Linear and Nonlinear Correlations in Order Aggressiveness of Chinese

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II. DATA DESCRIPTION with the same aggressiveness is a signal of the diagonal effect. The diagonal effect may vary from stock to stock. The Chinese stock market is the largest emerging market in the world and became the second largest stock 5 4 market after the USA market since 2009. It contains an 3 A-share market and a much smaller B-share market. A 2 1 ) t company listed on the A-share market can also issue B- ( 0 a -1 shares under certain conditions. We use the order flow -2 data of 32 A-share stocks and 11 B-share stocks traded on -3 -4 the Shenzhen Stock Exchange (SZSE) of China in 2003. -5 0 100 200 300 400 500 600 Our sample stocks belonged to the 40 constituent stocks t of the SZSE component index in 2003. The SZSE is open 5 4 for trading from Monday to Friday except the public hol- 3 idays and other dates as announced by the China Secu- 2 1 ) t rities Regulatory Commission. On each trading day, the ( 0 a -1 market opens at 9:15 and entered the opening call auction -2 -3 till 9:30. The continuous double auction operates from -4 9:30 to 11:30 and 13:00 to 15:00. Only the order flows -5 0 100 200 300 400 500 600 during the continuous double auction are considered in t this work. Orders can be classified into different types according FIG. 1. Example segments of the order aggressiveness time to their aggressiveness [1, 60]. We determine the order series ai(t) for an A-share stock 000720 (a) and a B-share stock 200541 (b). aggressiveness time series ai(t) for stock i as follows. The absolute value of ai(t) is larger if the order t is more aggressive and the sign of a (t) indicates the direction i In total, we have 43 order aggressiveness time series of order t such that a (t) is positive for buy orders and i {a (t): i = 1, 2, ··· , 43} for 43 Chinese stocks. The av- negative for sell orders. i erage length of the time series is 802106, the maximum If t is a partially filled buy order, ai(t) = 5, whose price length is 3151313 (A-share stock 000001), and the mini- is higher than the best ask price and whose quantity is mum length is 67649 (B-share stock 200541). These time greater than the amount of matched orders on the sell series are sufficiently long for DMA analysis and we do limit order book. If t is a partially filled sell order, ai(t)= not need to consider the short series effect or the finite- −5, whose price is lower than the best bid price and size effect [40, 64, 65]. whose quantity is greater than the amount of matched orders on the buy limit order book. These two types of orders are the most aggressive and have the highest immediate price impact [61–63]. III. LINEAR LONG-TERM CORRELATIONS If t is a filled buy order, ai(t) = 4, whose price is not lower than the best ask price and whose quantity is no less than the amount of matched orders on the sell limit A. The detrending moving average analysis order book. If t is a filled sell order, ai(t) = −4, whose price is not higher than the best bid price and whose We adopt the detrending moving average analysis to quantity is no less than the amount of matched orders on investigate the order aggressiveness for possible long- the buy limit order book. These two types of orders are term correlations. The DMA method has been invented also marketable orders and have a power-law immediate for this purpose and extensively studies [23–30]. The pro- price impact [61–63]. cedure of the centred DMA method is briefly described If order t is placed inside the spread such that its price below. is higher than the best bid price and higher than the best Step 1. Consider an order aggressiveness time series ask price, ai(t) = 3 for buy orders and ai(t)= −3 for sell orders. If order t is placed on the same best such that its a(t), t =1, 2,...,N, in which t stands for the t-th order price is equal to the best price on the same side, a (t)=2 and the subscript i for stock i has been dropped. We i construct the sequence of cumulative sums for buy orders and ai(t) = −2 for sell orders. If order t is placed inside the same-side limit order book, ai(t)=1 t for buy orders and ai(t)= −1 for sell orders. Figure 1 shows two segments of the order aggressive- y(t)= [a(j) − hai] , t =1, 2,...,N, (1) j=1 ness time series ai(t) for an A-share stock (Code: 000720) X and a B-share stock (Code: 200541). We choose these two segments to illustrate that the local patterns of ai(t) where N is the length of the time series a(t) for a given may change over time. The appearance of order bunches stock and hai is the sample mean of a(t). 3

Step 2. Calculate the moving average function y(t) in series as a function of the window size s for two A- a moving window of size s, share stock and two B-share stocks. Roughly, we ob- e server power-law relationships. However, there are also ⌈(s−1)(1−θ)⌉ 1 crossovers with s× ∈ (10, 100) such that y(t)= y(t − k), (2) n H1 k=−⌊(s−1)θ⌋ s , s ≤ s× X F2(s) ∼ H2 (8) e (s , s>s× where the operator ⌊x⌋ is the largest integer not greater than x, ⌈x⌉ is the smallest integer not smaller than x, The DMA exponent is larger when the window size s is and θ is the position parameter with the value varying larger than s×. The crossover behavior is ubiquitously re- in the range [0, 1]. Here, θ = 0, θ = 0.5 and θ = 1 ported in diverse systems and its possible origins contain corresponding respectively to the backward, centred and additive noise and linear and non-linear trends [8, 66–77]. forward DMA [46]. We use θ =0.5 in this work. Hence, For comparison, we also show the results for the shuﬄed Eq. (2) becomes time series. Basically, the DMA exponent is close to 0.5, ignoring the small curvature at large scale in Fig. 2(b). ⌈(s−1)/2⌉ 1 104 105 the original series (short range) the original series y(t)= y(t − k), (3) h1=0.6 h1=0.7751 4 shuffled seiries 3 the original series (long range) 10 n 10 h2=0.88816 h=0.49771 k=−⌊(s−1)/2⌋ shuffled seiries X ) h=0.50946 ) 103 s s

( 102 ( 2 2 e 2 Step 3. Detrend the cumulative time series by remov- F F 10 101 101 ing the moving average function y(t) from y(t), and ob- (a) (b) 100 100 tain the residual time series ǫ(t) through the following 100 101 102 103 104 105 100 101 102 103 104 105 s s equation e 104 104 the original series (short range) the original series (short range) h1=0.51816 h1=0.4957 ǫ(t)= y(t) − y(t), (4) 3 the original series (long range) 3 the original series (long range) 10 h2=0.70006 10 h2=0.80977 shuffled seiries shuffled seiries

) h=0.49638 ) h=0.51117 s s

( 102 ( 102 6 6 2 2 where n −⌊(s − 1)/2⌋ i N −⌊e (s − 1)/2⌋. F F Step 4. Divide the residual time series ǫ(t) into Ns 101 101 disjoint segments with the same size s, where N = (c) (d) s 100 100 100 101 102 103 104 105 100 101 102 103 104 105 ⌊N/s − 1⌋. Each segment can be denoted by ǫv such s s that ǫv(t) = ǫ(t +1) for 1 6 t 6 s, where l = (v − 1)s. The root-mean-square function Fv(s) with the segment FIG. 2. (color online) Power-law dependence of the over- size s can be calculated as follows all fluctuation functions F2(s) with respect to the scale s. The straight lines are the best power-law fits to the data. s 1 The lower curves show the results for the shuffled time series, F 2(s)= ǫ2(t). (5) v n v which have been translated vertically for better visibility. (a) t=1 X A-share stock 000539. (b) A-share stock 000720. (c) B-share stock 200024. (d) B-share stock 200625. Step 5. Determine the qth order overall fluctuation function Fq(s) as follows, We observe that all the 43 stocks exhibit the crossover

1 behavior. In order to estimate the two DMA exponents N 2 1 s H1 and H2, we need to determine ﬁrst the crossover scale F (s)= F 2(s) , (6) 2 N v s×. We adopt an objective method for the simultaneous ( s v=1 ) X determination of s×, H1 and H2. We rewrite Eq. (8) as follows Step 6. Determine the power-law relation between the function Fq(s) and the size scale s by varying the values c1 + H1 ln s, s ≤ s× of window size s, which reads ln F2(s)= (9) (c2 + H2 ln s, s>s× H F2(s) ∼ s , (7) Since the two straight lines should intersect at s×, that is, we have where H represents the DMA scaling exponent of the raw time series. c1 + H1 ln s× = c2 + H2 ln s×. (10)

To estimate the three parameters s×, H1 and H2, we minimize the following function B. Estimation of the Hurst indexes 2 O(s×,H1,H2,c2)= [ln F2(sj ) − H1 ln sj − c1] sj ≤s× Figure 2 illustrates the dependence of the overall ﬂuc- P 2 + [ln F2(sj ) − H2 ln sj − c2] . (11) tuation function F2(s) of the order aggressiveness time sj ≥s× P 4

hH1i =0.529 ± 0.026. For B-share stocks, the minimum H1 H2 TABLE I. Estimated Hurst indexes and of the order is H = 0.496 for stock 200088, the maximum value aggressiveness time series in the short and long terms whose 1,min is H1,max = 0.596 for stock 200009, and the average is scaling ranges have a crossover at scale s×. The left panel and the right panel show respectively the results for A-share hH1i = 0.538 ± 0.031. It shows that there is no signiﬁ- stocks and their corresponding B-share stocks (if available). cant correlations in the order aggressiveness time series Stock 000720 is an evident outlier without crossover. The in the short term and there is no evidence diﬀerence be- Omin values are the minimized O values. tween A-shares and B-shares.

A-share stocks B-share stocks We then turn to the long-term scaling behavior char- code H1 H2 Sx O code H1 H2 Sx O acterized by H2. For A-share stocks, the minimum value 000001 0.528 0.926 115.228 0.017 H2,min = 0.747 for stock 000429, the maximum value 000002 0.515 0.909 72.144 0.024 200002 0.522 0.754 30.148 0.017 000009 0.516 0.868 62.545 0.027 is H2,max = 0.926 for stock 000001, and the average is 000012 0.513 0.812 39.783 0.040 200012 0.541 0.713 22.000 0.024 hH2i =0.839 ± 0.041. For B-share stocks, the minimum 000016 0.511 0.800 38.799 0.011 200016 0.596 0.677 16.000 0.027 000021 0.513 0.825 46.482 0.026 value is H2,min = 0.677 for stock 200009, the maximum 000024 0.500 0.806 35.137 0.020 200024 0.518 0.700 16.000 0.040 value is H2,max = 0.810 for stock 200088, and the av- 000027 0.534 0.894 63.898 0.009 erage is hH i = 0.744 ± 0.044. On average, the order 000063 0.548 0.883 50.474 0.012 2 000066 0.514 0.826 42.541 0.020 aggressiveness of A-share stocks has stronger long-term 000088 0.560 0.874 47.563 0.015 correlations than B-share stocks. This can be attributed 000089 0.528 0.857 44.324 0.014 000406 0.522 0.804 39.578 0.019 to the fact that the proportion of institutional traders is 000429 0.481 0.747 24.955 0.019 200429 0.534 0.741 30.150 0.011 higher in the B-share market than the A-share market 000488 0.556 0.784 65.504 0.022 200488 0.507 0.789 38.243 0.026 000539 0.600 0.888 65.402 0.007 200539 0.528 0.805 46.609 0.020 and retailer traders are more likely to herd. 000541 0.550 0.826 48.255 0.005 200541 0.532 0.698 22.598 0.009 000550 0.518 0.819 37.722 0.024 200550 0.590 0.746 21.981 0.012 000581 0.585 0.859 70.329 0.052 200581 0.549 0.753 31.438 0.020 000625 0.531 0.828 53.185 0.018 200625 0.496 0.810 39.217 0.015 C. Determinants of the Hurst indexes 000709 0.538 0.841 51.977 0.013 000720 / 0.775 / / 000778 0.522 0.832 53.877 0.019 We proceed to investigate possible firm-specific charac- 000800 0.562 0.860 76.169 0.006 000825 0.531 0.862 62.382 0.039 ters that might impact the variations of the Hurst indexes 000839 0.518 0.854 62.964 0.015 H1 and H2. The firm-specific characteristics we use in- 000858 0.519 0.838 61.020 0.014 clude share trading volume X1, dollar trading volume in 000898 0.532 0.891 75.349 0.017 000917 0.476 0.775 26.300 0.014 RMB X2, annual turnover rate based on full shares X3, 000932 0.522 0.854 60.034 0.019 annual turnover rate based on tradable shares X4, aver- 000956 0.520 0.819 50.880 0.020 000983 0.530 0.816 43.593 0.010 age daily turnover ratio based on full shares X5, average daily turnover ratio based on tradable shares X6, full shares X7, tradable shares X8, yearly return X9, price earning ratio X10, earnings per share X11, return on eq- where c1 is not a free parameter since it is constrained by other four parameters according to Eq. (10). uity X12, operating profit per share X13, net asset value For each aggressiveness time series of the associated per share X14, and income per share X15. A linear model is set up as follows stock, we determine the three key parameters s×, H1 and H2, which are presented in Table I. The left panel 15 and the right panel show respectively the results for A- H1,2 = X0 + βiXi + ǫ, (12) share stocks and their corresponding B-share stocks (if i=1 available). Note that only part of the A-share stocks X have corresponding B-share stocks. All stocks exhibit a We use the stepwise regression to determine which de- crossover in the scaling behavior except for stock 000720 pendent arguments are statistically significant. (Shandong Luneng Taishan Cable Co. Ltd) which is an The results show that only the annual turnover rate evident outlier. Indeed, stock 000720 was well recognized based on full shares X3 has a significant influence on as a stock dominated by price manipulators and it ex- H1. The estimate of the coefficient is β3 = −2.52 × hibited distinct behaviors in many aspect: Its intertrade 10−4 with the p-value being 0.034. The adjusted R2 is durations do not exhibit a crossover in the DFA scaling 0.083. It means that the Hurst index H1 quantifying the [78], its relative order prices have a different DFA scaling short-term correlation decreases with increasing annual behavior during opening call auction [79], its immedi- turnover rate. However, an increase of 10% in the annual −5 ate price impact does not exhibit nice power-law scaling turnover rate will decrease H1 by only 2.52×10 , which especially for filled buy trades [61], its distribution of is negligible. We thus conclude that the investigated firm- inter-cancellation durations behaves differently [80], and specific characteristics do not influence the short-term so on. correlations in the order aggressiveness time series. If we Let’s consider the short-term scaling behavior charac- use only X6 and X9 to X15 as independent arguments in terized by H1. For A-share stocks, the minimum value the regression, no factors are statistically significant. is H1,min = 0.476 for stock 000917, the maximum value In contrast, the trading volume in RMB X2, the av- is H1,max = 0.600 for stock 000539, and the average is erage daily turnover ratio based on tradable shares X6 5

10 and the income per share X15 have signiﬁcant impact 10 1010 on H quantifying the long-term correlation. The esti- 2 6

) 10 6 ) 10 mates of the coeﬃcients are β2 =0.061, β6 = −0.033 and S S ( ( q q

β15 = −0.0012 with the p-values being 0.000, 0.003 and F 2 F 2 q=10 2 10 q=10 q=5 10 0.041 respectively. The adjusted R is 0.776. Because q=5 q=0 q=0 q=-5 q=-5 the turnover is a better measure than share volume and q=-10 q=-10 10-2 -2 1 2 3 4 5 10 dollar volume for trading activities [81] and the annual 10 10 10 10 10 101 102 103 104 105 turnover rates are not accurate, we keep X6 and remove s s X1 to X5, X7 and X8. We find that only X6 and X15 10 10 1010 are statistically significant. The associated coefficients 8 are β =0.32 and β = −0.0032 with the p-values being 10 108 6 15 6 2 10 106

0.020 and 0.0006. The adjusted R is 0.320. The ﬁrst F 4 F conclusion is that the order aggressiveness has stronger 10 104 2 long-term correlations if the turnover rate of the stock is 10 102 higher reﬂecting that traders exhibit stronger imitations 100 0 1 2 3 4 5 10 10 10 10 10 10 101 102 103 104 105 and herding. The result that β6 = 0.032 is consistent s s with the results that β2 = 0.061 and β6 = −0.033 such 10 that β2 + β6 =0.028 in which all Xi are included in the 10 1010 regression. The second conclusion is that the long-term 6

) 10 6 correlation is stronger for stocks with lower income per ) 10 S S ( ( q share, which reﬂects the irrational trading behavior of q F

2 F Chinese traders who like to speculate low-performance 10 102 stocks. 10-2 -2 1 2 3 4 5 10 10 10 10 10 10 101 102 103 104 105 s s IV. NONLINEAR LONG-RANGE CORRELATIONS FIG. 3. (color online) Power-law dependence of the fluctuation functions Fq (s) with respect to the scale s for q = −10, − In order to check if there are any nonlinear long- q = 5, q = 0, q = 5 and q = 10, using the backward range correlations in the order aggressiveness time series, (top row), centered (middle row) and forward (bottom row) MFDMA methods. The straight lines are the best power-law we perform the multifractal detrending moving average fits to the data. The results have been translated vertically (MFDMA) analysis [46]. The MFDMA analysis is a ex- for better visibility. The left column is for stock 000009 and tension of the DMA approach by generalizing the overall the right column is for stock 000024. fluctuation function F2(s) to Fq(s) of different orders, described as follows. Step 5. Determine the qth order overall fluctuation the curves from the centered MFDFA exhibit evident function Fq(s) as follows, crossovers. 1 According to the standard multifractal formalism, the N q 1 s multifractal scaling exponent τ(q) can be used to char- F (s)= [F (s)]q , (13) q N v acterize the multifractal nature, which reads ( s v=1 ) X where q can take any real value except for q = 0. When τ(q)= qh(q) − Df , (16) q = 0, an application of L’Hôspital’s rule results in where Df is the fractal dimension of the geometric sup- Ns 1 port of the measure [45]. We have Df = 1 for time series. F (s) = exp ln[F (s)] . (14) 0 N v If the mass exponent function τ(q) is a concave function ( s v=1 ) X of q, the measure has multifractal nature. Plots (a) and Step 6. Determine the power-law relation between the (b) of Fig. 4 show the mass exponent functions τ(q) for the two stocks. It is found that the three curves exhibit function Fq(s) and the size scale s, which reads discrepancy and do not overlap for each stock. For stock h(q) 000009, the τ(q) functions are seemingly concave. How- Fq(s) ∼ s . (15) ever, for stock 000024, the τ(q) functions show abnormal Figure 3 illustrate the dependence of the fluctuation curvatures around q = 0. Hence, the nonlinearity of the function Fq(s) with respect to the scale s for two stocks τ(q) function does not ensure that the time series has 000009 and 000024, using the backward, centered and for- a multifractal nature. One needs to investigate if the ward MFDMA methods. For the results from the back- singularity spectrum has a bell-like shape. ward and forward MFDMA methods, we observe nice The singularity strength function α(q) and the mul- power-law scalings without clear crossovers. In contrast, tifractal spectrum f(α) can be obtained numerically 6

10 10 to have richer dynamics than others [83]. 5 5 ) ) q q

( 0 ( 0 τ θ=0 τ θ=0 V. CONCLUSION -5 θ=0.5 -5 θ=0.5 θ=1 θ=1 -10 -10 In this work, we investigated the linear and nonlin- -10-8 -6 -4 -2 0 2 4 6 8 10 -10-8 -6 -4 -2 0 2 4 6 8 10 q q ear long-range correlations in the time series of order aggressiveness of 43 Chinese stocks, using the detrending 1 1 θ=0 θ=0 moving average analysis and the multifractal detrending θ=0.5 0.9 θ=0.5 moving average analysis. The DMA analysis identiﬁed 0.9 θ=1 θ=1 ) )

q q crossovers in the scaling behaviors of overall fluctuations ( ( 0.8 α α 0.8 and the order aggressiveness time series has been found 0.7 to exhibit linear long-term correlations. We found that, 0.7 0.6 the short-term correlation is independent of the key firm- -10-8 -6 -4 -2 0 2 4 6 8 10 -10-8 -6 -4 -2 0 2 4 6 8 10 q q specific characteristics, while the long-term correlation strength increases with daily turnover rate and decreases 1 with income per share. The stronger long-term correla- θ=0 1 0.8 θ=0.5 tions and the associated higher turnover rate are both θ=1 ) ) 0.8 caused by the stronger imitations among traders [1]. α α

( 0.6 ( f f θ=0 0.6 The multifractal detrending moving average analysis 0.4 θ=0.5 shows that the order aggressiveness time series exhibit θ 0.4 =1 nice scaling laws of the fluctuation functions and the 0.2 0.2 0.7 0.8 0.9 1 0.6 0.7 0.8 0.9 1 singularities broadly distributed. It means that these α α time series possess multifractality. However, the behaviour of some stocks (like stock 000024) are different FIG. 4. Estimated mass exponent functions τ(q), singularity strength functions α(q) and singularity spectra f(α) for stock from others (like stock 000009) in the sense that their 000009 (left column) and stock 000024 (right column). singularity spectra are not bell-shaped. It suggests that stock 000024 reveals much more dynamic behaviour with extreme events while stock 000009 is more stable and “quiet” [83]. The asymmetry of the multifractal spec- through the Legendre transform [82] trum towards the right side in Fig. 4(a) is observed in many empirical systems [84], which might implies different underlying mechanisms. However, the the situation is α(q) = dτ(q)/dq . (17) very complicated because the multifractal spectrum can f(q)= qα − τ(q) be right-sided in one market [85] but left-sided in another market [86]. In Fig. 4(c) and (d), we illustrate the singularity strength functions α(q). It is obvious that the shape of the α(q) functions is very different from the conventional curves ACKNOWLEDGMENTS of multifractal measures in which α(q) is a monotonically decreasing function. In Fig. 4(e) and (f), we show the We acknowledge financial supports from the Na- singularity spectra f(α). The f(α) curves seem normal tional Natural Science Foundation of China (71532009, for stock 000009 but abnormal for stock 000024. Overall, 71501072, 71571121 and 71671066) and the Funda- the results indicate that the order aggressiveness time mental Research Funds for the Central Universities series possess multifractal nature and some stocks seem (222201718006).

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