Asymmetric risk transmission effect of cross-listing stocks between mainland and Hong Kong stock markets based on MF-DCCA method

Physica A 526 (2019) 120741

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

Asymmetric risk transmission effect of cross-listing stocks between mainland and Hong Kong stock markets based on MF-DCCA method ∗

Guangxi Cao a,b , , Ling Zhou a a

School of Management Science and Engineering, Nanjing University of Information Science & Technology, Ningliu Road 219, Nanjing, Jiangsu 210044, PR China b Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Binjiang College, Nanjing University of Information Science & Technology, Ningliu Road 219, Nanjing, Jiangsu 210044, PR China

highlights • Apply the MF-ADCCA and DMF-ADCCA methods to A+H cross-listed shares. • Study the asymmetric cross-correlation of the A+H cross-listed markets. • Explores the price conduction direction of the A+H cross-listed share markets.

article

info

Article history: Received 11 December 2018 Received in revised form 10 March 2019 Available online 6 April 2019 Keywords: A+H cross-listing Asymmetric MF-DCCA Long memory Risk conduction

a b s t r a c t With the implementation of ‘‘Shanghai–Hong Kong Stock Connect’’ and ‘‘Shenzhen– Hong Kong Stock Connect,’’ the mainland and Hong Kong stock markets are becoming more closely linked. Based on the A+H cross-listed A-share and H-share market indices, this study employs asymmetric multifractal cross-correlation methods to analyze the asymmetric cross-correlation between the A-share and H-share markets from diverse perspectives of different ups and downs and various conduction directions with 79 sample stocks from January 1 a=2004 to May 26, 2017. Empirical results show that the A+H shares have long memory in different trends, which is stronger in the downward trend of stock price. It indicates that regardless of which market with A+H shares showing a downward trend are on, driving the future on the local market and the corresponding cross-listed market show a downward trend is easier than driving the rising trend. In addition, a bidirectional risk conduction effect exists between A and H shares, and the A-share market has a strong transmission effect on the H-share market. © 2019 Published by Elsevier B.V.

1. Introduction Since 2001, with the gradual opening of policies, the phenomenon of A+H cross-listing in Chinese companies has become increasingly common, and its impact on the mainland securities market has been growing. Cross-listing refers to the behavior of a company listed on two or more stock exchanges (usually in different countries or regions). According to the ‘‘one price law’’, the value of the company’s stock on different stock exchanges should be consistent. However, ∗ Corresponding author at: School of Management Science and Engineering, Nanjing University of Information Science & Technology, Ningliu Road 219, Nanjing, Jiangsu 210044, PR China. E-mail address: [email protected] (G. Cao). https://doi.org/10.1016/j.physa.2019.03.106 0378-4371/© 2019 Published by Elsevier B.V.

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G. Cao and L. Zhou / Physica A 526 (2019) 120741

substantial domestic and foreign literature has found that prices traded on domestic exchanges often has a more common discount than those on overseas exchanges. Price differentials lead to the transmission problem of price fluctuations, which has been confirmed by many scholars. Xu and Fung (2002) [1] studied the information flow patterns of Chinese stocks cross-listed in Hong Kong and the United States and found that the domestic market plays a more important role in the pricing process than foreign markets, which have a dominant right in volatility spillovers. Su (2007) [2] studied the price discovery function of Chinese stocks cross-listed in the United States and Hong Kong and verified that stocks listed in Hong Kong contributed more to the price discovery process. Cheung and Lam (2015) [3] found that casino gaming companies listed in Hong Kong have higher excess returns than those listed in the United States. Agarwal et al. (2007) [4] drew similar conclusions. Their research found that the same stocks traded in London and Hong Kong are asymmetrical. Hong Kong trading mainly determines the price discovery in London, and the impact of London trading on Hong Kong price discovery is extremely small. Hui and Chan (2018) [5] investigated the impact of Shanghai–Hong Kong Stock Connect on the H-share discounts of 12 stocks cross-listed in Shanghai and Hong Kong through linear regression. Results showed that the trading activity in the mainland market has a more significant impact on the AH premium. Hence, investigating the cross-correlation of two stock markets and the risk conduction effect caused by price fluctuations can be helpful to reduce investment risks. At present, regarding A-share and H-share price fluctuation conduction problems, co-integration and Granger causality tests are applied for the analysis of the transfer effect and leading lag relationship between prices. Dong and Wu (2008) [6] utilized a co-integration test and error correction and multivariate GARCH models to study A+H stocks. They uncovered that although the price trends of A-share and H-share are different, they have a long-term stable equilibrium relationship, and the H-share has a significant one-way lead relationship with A-share. Huang and Kuo (2015) [7] employed a trivariate BEKK-GARCH model to explore the relationships among Mainland China, Hong Kong, and Taiwan markets from the perspective of information transmission. They confirmed that the Chinese mainland stock market has a major impact on the Hong Kong and Taiwan markets due to volatility spillovers, playing a lead role in information transmission. Feng and Duan (2016) [8] studied the linkage effect between Shanghai and Hong Kong stock markets through the Granger causality test and GARCH-BEKK model. Results showed that the implementation of the ‘‘Shanghai–Hong Kong Stock Connect Program’’ enhanced the linkage effect of the Shanghai–Hong Kong stock market, and the volatility spillover effect of the Shanghai stock market on Hong Kong was remarkably improved. Using the co-integration and Granger causality tests and the GARCH model, Huo (2017) [9] concluded that the volatility spillover effect from Shanghai to Hong Kong was strengthened, and a weak and unstable cointegration relationship existed between the two. However, financial market data tend to be a thick-tailed distribution rather than typical normal distribution. Hence, Ke (2014) [10] employed a variety of Copula models to analyze the linkage of the A+H stock markets from the cross-listing perspective and indicated that in recent years, the impact of A-share on H-share has been considerably enhanced. Nevertheless, the above methods that are based on the efficient market hypothesis have certain defects because the distribution of financial asset yields exhibit evident autocorrelation and long memory characteristics. On the basis of the foregoing characteristics, behavioral finance, nonlinear dynamics, and fractal theories are applied to financial market research. Peng (1994) [11] proposed the use of detrended fluctuation analysis (DFA) method in the study of the typing structure of DNA molecular chains for the observation of the long-range correlation between nonstationary time series and the phenomenon of removing pseudo correlation. In 2002, basing on the DFA, Kantelhardt (2002) [12] first proposed the multifractal detrended fluctuation analysis (MF-DFA) method to characterize the multifractal characteristics of time series under different time scales [13–16]. Podobnik et al. (2008) [17] also proposed the detrended cross-correlation analysis (DCCA) method, which can be used to measure the long-term correlation of two nonstationary time series. Gvozdanovic et al. (2012) [18] employed DFA and DCCA to analyze the auto-correlations and the cross-correlations in absolute returns of the 30 Dow Jones Industrial Average. To reveal the multifractal characteristics of two cross-correlated nonstationary signals, Zhou (2008) [19] proposed the multifractal detrended cross-correlation analysis (MF-DCCA) method, merging the MF-DFA and the DCCA methods. Furthermore, substantial literature has used developed methods, such as MF-DXA [20], MF-X-DMA [21], MF-X-WT [22], MFCCA [23], MF-DPXA [24], MF-HXA [25], MF-X-PF [26–28], MF-X-WL [29] to conduct empirical research. Among them, the MF-DCCA method is the most widely used, such as in Wang et al. (2010) [30], He and Chen (2011) [31], Ma et al. (2013) [32], and Cao et al. (2012; 2014) [33,34], but these empirical analyses ignored the asymmetric features that exist between financial markets. Therefore, Cao et al. (2014) [35] proposed an asymmetric multifractal detrended cross-correlation analysis method (MF-ADCCA) based on different trends (rise or fall) and studied the asymmetric cross-correlation between Chinese and American stock markets and exchange rate markets; Cao et al. (2017) [36] also proposed an asymmetric MF-DCCA method based on different risk conduction directions (DMF-ADCCA) to study the asymmetric risk conduction relationships of stock markets in China, the United States, India, and Brazil. Although Ruan et al. (2018) [37] and Zhang et al. (2018) [38] employed MF-DFA and MF-DCCA to conduct multifractal analysis of the mainland and Hong Kong markets before and after the Shanghai–Hong Kong Stock Connect, research on the risk conduction relationship between A+H cross-listed stocks using the asymmetric MF-DCCA method is still limited. Therefore, this study pioneers the application of the MF-ADCCA and DMF-ADCCA methods to investigate the asymmetric cross-correlation features of A+H shares and constructs the market index of A+H cross-listed A-share and H-share via the compilation method of the Shanghai Composite Index. On the one hand, from the perspective of different ups and downs, this study uses the MF-ADCCA method to study the asymmetric long memory of the cross-listed A-share market and H-share market under different trends. On the other hand, from the perspective of risk conduction, this study explores the price conduction direction of the cross-listed A-share market and H-share market based on the DMF-ADCCA method. Finally, some suggestions for evading or reducing risks are proposed for investors who invest in A+H shares.

G. Cao and L. Zhou / Physica A 526 (2019) 120741

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2. Methodology 2.1. MF-ADCCA method The MF-ADCCA method is proposed by Cao et al. (2014) [35] on the basis of combining MF-DCCA [19] with asymmetric detrended fluctuation analysis [39] to characterize cross-correlation features in rising or falling trends. Suppose two time series {x(1) (t)} and {x(2) (t)} exist, where t = 1, 2, K , N, and N is the length of the sequence. Step 1: Build the contour sequence y(i) (j) =

j ∑

(x(i) (t) − x(i) ),

j = 1, 2, K , N , i = 1, 2,

(1)

t =1

(i) where x(i) = N1 t =1 x (t). Step 2: The total length N is not always an integer multiple of n, thus, to consider all data, the sequence {x(i) (t)} and the contour sequence {y(i) (t)} (i = 1, 2) should be divided into noncoincident subsequences of the same length n (i) (i) in ascending and descending orders; then, we obtain a total of 2Nn segmentation intervals.Sj = {sj,k , k = 1, K , n} is

∑N

(i)

defined as the jth subinterval sequence of length n, and Yj j = 1, 2, K , 2Nn , k = 1, 2, . . . , n. We can generalize that and (i)

(i) sj,k

= {y(j,ik) , k = 1, K , n} is the jth contour subsequence. Where = x(i) ((j − 1)n + k), y(j,ik) = y(i) ((j − 1)n + k), where j = 1, 2, K , Nn ,

(i)

sj,k = x(i) (N − (j − Ns )n + k), yj,k = y(i) (N − (j − Ns )n + k),

(2)

where j = Ns + 1, K , 2Nn . Referring to Peng et al. (1994) [11], the value of n is usually set at 5 ≤ n ≤ N /4. (i) (i) (i) (i) Step 3: Calculate the local least squares fit model LS (i) (k) = aS + bS k and LY (i) (k) = aY + bY k of the corresponding (i)

sequence Sj

j

j

(i)

j

j

j

j

and Yj , representing the linear trend, where k is the horizontal coordinate, i = 1, 2. The slope bS (i) is used (i)

to distinguish the trend of Sj

j

(i)

up or down, and LY (i) (k) is used to eliminate the trend of Yj . Then, define the following j

fluctuation function: n ⏐ ⏐ ⏐ 1 ∑ ⏐ (1) ⏐y − L (1) (k)⏐ · ⏐y(2) − L (2) (k)⏐ , j,k j,k ⏐ ⏐ ⏐ ⏐ Y Y j j n

⏐

⏐ ⏐

⏐

Fj (n) =

(3)

k=1

where j = 1, 2, K , 2Nn in each segmentation interval. Step 4: Calculate the average fluctuation function when the time series x(i) has different linear trends, and bS (i) > 0 (i)

j

(or bS (i) < 0) indicates that the sequence x(i) has an upward (or downward) trend on the sequence Sj , then, the q-order j

average fluctuation function is as follows:

⎛ +

Fq (n) = ⎝

⎛ Fq− (n) = ⎝

2Nn sgn(b (1) ) + 1 Sj 1 ∑

M+

2

j=1

⎞1/q [Fj (n)]

2Nn −[sgn(b (1) ) − 1] Sj 1 ∑

M−

∑2N

2

j=1 sgn(b (1) )+1 S

q/2

,

⎠

(4)

⎞1/q [Fj (n)]q/2 ⎠

∑2N

,

(5)

−[sgn(b (1) )−1] S

j j n n where M + = and M − = respectively stand for the number of samples of the j=1 j=1 2 2 subsequence in the rising and falling trends. For all j = 1, 2, K , 2Nn , M + + M − = 2Nn if bS (1) ̸ = 0. j

Similarly, the average volatility function of the traditional MF-DCCA method proposed by Zhou (2008) [19] is

⎞1/q 2Nn ∑ 1 [Fj (n)]q/2 ⎠ . Fq (n) = ⎝ ⎛

2Nn

(6)

j=1

If the power law cross-correlation exists, the scale relationship satisfies the following conditions: +

−

Fq (n) ∼ nH12 (q) ; Fq+ (n) ∼ nH12 (q) ; Fq− (n) ∼ nH12 (q) ,

(7)

+ − where H12 (q), H12 (q), and H12 (q) respectively represent the overall, rising, and falling scale indices of the sequence {x(1) (t)} and are determined by the analysis of the double logarithmic plots of Fq (n), Fq+ (n), and Fq− (n) with n; thus, Eq. (7) has the following forms:

log Fq (n) = H12 (q) log(n) + log A1 ,

(8)

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G. Cao and L. Zhou / Physica A 526 (2019) 120741 + (q) log(n) + log A2 , log Fq+ (n) = H12

(9)

− (q) log(n) + log A3 . log Fq− (n) = H12

(10)

H12 (q) is a generalized cross-correlation scale index representing the power-law relations of two time-dependent sequences. If H12 (2) > 0.5, the correlation between the two sequences has long memory; if H12 (2) < 0.5, an antipersistent feature exists; if H12 (2) = 0.5, the two sequences are irrelevant or have a short-term correlation at most. + − If H12 (q) = H12 (q), the cross-correlation between the two sequences is symmetrical; otherwise, it is asymmetrical. For the measurement of the asymmetric degree of cross-correlation, we have the following definitions: − + (q). (q) − H12 ∆H12 (q) = H12

(11)

For a constant q, the larger the absolute value of ∆H12 (q), the stronger the asymmetry. If ∆H12 (q) > 0 (∆H12 (q) < 0), the persistence of cross-correlation under the rising (falling) trend of the sequence {x(1) (t)} is stronger than that under the falling (rising) trend. If ∆H12 (q) does not significantly deviate from 0, the cross-correlation of the sequence under different trends is symmetrical. 2.2. DMF-ADCCA method The DMF-ADCCA method is briefly described as follows [36]: construct two time series, {x(1) (t)} and {x(2) (t)}; with the same length N and sequence {x(2) (t)} lag ∆t, the contour sequences can be recorded as y(1) (m) =

m ∑

(x(1) (t) − x(1) ),

(12)

(x(2) (t + ∆t ) − x(2) (t + ∆t)),

(13)

t =1 m

y(2) (m) =

∑ t =1

where m = 1, 2, . . . , N − ∆t. The remaining steps are the same as Steps 2 to 5 in the literature [40]. So that the effect of {x(1) (t)} on {x(2) (t)} can be explored, but the method ignores the autocorrelation effect of {x(2) (t)}, and it may not be effectively tested because of the existence of autocorrelation. Therefore, we add a noise during construction y(2) (m), which weakens the influence of autocorrelation on the sequence, and refer to Granger causality test [41–43]. We record Sequence 1, which lags ∆t1 as {x(1) (t + ∆t1 )}, and Sequence 2, which lags ∆t2 as {x(2) (t + ∆t2 )} (∆t1 ̸ =∆t2 ). Then, we can control the influence of the noise on the {x(2) (t)} by changing ∆t1 . The new contour sequences are as follows: y(1) (m) =

m ∑

(x(1) (t) − x(1) ),

(14)

(x(2) (t + ∆t2 ) − x(2) (t + ∆t2 ) + α x(1) (t + ∆t1 ) − α x(1) (t + ∆t1 )),

(15)

t =1 m

y(2) (m) =

∑ t =1

where ∆t1 ≥ ∆t2 , m = N − ∆t1 ; ∆t1 < ∆t2 , m = N − ∆t2 . To keep the overall characteristic of {x(2) (t)}, we introduce parameter α before {x(1) (t + ∆t1 )}, and the value of α must be small enough. Taking the same approach in the literature [36],

⎧ ⎪ ⎪ ⎪ ⎨α = ⎪ ⎪ ⎪ ⎩α =

∑m

(2)

t =1 (x

(N − ∆t1 )

∑m

t =1 (x (2)

(1) (t

+ ∆t))2

(t + ∆t))2

t =1 (x

(N − ∆t2 )

(t + ∆t))2

∑m ∑m

t =1 (x

(1) (t

+ ∆t))2

∆t1 ≥ ∆t2 , m = N − ∆t1 (16)

∆t1 ≤ ∆t2 , m = N − ∆t2 .

The value of α is small, therefore, y(2) (m) emphasizes the influence of autocorrelation caused by {x(2) (t)}. The remaining steps are the same as Steps 2–5 in the previous section. When {x(1) (t)} lags, the result shows the effect of {x(1) (t)} on {x(2) (t)}. Similarly, when {x(1) (t)} lags, we check the influence of {x(2) (t)} on {x(1) (t)}. 3. Multifractal detrended cross-correlation analysis 3.1. Sample selection and descriptive statistical analysis As of May 26, 2017, a total of 95 A+H shares in China are cross-listing simultaneously in the mainland and Hong Kong markets. We select stocks that have been listed for three years in both places (listed before May 26, 2014) as the research object. After eliminating four ST stocks, 79 sample stocks satisfy the conditions with the sample interval from January 1, 2004 to May 26, 2017, with 3161 data in total. To facilitate the research, we regard the A+H cross-listed A-share and

G. Cao and L. Zhou / Physica A 526 (2019) 120741

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Fig. 1. Trend chart of A-share and H-share index closing price sequences over time. Table 1 Descriptive statistical properties of the logarithmic yield. rA rH

Mean

Max

Min

S.D

Ske

Kur

J–B

1.30E−04 6.02E−04

0.121387 0.330082

−0.098386 −0.33943

0.017488 0.031478

−0.066906 −0.0297564

7.699598 22.48093

2911.297*** 50 030.83***

Note: ‘‘max,’’ ‘‘min,’’ ‘‘SD,’’ ‘‘ske,’’ and ‘‘kur’’ respectively represent maximum, minimum, standard deviation, skewness, and kurtosis; J–B represents the Jarque–Bera statistic, which tests the hypothesis of normal distribution. ***Indicates that the statistic value is significant at the 1% significance level.

H-share as two separate stock pools and compile the corresponding stock index. This study adopts the index method of the Shanghai Composite Index: capitalization weighting. The formula is as follows: Daily index =

Total market value of reporting stocks Base period

× Base period index,

(17)

where Total market value = Σ (Stock price × Issuance of shares) and the base period index are the base point. According to the new Shanghai Composite Index, we take 1000 points as the base point; the base period is set to January 1, 2014. Looking back at the news, the A-share weakened in 2004, whereas the H-share strengthened, visually indicating the asymmetry of the AH stock market. These data are from Tongdaxin Securities Trading Software. As shown in Fig. 1, the data fluctuation range of the Hong Kong stock market is considerably larger than that of A-share because the A-share market has a price limit, whereas the Hong Kong stock market does not, and the magnitude of the data is also different. Considering that the two market currency units are different, this study uses the daily logarithmic yield to describe the volatility correlation between the two indices: rt = log(Pt ) − log(Pt −1 ), where Pt is the closing price of the index at time t. Table 1 shows that the skewness of the two yield series deviates from 0, and the kurtosis deviates significantly from 3, which means that the sequence is non-normal. In addition, the J–B statistic is significant at the 1% significance level, indicating that the two sequences have typical long memory characteristics. 3.2. Cross-correlation analysis In this section, we first employ the cross-correlation test proposed by Podobnik et al. [44] to quantify the crosscorrelation between the mainland market and Hong Kong market. For two time series {x(1) (t)} and {x(2) (t)} sharing the same length N, the cross-correlation statistic Qcc (m) is defined as follow: Qcc (m) = N 2

m ∑ Ct2 , N −t

(18)

t =1

where the cross-correlation function Ct is defined as: (1) (2)

∑N

k=t +1

Ct = √

∑N ( k=1

(1)

Xk

Xk Xk−t

)2 ∑

N k=1

(

(2)

Xk

)2 .

(19)

Qcc (m) is approximately χ 2 (m) distributed with m degrees of freedom. If the cross-correlation statistic exceeds the critical value of χ 2 (m) distribution, then the cross-correlations are significant. Fig. 2 plots the estimated values of crosscorrelation statistics for the returns of the A-share index and H-share index. As we can see, the Qcc (m) values are

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G. Cao and L. Zhou / Physica A 526 (2019) 120741

Fig. 2. Q statistics of the cross-correlation test.

Fig. 3. log Fn (q) with the log(n) trend graph of A-share and H-share return series.

larger or close to the critical value of χ 2 (m) at 5% significance level, suggesting that relatively significant long-range cross-correlations exists between the mainland and Hong Kong markets.

3.3. Multifractal features analysis

On the basis of confirming the cross-correlation between A-share and H-share, we continue to quantitatively analyze the relevant index of the yield series. Fig. 3 shows the logarithmic plots of log Fq (n) as a function of log(n) for the yield series of A-share and H-share for q = −10, −8, . . ., 8, 10 corresponding to the curve from the bottom to the top. And the polynomial order k is taken as 3. When k = 1, 2, 4, 5, the empirical analysis results are similar, so this process is ignored. It can be seen from Fig. 3 that for different q, each curve is linear, which illustrates that there is a powerlaw relationship between the A-share index and the H-share index yield series, which further illustrates the power-law relationship between the mainland stock market and Hong Kong stock market. Additionally, we also employ another method proposed by Podobnik et al. [45] to affirm our results. The DCCA cross2 correlation coefficient (ρDCCA )is defined as the ratio between the detrended covariance function FDCCA and two detrended variance functions FDFA expressed as follows:

ρDCCA =

2 FDCCA (n)

FDFA1 (n)FDFA2 (n)

.

(20)

The values of ρDCCA ranges from −1 to 1. For, 0 < ρDCCA ≤ 1 the two series are cross-correlated. If the value equals 0, there is no cross-correlation. For −1 ≤ ρDCCA < 0, an anti-cross-correlation exists. The calculation results are shown in Fig. 4 All the critical value for the DCCA cross-correlation coefficient for each window size n are larger than 0, which confirms the results of the Qcc (m) test’’.

G. Cao and L. Zhou / Physica A 526 (2019) 120741

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Fig. 4. The DCCA cross-correlation coefficient ρDCCA versus window size n (k = 1).

Fig. 5. log F2 (n), log F + 2 (n), and log F − 2 (n) with the log(n) trend graph of the yield sequence rA . Table 2 Scale indicators under different ups and downs. Sequences under different trends

H12 (2)

H + 12 (2)

H − 12 (2)

△H12 (2)

| △H12 (2) |

r A ∼r H rH ∼rA

0.668658 0.668658

0.586316 0.579580

0.697985 0.720180

−0.111669 −0.140601

0.111669 0.140601

4. Asymmetric multifractal detrended cross-correlation analysis The following is an empirical analysis of the A+H cross-listed A-share and H-share via the MF-ADCCA and DMF-ADCCA methods, respectively, to explore the asymmetry under different ups and downs and the asymmetric risk conductance of different conduction directions of the A+H stock market cross-correlation. 4.1. Asymmetric correlation analysis under different ups and downs Figs. 2 and 3 show the logarithmic plots of log F2 (n), log F + 2 (n), and log F − 2 (n) as a function of log(n) for the yield series of A-share and H-share under different trends. The overall, rising, and falling trends are indicated in the legend. As can be seen from Figs. 5 and 6, with log(n) equal to 2.4 (i.e., n ≈11), when the time scale n is less than 11, no obvious asymmetry can be observed in the cross-correlation between the A+H cross-listed stocks; on the contrary, when n is greater than 11, the asymmetry feature is evident. Table 2 lists the scale indices H12 (2), H + 12 (2), and H − 12 (2) and asymmetry indicator ∆H12 (q) for the overall, ascending, and descending trends under different ups and downs, where rA ∼rH represents the scale index of the cross-correlation of A-share in different trends, and rH ∼rA represents the scale index of the cross-correlation of H-share in different trends in the A+H cross-listing. Table 2 shows that H12 (2) is greater than 0.5, indicating that the cross-correlation of the A-share and H-share yield sequences in the A+H cross-listing has long memory. Moreover, H + 12 (2) and H − 12 (2) are greater than 0.5, implying that the cross-correlation of the two has a long memory characteristic regardless if under the rising or falling trend. Nevertheless, the cross-correlation is asymmetric because H + 12 (2) is not equal to H − 12 (2). Regardless of which trend the

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G. Cao and L. Zhou / Physica A 526 (2019) 120741

Fig. 6. log F2 (n), log F + 2 (n), and log F − 2 (n) with the log(n) trend graph of the yield sequence rH .

Fig. 7. Hurst index for different q variations under different trends: (a) Hurst index with q changes of A-share under the overall, rising, and falling trends; (b) Hurst index with q changes of H-share under the overall, rising, and falling trends.

A+H cross-listed A-share or H-share is under, the values of H − 12 (2) are both greater than those of H + 12 (2). It indicates that no matter in which market the cross-correlation persistence of A+H cross-listed stocks where price falls is higher than that when the price increases. That is, in any market where A+H shares show a downward trend, driving a downward trend in the local market and the corresponding cross-listed market is easier in the future than driving the rising trend. In the real market, it usually shows that when the bear market comes, the inertia caused by falling stock prices is often greater than that caused by falling stock prices in the bull market. In addition, when the H-share in the A+H cross-listing undergoes different trends, |∆H12 (2)| is larger than |∆H12 (2)| when the A-share undergoes different trends, explaining that the cross-correlation asymmetry of A+H shares is stronger when they have different trends in the Hong Kong market. Fig. 7 displays the variation of the Hurst index with q under different trends in the A-share or H-share market. As shown in the graph, H12 (2), H + 12 (2), and H − 12 (2) are not constants with the change of the scale q. Although there is a slight increase, they generally show a downward trend. It indicates that the asymmetric cross-correlation has multiple fractal features whether the mainland or Hong Kong market is in the uptrend or downtrend, which is consistent with the results of Refs. [37,38]. Regardless of the value of q, the Hurst index under the falling trend of A-share and H-share are both greater than the Hurst index under the rising trend. This indicates that the stock price in the down phase is more inertial, causing a negative impact, which leads to the corresponding stock market to fall. We calculate the asymmetric degree |∆H12 (2)| under different scales q for further observation. Fig. 8 reports that when A+H shares have different trends in the mainland market, the value of |∆H12 (2)| when q > 0 is greater than |∆H12 (2)| when q < 0, suggesting that the asymmetric degree of cross-correlation caused by large fluctuations is stronger than that caused by small fluctuations. On the contrary, when A+H shares have different trends in the Hong Kong market, the value of |∆H12 (2)| when q > 0 is less than |∆H12 (2)| when q < 0, indicating the asymmetric degree of cross-correlation brought by small fluctuations is stronger than that caused by large fluctuations.

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Fig. 8. (a) |∆H12 (2)| of A-share for different q variations under different trends; (b) |∆H12 (2)| of H-share for different q variations under different trends. Table 3 Multifractal degrees of the original, shuffled, and surrogated series. rA

rH

Original

Shuffled

Surrogated

Original

Shuffled

Surrogated

Overall

Hmax Hmin ∆H

0.7045 0.6121 0.0924

0.6188 0.4518 0.1670

0.5833 0.5459 0.0374

0.7045 0.6121 0.0924

0.6188 0.4518 0.1670

0.5833 0.5459 0.0374

Rising

Hmax Hmin ∆H

0.7281 0.4692 0.2589

0.6244 0.4390 0.1854

0.5917 0.5844 0.0073

0.6715 0.4649 0.2066

0.6222 0.4069 0.2153

0.5884 0.5861 0.0023

Falling

Hmax Hmin ∆H

0.7913 0.6320 0.1593

0.6226 0.4523 0.1703

0.6245 0.5103 0.1142

0.8046 0.6401 0.1645

0.6134 0.4719 0.1415

0.6285 0.5065 0.1220

It is generally recognized that the origins of the multifractality are two factors, namely fat-tailed probability distribution and long-range correlations of large and small fluctuations [12]. By comparing the multifractal degrees between the original and the surrogated series, we can quantify the contribution of fat-tail. The surrogated series can be created by the phase randomization. And the classic method of quantifying the contribution of long-range correlations is comparing the multifractal degrees between the original series and randomly shuffled series. Therefore, we calculate the scale index of the original shuffled, and surrogated series, employ the following measure [33,46] to quantify the degree of multifractality:

∆H = Hmax − Hmin .

(21)

The greater the ∆H is, the stronger the degree of multifractality becomes. For convenience of description, we use ∆Horig , ∆Hshuf , and ∆Hsurr to denote ∆H for the original series, shuffled series, and surrogate series, respectively. Table 3 provides the multifractal degrees of the original, shuffled, and surrogated series for rA and rH . When A-share or H-share are in an overall trend, ∆Hsurr is smaller than ∆Horig , it indicates that the multifractal feature of the asymmetric cross-correlation is caused by the fat-tail distribution. Also, we can conclude the same conclusion when A-share is in a downward trend or H-share is in the rising trend. It can also be seen from Table 3, when the stock market is going up for A-share, or is going down for H-share, the values of ∆Hshuf and ∆Hsurr are smaller than those of ∆Horig , ∆Hsurr is smaller than ∆Hshuf . This result means that although the fat-tailed distribution and long-range correlations are both the causes of the multifractality, the multifractal feature of the asymmetric cross-correlation is more attributed to the fat-tailed distribution. 4.2. Asymmetric risk conduction analysis The influence of time lag on the cross-correlation of A+H cross-listed A-share and H-share and the risk conduction characteristics between the two are analyzed through the asymmetric MF-DCCA method based on risk conduction direction. After contour sequence y(2) (m) is constructed according to Eq. (15), if the two {x(1) (t)} and {x(2) (t)} are completely linear, ∆t1 equals ∆t2 is meaningless because the autocorrelation of {x(2) (t)} cannot be weakened. However, in the analysis of Section 4.1, the cross-correlation between A-share and H-share is nonlinear. Thus, for convenience, lag time ∆t1 is made

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Fig. 9. H∆rA∼rH (2) and HrA∼∆rH (2) under different lag time variations.

equal to ∆t2 , and the value is taken from 1 to 50 (about a two-month trading day), which means that the time lags for one day to two months. When q = 2, the former of rA ∼rH lags and the latter remains unchanged, scilicet the A-share yield series lags and the H-share remains unchanged, the long memory index Hurst is recorded as H∆rA∼rH (2). On the contrary, when the H-share yield series lags, and the A-share unchanged, recording the Hurst as HrA∼∆rH (2). Fig. 9 shows the trends of H∆rA∼rH (2) and HrA∼∆rH (2) at different lag time variations. Fig. 9 shows that H∆rA∼rH (2) and HrA∼∆rH (2) are greater than 0.5, indicating that in the A+H cross-listed stocks, both have long memory regardless of which market lags. If one A+H cross-listed stock falls in the mainland market ∆t1 days later due to positive sustainability, investors assume that the stock in the Hong Kong market will also fall ∆t1 days later, thereby selling substantial stocks in the Hong Kong market, which causes the price of the H-share to fall, reflecting the conductivity of the stock, and vice versa. This condition indicates that the risk transmission effect of A+H shares between the mainland and Hong Kong markets is bidirectional. The empirical results show that the bidirectional risk conduction effect lasts for 17 days. When ∆t1 (∆t2 ) is at 17–25 days, H∆rA∼rH (2) and HrA∼∆rH (2) have almost no fluctuations, implying that the investors’ reaction to the stock in another market is indistinguishable when A+H stocks lag at 17 to 25 days in a certain market. Besides, with the extension of lag time ∆t1 (∆t2 ), H∆rA∼rH (2) and HrA∼∆rH (2) generally show a fluctuating upward trend, suggesting that the long memory of A+H cross-listed stocks in two markets will also become stronger over time; that is, the longer a stock lags in the mainland or Hong Kong market, the more investors react to it in another market at the same time in the future. This situation makes the stock show similar volatility in the market. Fig. 9 shows that when ∆t1 (∆t2 ) is at 17–25 days, H∆rA∼rH (2) and HrA∼∆rH (2) have almost no fluctuations, implying that the investors’ reaction to the stock in another market is indistinguishable when A+H stocks lag at 17 to 25 days in a certain market. We provide the following indicators to study the strengths and weaknesses of the two-way risk conduction effect of the stock market: H ∆T (2) = H∆rA∼rH (2) − HrA∼∆rH (2),

(22)

where ∆T = ∆t1 = ∆t2 = 1, 2, 3, K , 50. H ∆T (2) < 0 indicates that the risk conduction effect of the mainland market on the Hong Kong market is stronger than that of the other way around; otherwise, we conclude the opposite. Fig. 10 shows H ∆T (2) under different lag time variations. The values of H ∆T (2) are all less than zero, indicating that the risk conduction effect of the mainland market on the Hong Kong market is stronger than that of the other way around. This result may be due to A+H listed companies being mainland enterprises (most of them are large state-owned enterprises). Although the Hong Kong market is as mature as many developed financial markets, the mainland market is more sensitive and reacts more quickly to all information, such as the fundamentals of the company, compared with the Hong Kong market; thus, price fluctuations in the mainland are transmitted to the Hong Kong market. Furthermore, The trend of the Hong Kong market is increasingly affected by domestic policies and fluctuations in the domestic stock market. All of the above can explain why the risk transmission effect of A+H shares in the mainland market is stronger than in the Hong Kong market. 5. Conclusions and policy recommendations We used the asymmetric MF-DCCA and asymmetric MF-DCCA methods based on risk conduction direction to analyze empirically the cross-correlation and risk conduction effect characteristics of A+H cross-listed stocks in the mainland and Hong Kong markets. The following conclusions are obtained: (1) The A+H cross-listed stocks have a long memory of the yield series in the mainland and Hong Kong markets. Regardless of different trends in A-share or H-share, cross-correlation has long memory characteristics and multifractal

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Fig. 10. H ∆T (2) under different lag time variations.

features. Besides, the multifractality is more attributed to the fat-tailed distribution. The cross-correlation of the A+H shares yield series in the mainland and Hong Kong markets is asymmetric, and the cross-correlation becomes increasingly asymmetric when different trends exist in the Hong Kong market. When A+H shares have different trends in the mainland market, the asymmetric degree of cross-correlation caused by large fluctuations is stronger than that caused by small fluctuations. By contrast, when different trends exist in the Hong Kong market, the asymmetric degree of cross-correlation caused by small fluctuations is stronger than that caused by large ones. Investors who invest in A+H cross-listed stocks can timely adjust their investment strategies in the local market according to the stock’s fluctuations in the corresponding cross-market. If the stock has a downward trend in the corresponding cross market, investors need to sell the stock in time to avoid losses. Investors should accurately analyze their financial situation and risk preferences, understand the information in all aspects, and establish appropriate profit points or stop-loss points to avoid the herd effect. Furthermore, investors are not advised to invest in the same stock in Hong Kong and the mainland market simultaneously because risks cannot be well dispersed. (2) A+H cross-listed stocks have a long memory when they lag in the mainland or Hong Kong market and continue to strengthen as the lag period increases. Moreover, the transmission effect of risk is bidirectional; the fluctuation crosscorrelation between A+H stocks does not change when a lag of 17 to 25 days occurs in a certain market, demonstrating that the risk conduction effect at this time is not obvious, and A-share and H-share have almost no influence on each other. (3) The risk conduction effect of A+H shares in the mainland market is stronger than that in the Hong Kong market. The impact of the Hong Kong market on the A+H cross-listed stocks when the mainland market lags is weaker than the other way around. We suggest that the investors who invest in H-share should pay more attention to the fundamentals of the company and the dynamics of stock prices in the mainland market, so that they can take actions in a timely manner to adjust their investment strategy in the Hong Kong market. Acknowledgments We thank for the financial support from National Natural Science Foundation of China (No. 71371100, 71271118), and the Humanities and Social Sciences Fund sponsored by the Ministry of Education of the People’s Republic of China (No. 13YJCZH007). This work also sponsored by the Program for Innovative Research Team of Shanghai University of Finance and Economics, Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, Project Funded by the Flagship Major Development of Jiangsu Higher Education Institutions (2015) and the six talent peaks project in Jiangsu Province (2016). References [1] X.Q. Xu, H.G. Fung, Information flows across markets: evidence from china–backed stocks dual–listed in hong kong and new york, Financ. Rev. 37 (2002) 563–588. [2] Q. Su, T.L. Chong, Determining the contributions to price discovery for chinese cross-listed stocks, Pac.-Basin Finance J. 15 (2007) 140–153. [3] M.Y. Cheung, D. Lam, Comparing the price of sin: Abnormal returns of cross-listed casino gaming stocks in the hong kong and us markets, Int. J. Hosp. Manage. 45 (2015) 73–76. [4] S. Agarwal, C.L. Liu, S.G. Rhee, Where does price discovery occur for stocks traded in multiple markets? evidence from hong kong and london, J. Int. Money Finance 26 (2007) 46–63. [5] C.M. Hui, K.K. Chan, Does the shanghai–hong kong stock connect significantly affect the A-H premium of the stocks?, Physica A 492 (2018) 207–214. [6] X.L. Dong, R.S. Wu, Stock cross-listing and price discovery——empirical evidence from China’s A+H stocks, J. Appl. Stat. Manage. 27 (2008) 1080–1088, (in Chinese).

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