The Liquidity Spillover Effects Between the Stock Index Futures and Spot Under the Fractal Market Hypothesis

: In recent years, the extreme risk events occurred frequently in the financial market have not only brought huge losses to investors and inflicted heavy losses on the market, but also posed a severe challenge for the traditional effective market hypothesis. These extreme risk events are often accompanied by sudden plummeting of liquidity. Different from the efficient market hypothesis(EMT), firstly, this paper studies the nonlinear fluctuation characteristics and causes of contracts with different maturity periods in China stock index futures market under the framework of fractal market theory and using the multifractal detrended fluctuation model Secondly, under the framework of the fractal market theory, the existence of the liquidity spillover effect between the stock index futures and spot is tested, the direction, intensity, and contribution of spillover between stock index futures and spot are analyzed. Finally, there is a robustness test. The study finds that both stock index futures and stock index spot in China have obvious nonlinear fractal fluctuation characteristics, and stock index futures have higher degree of multifractal , the characteristics are related to correlated multifractal and distributed multifractal; the longer the maturity period of the stock index futures contract, the lower the multifractal degree; there are significant asymmetric liquidity spillover effects between the stock index futures and spot; the multifractal degree has an important influence on the intensity and contribution of the liquidity spillover effect, and the multifractal degree is inversely proportional to the intensity of liquidity spillover and the contribution of spot to futures fluctuations.


Introduction
As the first stock index futures product in Chinese mainland, CSI 300 index futures have attracted extensive attention from investors since its listing. Stock index futures are the assets of stock index spot, the stock market price fluctuations will directly affect the price fluctuation of stock index futures. Because of the hedging, risk-averse and other functions, the stock index futures will in turn indirectly affect the stock market.
Therefore, stock index futures and spot markets are highly dependent on each other.
At present, there are three different views on the impact of stock index futures on price volatility in the spot market: Firstly, stock index futures increase the degree of volatility of the spot market (Harris (1989), Brorse (1991), Kamara, et al (1992), Bae, et al (2004), Kittiakarasakun, et al (2013), Zhou, et al (2015)). By studying the S & P 500, South Korea KOSPI200, and China CSI 300 index, the high leverage may increase the intensity of market speculation and investment risk, generate more "noise" trading signals, and an increase in the trading behavior of uninformed traders, the volatility of stock market will be further increased Secondly, stock index futures enhances market price discovery function and improves market efficiency and liquidity of the stock, these can effectively reduce the volatility of the stock market (Tu and Guo (2008), Kasman and Kasman (2008), Bessembinder  In financial markets, whether it is financial derivatives market or basic securities market, extreme abnormal events such as the flash crash in the US securities market, China's securities market of a thousand shares fell by the daily limit, Guangda Group 8.16 incident, and the Shanghai 50ETF options have skyrocketed in 5.25 are still fresh in our memory, the impact caused by them also yet to fade away. In fact, these extreme events always are accompanied by a temporary and extreme lack of market liquidity. As Amihud said: "In the market, liquidity is everything." (1980) In addition, the frequent occurrence of these financial visions also makes scholars question the traditional financial theory, especially the effective market hypothesis (Mandelbrot (1963), Mantegna and Stanley). At present, most studies on the spillover effects of stock volatility are based on the effective market hypothesis (EMH). However, Mandelbrot (1999) creatively proposed the fractal theory, then more and more studies (Peters (1991), Yuan and Zhuang (2008), He, et al (2014), Delbianco F, et al (2016), jiang ZQ, et al (2017)) showed that the complex nonlinear structure of the securities market is obvious fractal.
The main purpose of the launch of China stock index futures is not only to realize the diversification of trading varieties in the financial capital market and the effective prevention and control of risks, but also to meet the needs of investors' risk management. Therefore, the deep characteristics of liquidity fluctuations, and the liquidity spillover effects are particularly important. In fact, not just the stock market, the stock index futures market also does not meet the efficient market theory, the complex process of price change is the result of the combination of factors in different dimensions. Yin and Hua (2017), Tang and Zhu find that China's securities market, as well as CSI 300 index futures have multifractal characteristics, and there are significant differences in long memory and market risks. The so-called multifractal refers to an infinite set of different scaling indices on fractal structures, which is used to describe different scaling properties of the mass distribution over different regions. It has the characteristics of multiscale and the transition of scaling. As for multifractal research model, Peng, et al (1994) first proposed detrended fluctuation analysis (DFA), which is mainly used for long-range correlation analysis of time series. Trend components of all order in the sequence can be efficiently filtered, and long-range correlations containing noise and superimposed polynomial trend signals can be analyzed. However, this model can't be used to describe the multi-scale and fractal time series subset. On this basis, Kantelhardt, et al (2002) proposed multifractal detrended fluctuation model (MF-DFA). The use of multifractal to describe different levels of the fractal market enables not only to analyze the long-range correlations of non-stationary sequences, but also to avoid misjudgments (Norouzzadeh and Jafari).
Although a lot of existing researches have studied the fractal structure of the securities market and stock market volatility spillover, but we find that at present no one has been involved in the fields of the multifractal characteristics of liquidity and liquidity spillover effect between the stock index futures and spot. The traditional linear theory is difficult to deeply reveal the mechanism of fluctuations, and accurately predict its change trends. Therefore, this paper will study the liquidity spillover effect of stock index futures and spot market based on the fractal theory, aiming to answer the following questions: whether the fractal characteristics in liquidity of stock index futures and spot markets exist? What kind of impact will the maturity period have on the fractal degree of the liquidity of stock index futures contracts? Under the fractal market hypothesis (FMH), whether there is a spillover effect exists between the liquidity of stock index futures market and stock market? Whether the multifractal degree of market will influence this spillover effect?
Based on this, firstly, this paper uses the historical data of China stock index futures market and spot market, establishes the indicators of liquidity, tests the multifractal characteristics of China stock index futures market and spot market with the MF-DFA model, measures the multifractal degree and judges the causes of multifractal. Secondly, the influence of the multifractal degree on the correlation, spillover direction, strength, and contribution are studied respectively.

Multifractal detrended fluctuation model (MF-DFA)
It is assumed that a limited time sequence (1) Construct the cumulative discrepancy columns: Among them, x is the mean of time sequences (2) Equally space the column   , the time series obeys the random walk, that is, there is no cross-correlation; , there is a positive persistence in the time series, which is also known as long memory, that is, the current time series has an upward trend, then presents a rising trend.
It is known that the dimension ( h  ) of multifractal forms in the time series is shown: Obviously, the larger the value of h  , the larger the multifractal degree of the time series, and the volatility of the time series is also deviated from the random walk.

Data source
This paper takes China's CSI 300 stock index futures and CSI 300 index as the research object, in which CSI 300 index futures mainly have four kinds of contracts, such as IF00, IF01, IF02 and IF03, and also uses the data of 2724 trading days from April 16,2010 to June 30,2021. This paper studies the multifractal fluctuation characteristics of the market liquidity and the liquidity spillover effect between the stock index futures and spot. All data are from iFinD Financial Database.

Indicator description
In the indicators to measure the market liquidity, most scholars (  In addition, some scholars have pointed out that there is an intrinsic relevance between the volume of transactions and liquidity, and the liquidity can also be reflected.

Descriptive statistics
According to the formula (10) and formula (11), this paper obtains the descriptive statistical results of the liquidity ( 1 H ) of CSI 300 index futures and CSI 300 index, as shown in Table 1. As can be seen from Table 1, CSI 300 index futures and CSI 300 index significantly do not meet the assumptions of normal distribution, showing a significant asymmetric peak-tail feature. In addition, the ADF test is used to test the time series variable, and it is found that the liquidity of the stock index futures and spot are stable time series. At the same time, from the perspective of mean and standard deviation, the liquidity indicator (Amihud non-liquidity indicator) of CSI 300 index is significantly lower than CSI 300 index futures, and the fluctuations are smaller. In other words, compared with CSI 300 index futures, the stock index liquidity is better, and less volatile. This paper is based on liquidity indicator 1 H to draw the liquidity circular timing chart of CSI 300 index futures and CSI 300 index respectively, the results are shown in  The descriptive statistical results of the liquidity indicator 2 H are shown in Table 2.
It can be observed that the bias is not 0, the peak degree is not 3, and the accompanying probability of JB statistics is 0, the sequence is stable and statistical characteristics. The descriptive statistical results of indicator 2 H is consistent with the indicator 1 H . Table 2 The descriptive statistics of the liquidity of CSI 300 index futures and CSI 300 index Based on transaction volume to structure the liquidity indicator 2 H , its corresponding circular timing chart of CSI 300 index futures and CSI 300 index are shown in Fig.3 and Fig.4.  and CSI 300 index with significant multifractal characteristics will be analyzed in the following part. When q takes a specific value, the value of the evaluation scale index () hq and h  of CSI 300 index futures and CSI 300 index are showed in Table 3. As can be seen from in CSI 300 index, indicating that the multifractal degree of the stock index futures market is stronger than the spot market, that is, the liquidity of the stock index futures market is more complex and more deviated from random walk; Finally, the longer the expiry period of the stock index futures contract, the smaller the multifractal degree.

Multi -fractal test and cause analysis
This is because the longer the time, the greater the uncertainty of the future, the less susceptible the contract is to external manipulation, so the multifractal degree is smaller, and the contract is more biased towards the random change.   Fig.6 and Table 3 are multifractal features and values of () hq of the reset flow sequence. As can be seen from Fig. 6, the reset liquidity sequence still has significant multifractal characteristics. At the same time, it is known that the reset sequence also has obvious long-term memory, and the generalized Hurst indexes before and after reset are not equal. According to the multifractal theory, when

Existence of the liquidity spillover effect in stock index futures and spot market
According to the ADF test results above, CSI 300 index futures and CSI 300 index liquidity sequence are all stable time series. Therefore, it is possible to study the dynamic related relationship between CSI 300 index futures and CSI 300 index, and determine whether there is liquidity spillover effect.
This section utilizes the VAR model. First of all, we need judge the optimal lag order of CSI 300 stock index futures and CSI 300 index based on the Likelihood Ratio (LR),

Final Prediction Error (FPE), Akaike Information Criterion (AIC), Schwarz Criterion
(SC), and Hannan-Quinin Guidelines (HQ), then find that the optimal lag order is 8, and the VAR models are showed in Table 4. First, CSI 300 index has a one-way liquidity spillover effect to the stock index futures.
In the current period, the IF00 liquidity is significantly positively affected by its own lag period 1 to 8, and also by the significant negative impact of lag period 1 and 5 of the CSI 300 index, it shows that the IF00 liquidity is mainly adjusted according to its own past liquidity, and affected by the CSI 300 Index to a lesser extent. In addition, the current liquidity of the CSI 300 index is only significantly positively affected by its own lag periods; Second, there is a two-way liquidity spillover effect between IF01 and CSI 300 index.
In the current period, IF01 liquidity is significantly affected by its lag periods except lag period 6, while the liquidity of the CSI 300 index in lag period 1 and 2 has a significant negative impact on IF01, it shows that IF01 liquidity is adjusted according to its own past liquidity, and affected by the liquidity lag periods of the CSI 300 index.
At the same time, the liquidity of CSI 300 index is subject to the significant positive impact of its own past liquidity and the significant negative impact of IF01 in lag period 1 and 8; Third, there is a two-way liquidity spillover effect between next IF02 and CSI 300 index quarter. The current IF02 liquidity is significantly affected by its other lag periods except lag period 6, while the liquidity of the CSI 300 index in lag period 1, 2 has a significant negative impact on IF02, it shows that the IF02 liquidity will be adjusted according to its own past liquidity, while being affected by the liquidity lag periods of the CSI 300 index. At the same time, the liquidity of CSI 300 index will be subject to the significant positive impact of all lag period of their own liquidity and the significant negative impact of IF01 in lag period 1; To sum up, there is a significant liquidity spillover effect between CSI 300 index futures and CSI 300 index. The longer the maturity period of the futures contract, that is, the lower the multifractal degree, the more significant the liquidity spillover effect between them.

The direction of the liquidity spillover effect
The above studies have shown that there is a significant liquidity spillover effect between CSI 300 index futures and CSI 300 index, in this part, the Grander causality test is used to analyze the direction of liquidity spillover effect. Considering that different lag period can cause resulting errors, the lag period is selected as the same in the VAR model, that is 8. The Grange causal test results are shown in Table 5. index is not significant. In the above section, using the VAR model can also prove that there is a significant liquidity spillover effect, which explains that the significance of the corresponding coefficient of the CSI 300 index in the VAR model of the CSI 300 stock index futures is much greater than that of CSI 300 index futures in the CSI 300 index, so the conclusion of Granger causality test and the regression result of the VAR model are basically consistent.

Intensity of the liquidity spillover effect
The following uses the impulse response function to verify the intensity of the liquidity spillover effect from the spot to stock index futures. First, the root of the AR is used to perform a stationary test. It is found that the roots of the model obviously locate in the unit circle, indicating that they are stable.
The results of the impulse response function are shown in Fig.7. As can be seen from the figure, the liquidity spillover effect of the spot to stock index futures is positive, and the impact has a long time. As the expiry period of the stock index futures grows, the intensity has increased, that is, the contracts' expiry period and the intensity spillover intensity are proportional. In addition, the expiry period of the stock index futures contract is inversely proportional to the multifractal degree, so it is true that the spillover intensity is inversely proportional to the degree of multifractal. The above conclusions can be explained that the lower the multifractal degree of the market, the closer to the random walk distribution, thus fluctuating becomes more unpredictable, 'and market volatility cannot be controlled and reduced in advance, and the larger volatility means that the greater intensity of liquidity spillover across markets. Fig.7 The intensity of the liquidity spillover effect from CSI 300 index to CSI 300 index futures

Contribution to liquidity spillover effect
This paper further utilizes Hasbrouck variance decomposition to analyzes the difference of the contribution of CSI 300 index to the liquidity spillover effect of stock index futures contract with different maturities. The variance decomposition results are shown in Table 6. Response of H to F3 the multifractal degree is, the less the market changes fit the random walk in the effective market hypothesis, and this improves the investor or government's predictive effeciency on future liquidity risk infection, so they can take the corresponding measures in advance to inhibit negative effects. In CSI 300 index futures, due to the most predictability of IF00, it is possible to lock in its liquidity risk sources in advance, namely CSI 300 index, and reduce the infection strength of CSI 300 index.

Robustness test
In order to verify the effectiveness of the above conclusion, the liquidity indicator 2 H is used for a robustness test. First, the multifractal characteristics and degree of CSI 300 index futures and CSI 300 index are verified according to the MF-DFA method, and the results are shown in Fig.8 and Table 7. It can be seen that IF00, IF01, IF02 and CSI 300 index have significant multifractal characteristics, and the complexity of the stock index futures is much larger than the spot.     Fig.9 shows the impulse response function of the liquidity indicator 2 H , it can be seen that as the liquidity spillover party, the CSI 300 index has a significant positive relationship between the contract expiry period and the spillover intensity, that is, as the contract expiry period of stock index futures grows, its spillover intensity also increases.
(a) (b) (c) Fig.9 The intensity of the liquidity spillover effect from CSI 300 index to CSI 300 index futures ( 2 H ) It can be seen from Table 9 that the liquidity spillover contribution of the CSI 300 index to the CSI 300 index futures increases with the increase of the contract expiry period, that is, the liquidity contribution of the spot to the stock index futures is inversely proportional to the multifractal degree. In summary, whether it is multifractal characteristics, or the existence of the liquidity spillover effects, spillover direction, intensity, and contribution, the result of the new liquidity indicator 2 H is consistent with the above research, it shows that this study has great robustness.

Conclusion
In recent years, there have been many extreme events such as Guangda Group 8.16 incident, China stock market crash in 2016, significantly affecting investors and market regulatory authorities, and posed a serious challenge to traditional effective market hypothesis. In this context, this paper uses the fractal market hypothesis, combined with CSI 300 index futures and CSI 300 index in China, first tests whether China stock index futures and spot market meet the effective market hypothesis; secondly, using the MF-ADF model to confirm that CSI 300 index futures and CSI 300 index have significant multifractal characteristics, and the financial market has more complex non-normal distribution. CSI 300 index futures have high multifractal degree, the multifractal degree is inversely proportional to the contracts' expiry period; then, using the VAR model, Granger causality test, impulse response functions and variance decomposition to prove the existence of the liquidity spillover effects, spillover direction, intensity, and contribution. Research finds: there is a significant asymmetric liquidity spillover effect between the stock index futures market and spot market, and the spillover strength and contribution are significantly affected by multifractal characteries; the intensity of liquidity overflow is proportional to the expiry period of the stock index futures contract, but inversely proportional to the multifractal degree of the stock index futures contract; the contribution of the spot is inversely proportional to the multifractal degree. Finally, the robustness test indicates a good robustness in this study. The relevant research results can not only contribute to a deeper understanding of the liquidity relationship between the stock index futures and spot theoretically, but also provide guidance for investors' investment practice and decision-making reference for the daily supervision of the construction and management departments.