Podobnik et al. (2009) [15] proposed a new cross-correlation test to test whether the cross-correlation of two sequences is significant. Suppose { x i , i = 1 , 2 , ⋯ , N } and { y i , i = 1 , 2 , ⋯ , N } are two time series of the same length N. Formulas are defined as follows:

C i = ∑ k = i + 1 N x k y k − i ∑ i N x k 2 ∑ i N y k 2 (1)

Q c c ( m ) = N 2 ∑ i = 1 m C i 2 N − i . (2)

The cross-correlation test statistic Q c c ( m ) approximately obeys χ 2 ( m ) distribution, The interaction correlation between the two sequences is significant if the interaction correlation test statistic exceeds the critical value of the chi-square distribution.

According to Equation (1), the calculation principle of lag correlation coefficient C_i and horizontal correlation coefficient is the same, it can only measure the linear correlation of stationary sequences. In order to overcome the above shortcomings, Zebende (2011) proposed a DCCA interaction correlation coefficient method [16]. This method can effectively measure the correlation between stationary and non-stationary sequences. The cross correlation coefficient is defined as the ratio of the detrend covariance function F D C C A 2 to the detrend variance function, i.e., Equation (3).

ρ D C C A ( s ) = F D C C A 2 ( s ) F D F A { x ( i ) } ( s ) F D F A { y ( i ) } ( s ) . (3)

Given two time series { x ( i ) } and { y ( i ) } , i = 1 , 2 , ⋯ , N , N are the lengths of this time series, the MF-DCCA method can be described as follows:

Step 1: Generate a cumulative dispersion sequence { X ( i ) } and { Y ( i ) } :

X ( i ) = ∑ t = 1 i ( x ( t ) − x ¯ ) ,       Y ( i ) = ∑ t = 1 i ( y ( t ) − y ¯ ) ,   i = 1 , 2 , ⋯ , N . (4)

where x ¯ = 1 N ∑ i = 1 N x ( i ) , y ¯ = 1 N ∑ i = 1 N y ( i ) .

Step 2: Divide new sequences into N_s intervals, each of which is continuous and non-overlapping with length s. Since N is not necessarily an integer multiple of s, in order to make full use of the remaining data of the tail, the same operation is repeatedly performed from the end, thereby obtaining the 2N_s intervals. We set the window s range to 10 < s < N/4. This paper uses a total of 15 years of data from 2002 to 2017, where N is the sample size. Considering the short-term behavior and the need for a suitable number of samples per interval to calculate the covariance function, we set the lower limit of s to 10; In addition, considering that N/4 means about 45 months, the long-term relationship between the two sequences can be sufficiently included, so we set the upper limit of s to N/4. In fact, there is no significant difference in results, regardless of whether the upper limit is larger or the lower limit is smaller.

Step 3: Fit the local trend function in each subinterval v, v = 1 , 2 , ⋯ , 2 N s by least squares method and get the fitted polynomial X ˜ v ( i ) , Y ˜ v ( i ) . Then, the detrend covariance function can be calculated. For each subinterval v, v = 1 , ⋯ , N s , the covariance function is calculated as follows:

F 2 ( s , v ) = 1 s ∑ i = 1 s | X ( ( v − 1 ) s + i ) − X ˜ v ( i ) | ⋅ | Y ( ( v − 1 ) s + i ) − Y ˜ v ( i ) | . (5)

and for each subinterval v, v = N s + 1 , ⋯ , 2 N s , the covariance function is calculated as follows:

F 2 ( s , v ) = 1 s ∑ i = 1 s | X ( N − ( v − N s ) s + i ) − X ˜ v ( i ) | ⋅ | Y ( N − ( v − N s ) s + i ) − Y ˜ v ( i ) | . (6)

Step 4: Calculate the q-order detrend covariance function. For q ≠ 0 , the q-order detrend covariance function is defined as follows:

F x y ( q , s ) = { 1 2 N s ∑ v = 1 2 N s [ F 2 ( s , v ) ] q / 2 } 1 / q , (7)

and for q = 0 ,

F x y ( 0 , s ) = exp { 1 4 N s ∑ v = 1 2 N s ln [ F 2 ( s , v ) ] } . (8)

Finally, we show a double logarithmic plot of F_xy(q) versus s at each fixed q. If two time series {X(i)} and {Y(i)} are long-range cross-correlation, F_xy(q,s) will increase with s in power law,

F x y ( q , s ) ∼ s h x y ( q ) (9)

The above formula can also be expressed as

log F x y ( q , s ) = h x y ( q ) log ( s ) + log A (10)

where h_xy(q) is the scale index, describes the power-law relationship of two time series. In particular, when q = 2, especially when time series {X(i)} is the same as {Y(i)}, MF-DCCA method is MFDFA, otherwise , MF-DCCA is the standard DCCA method. When h_xy(q) > 0.5, the interaction exhibits long-range persistence; and when h_xy(q) < 0.5, the cross-correlation exhibits anti-persistence; if h_xy(q) = 0.5, there is no interaction or at most short-range interaction between the sequences.

Yuan and Zhuang (2009) proposed that Δ h = h max ( q ) − h min ( q ) can be used to measure multifractal strength. The larger Δh, the stronger the multi-fractality and the greater the market risk [17]. So according to this idea, use h_xy(q) instead of h(q), then Δh_xy can be used to measure the multifractal strength of the cross-correlation between the two sequences and the risk of the market.

Shdkhoo and Jafari (2009) propose an interaction correlation index [18]. The relationship between h_xy(q) and multifractal quality index τ x y ( q ) is as follows:

τ x y ( q ) = q h x y ( q ) − 1. (11)

If τ x y ( q ) is a linear function of q, then the interaction between the two sequences is single-fractal, otherwise it is multifractal. Carry out the Legendre transformation on the Equation (11) to obtain the following relationship:

α x y = τ ′ x y ( q ) = h x y ( q ) + q h ′ x y ( q ) . (12)

f x y = q α x y − τ ( α x y ) = q ( α x y − h x y ( q ) ) + 1. (13)

Among them, f_xy(q) is a multifractal spectrum, and α is a singular index, which is used to describe the singularity of each interval. The smaller the α, the larger the singularity, vice versa.

In this paper, we select the US Standard & Poor’s S & P 500 Index and the daily closing price data of the Shanghai Composite Index as the research object, from January 4, 2002 to March 14, 2017. After excluding different trading days, we have 3561 samples. Since the global financial crisis broke out between 2007 and 2008, January 1, 2007 and December 31, 2008 were set as the time points for segmentation. The full sample period is divided from January 4, 2002 to December 29, 2006, January 1, 2007 to December 31, 2008, and January 5, 2009 to March 14, 2017 which representing the period before, during and after the crisis. The logarithmic yields of these two indices are shown in Figure 1. We use a logarithm with a base of 2 here.

It can be seen from Figure 1 that both yield series exhibit the characteristics of volatility aggregation, that is, large fluctuations are followed by large fluctuations, and small fluctuations are followed by small fluctuations. It can also be found that the SSE index yield series is more volatile than the S & P500, which means that the SSE index yield sequence is more risky. Table 1 gives descriptive statistics of the yield series of two indices in three subsamples. In general, sample

Note: ***indicates rejection of the null hypothesis at the 1% significance level; The symbols “Mean”, “Max”, “S.D”, “Ske”, “Kur” indicate mean, maximum, minimum, standard deviation, skewness, kurtosis, respectively; JB represents the Jarque-Bera statistic.

2, i.e., the period of financial crisis, the standard deviation of the two index returns is greater than the other two periods; The skewness of the three sub-sample is not zero, and the kurtosis is greater than 4, indicating that during these three periods, the two yield series have the characteristics of sharp peaks and thick tails; The J-B statistic for both yield series is significant at the 1% level and therefore does not obey the normal distribution for the three periods. We mainly use Ske, Kur, S.D, and J-B indicators to describe the sequence, these formulas are as follows:

K u r = 1 n ∑ i = 1 n ( x i − x ¯ ) 4 ( 1 n ∑ i = 1 n ( x i − x ¯ ) 2 ) 2 , S k e = 1 n ∑ i = 1 n ( x i − x ¯ ) 3 ( 1 n ∑ i = 1 n ( x i − x ¯ ) 2 ) 3 / 2 , S . D = 1 n ∑ i = 1 n ( x i − x ¯ ) 2 , J - B = n 6 [ S k e 2 + ( K − 3 ) 2 4 ] (14)

where x_i represents the i-th sample value of a specific sequence.

Figure 2 shows the logarithmic plot of the cross-correlation test statistic for the different periods of the SSE yield series and the S & P 500 index yield series. The ordinate is the logarithm of the cross-correlation test statistic, i.e., the logarithm of Equation (3). The abscissa is the logarithm of the degree of freedom m (from 1 to 1000). The critical value of the chi-square test at the 5% significance level is also shown in Figure 2. In order to better display the graph, we use the logarithm with a base of 10.

It can be seen from Figure 2 that the Q_cc(m) curve of the cross-correlation test statistic for the full sample period (2002/1/4-2017/3/14) is almost above the critical value, thus, the null hypothesis that suppose no cross-correlation is rejecting.

That is, the SSE Index and the S & P 500 Index have significant long-range interactions throughout the sample period. However, before the 2007 financial crisis, sample 1, the cross-correlation test statistic for the two yield series was below the critical value curve, so the cross-correlation of the two indices was weak before the financial crisis. During the financial crisis, sample 2, whether the cross-correlation of these two sequences is significantly related to the degree of freedom. When log₁₀(m) < 1.8 or m < 63, the cross-correlation test statistic Q_cc(m) curve is above the critical value curve; however, if m > 63, the opposite is true. It shows that if the lag period is too long, there is no significant linkage between the Chinese and US stock markets, but it is highly correlated in the short term. After the financial crisis, sample 3, the Q_cc(m) curves of these two sequences are almost all above the critical value curve. In summary, the correlation between the two sequences is gradually enhanced, from the weak correlation before the crisis to the short-term correlation during the crisis. In particular, the outbreak of the financial crisis has significantly strengthened the linkage between the two stock markets in China and the United States, which is consistent with the conclusions of most documents.

The test above can only analyze whether there is linear interaction correlation and if it’s significant. In order to investigate the nonlinear cross-correlation of the two index’s yield series in three subsamples, we use Equation (3) to calculate the correlation coefficient of detrending and volatility. The correlation coefficients before, during and after the financial crisis were 0.4761, 0.5337 and 0.5529, respectively. This result further illustrates that the correlation of the two sequences in the three subsamples is gradually enhanced. The reason may be the outbreak of the financial crisis, which has led to panic psychological expectations of investors and the herd effect of investors. This negative impact has spread to other countries’ markets, leading to a significant increase in the relevance of financial markets in countries during the crisis.

The above results show that there is a significant interaction between the Shanghai Securities Composite Index yield series and the S & P 500 index yield series during the whole sample period, crisis period and crisis; however, it cannot further explain how the long-term interaction of the two sequences is and whether this interaction is single-fractal or multifractal. Are the multifractal characteristics of interactions the same in the short and long term? To this end, the MF-DCCA method is used to examine the specific characteristics of the interaction relationship.

Figure 3 is a double logarithmic plot of the covariance function and the time scale plotted by the MF-DCCA method, q = −10, −8, ..., 10, the order of the fitted polynomial is 2, x represents the Shanghai Securities Composite Index, and y represents Standard & Poor’s Index. It can be seen from Figure 3 that when q takes different values, the log-covariance function log₁₀F_xy(q) will change with the change of q, so the interaction relationship between the two indices has multi-fractal features and multiple fractal dimensions. Secondly, although there is no simple linear relationship between log₁₀F_xy(q) and log₁₀(s), log₁₀F_xy(q) linearly increases with the increase of log₁₀(s) from the trend or the whole. Therefore, there is a long-range power law interaction between the Shanghai Securities Composite Index and the Standard & Poor’s Index. Note that in order to better display the graph, we use a logarithm with a base of 10 here.

Ma and Wei (2013) pointed out that if there is a time scale s*, when s > s*, the linear trend or slope of the double logarithm graph changes or even structurally changes, and then this point is called crossover point. This crossover point s* can be used to distinguish the long-term and short-term nature of the time

series. The scale index for s < s* indicates short-term interaction behavior, which is easily affected by external factors of the market, such as political factors and economic factors; the scale index of s* indicates long-term interaction behavior, which is mainly determined by internal factors of time series [12]. Figure 3 shows that there is such a crossover point s* makes the slope of the linear relationship change on this time scale, which is roughly located at log₁₀(s*) = 2.193, or s^* = 156. Specifically, when the time scale s is small, the covariance function of different q is very divergent; as s becomes larger, the covariance functions of different q become closer and closer until log₁₀(s*) = 2.193, That is, when s* = 156, the covariance functions of different q have the smallest difference; however, when s > 156, the covariance function of different q becomes divergent again.

Figure 4 further shows the slopes of logF_xy(q) and log(s) at two scales, i.e. the cross-correlation index h_xy(q) at different q, including the generalized Hurst exponent of the Shanghai Securities Composite Index yield series, and that of the Standard & Poor’s yield series, as well as the average Hurst exponent of the two sequences.

Figure 4 contains a very rich amount of information. First, whether in the long run or in the short term, the cross-correlation index h_xy(q) depends on the q value and changes with the q value, which further illustrates the interaction between the Shanghai Securities Composite Index and the S & P index yield series. The relationship is multifractal. Second, in the case of small scales (s < s*) or short-term, the cross-correlation index varies more, from about 0.7 to 0.3, h_xy = 0.4143; in the case of large scales (s > s*), the cross-correlation index h_xy(q) is less dependent on q, and the range of variation is smaller and more stable in a long period of time and h_xy is only 0.1338. These two results show that the multifractal intensity of interaction in the short-term is more complicated than that of the long-term and multifractal features; while the interaction in the long-term is simpler, but it has stronger persistence (when q changes, H_xy is close to 0.6 or greater than 0.6, and no less than 0.5). Obviously, this is in line with reality. In the long run, the China-US stock market is mainly affected by international common factors and its long-term running trends and laws. These factors have far-reaching and stable effects on the two stock markets, so long-term interaction relationships are also more persistent and stable. And being more stable

means that the interaction is simpler and the strength of multiple fractals is relatively shorter in the short term. In the short-term, besides being affected by common factors, they are more affected by their domestic macroeconomic, policies, and behavioral habits of investors. Although there are certain interrelationships in the short term, they are affected by their specific factors. This interaction is not stable and the persistence is weaker but at the same time, it is more complex and multifractal features are more obvious.

Figure 4 (right) inadvertently answers the debate over whether the US stock market has long-term memory. It can be seen from the figure that the Hurst index of the Standard & Poor’s index, i.e. h_xx(q), not only fluctuates less, but also fluctuates around 0.5; on the contrary, the Hurst index of the Shanghai Securities Composite Index, h_xx(q), not only fluctuates greatly, but is much larger than 0.5 with the minimum value of around 0.6. This result shows that the Standard & Poor’s index has no long-term memory. The US stock market represented by the Standard & Poor’s index is more efficient and more in line with the efficient market hypothesis. The Shanghai Securities Composite Index has long-term memory, and its market pricing efficiency is lower, and is closer to fractal market hypothesis.