5 reports the result of Granger causality test with lag length 1 for the four variables. It is seen from Tab. 5 that there is two鄄way Granger causality relation between the complexity and efficiency for Shanghai stock market and so is for Shenzhen stock market. Moreover, it is also seen from Tab. 5 that there are two鄄way Granger causality relations between not only the efficiencies but also the complexities for the two stock markets. Tab. 5摇 Granger causality test Null hypothesis F-statistic P-value Decision COMsh does not Granger cause EFFsh 34. 541 2 0. 000 0 Reject EFFsh does not Granger cause COMsh 15. 639 7 0. 000 0 Reject COMsz does not Granger cause EFFsz 10. 526 2 0. 001 2 Reject EFFsz does not Granger cause COMsz 9. 974 05 0. 001 7 Reject EFFsz does not Granger cause EFFsh 32. 262 4 0. 0000 Reject EFFsh does not Granger cause EFFsz 33. 070 5 0. 000 0 Reject COMsz does not Granger cause COMsh 4. 683 65 0. 030 8 Reject COMsh does not Granger cause COMsz 9. 009 75 0. 002 8 Reject 3. 4摇 Variance decomposition analysis To study how to influence each other between the complexity and efficiency of stock markets, we establish 5第 1 期 顾荣宝:股票市场的复杂性与有效性(英文) a VAR model with four variables, which are EFFsh, EFFsz, COMsh and COMsz. Tab. 6 reports the results of the variance decomposition for the four variables. Tab. 6摇 Variance decomposition results Period EFFsh EFFsz COMsh COMsz EFFsh 1 100. 000 0 0. 000 0 0. 000 0 0. 000 0 2 99. 402 4 0. 000 3 0. 590 0 0. 007 2 5 94. 262 2 4. 022 7 0. 727 0 0. 987 9 10 91. 863 8 6. 386 3 0. 714 8 1. 034 9 20 89. 017 4 8. 699 8 1. 237 9 1. 044 7 EFFsz 1 0. 814 2 99. 185 7 0. 000 0 0. 000 0 2 0. 755 0 99. 197 8 0. 047 1 0. 000 0 5 1. 580 0 97. 506 8 0. 803 1 0. 109 9 10 3. 554 7 93. 846 4 2. 315 3 0. 283 4 20 7. 079 3 87. 334 3 5. 066 6 0. 519 6 COMsh 1 3. 932 6 0. 169 6 95. 897 7 0. 000 0 2 2. 604 5 0. 595 0 96. 792 8 0. 007 5 5 1. 608 3 0. 470 9 96. 861 1 1. 059 4 10 1. 086 1 0. 322 0 96. 503 4 2. 088 3 20 0. 913 2 0. 255 6 94. 907 4 3. 923 6 COMsz 1 0. 102 3 0. 535 2 0. 115 3 99. 247 0 2 0. 243 5 0. 437 6 0. 911 1 98. 407 6 5 0. 354 7 0. 637 6 1. 018 3 97. 989 2 10 0. 302 4 0. 473 5 0. 909 2 98. 314 7 20 0. 417 8 0. 421 5 0. 849 3 98. 311 2 摇 摇 According to Tab. 6, the initial change of EFFsh is all from own contribution and the own contribution reduces gradually with an increasing lag. The contributions of other EFFsz, COMsh and COMsz to EFFsh appear and increase gradually after period 2, and they are stabilized up the level of 8. 6% , 1. 2% , and 1% , respectively. This shows that the efficiency of Shanghai stock market is affected remarkably by the efficiency of Shenzhen stock market and is affected weakly by its own complexity. The contributions of EFFsh, COMsh and COMsz to EFFsz increase gradually and are stabilized about 7% , 5% , and 0. 5% , respectively. This shows that the efficiency of Shenzhen stock market is influenced remarkably by the efficiency and complexity of Shanghai stock market, and the affection caused by efficiency is larger than that by the complexity, but it is almost not affected by its own complexity. COMsh is remarkably affected by EFFsh in the beginning and reduces gradually with an increasing lag. The contribution of EFFsz to COMsh reduces gradually after its maximum at period 2 and the contribution of COMsz to COMsh appears and increases gradually after period 2. This shows that the complexity of Shanghai stock market is sensitive on the change of its own efficiency. With an increasing lag, the contribution of EFFsh to COMsz increases gradually and EFFsz and COMsh are decreasing gradually after their maximum at period 4. Note that contributions of them to COMsz are all below 1% , showing the influence is quite weak that caused by the efficiency and complexity of Shanghai stock market and the efficiency of Shenzhen stock market to the complexity of Shenzhen stock market. 4摇 Conclusion The study on the complexity of financial markets is quite difficult. Financial markets display different complex characteristics from different points of view. Horgan[22] introduced over 45 kinds of definitions about complexity. 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