# Research on Chinese Stock Market during COVID-19—Based on Random Matrix Theory

Authors: Wu Yaojie; Fang Tianhui
Journal: Complexity
 (2023)

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## Abstract

This paper focuses on the three industries that are greatly impacted by COVID-19, including the consumption industry, the pharmaceutical industry, and the financial industry. The daily returns of 98 stocks in the consumption industry, the pharmaceutical industry, and the financial industry in the 100 trading days from January 2, 2020, to June 3, 2020, are selected. Based on the random matrix theory, it first analyzes whether the stock market conforms to the efficient market hypothesis during the epidemic period, and second it further studies the linkage between the three industries. The results show that (1) the correlation coefficient is approximately a normal distribution, but the mean value is greater than 0, which is greater than that of the more mature markets such as the United States. (2) There are three eigenvalues greater than the prediction value, of which the maximum eigenvalue is about 11.18 times larger than the largest eigenvalue of the RMT. (3) There is a significant positive relationship between the maximum eigenvalue and the correlation coefficient. The specific market performance is that the stock price fluctuations show a high degree of consistency. (4) In the sample interval, the financial industry has a restraining effect on the consumption industry in the short term, and the pharmaceutical industry has a promoting and then restraining effect on the consumption industry in the short term. The consumption industry has a promoting effect on the financial industry in the short term, and the pharmaceutical industry has a promoting and then restraining effect on the financial industry in the short term. The consumption industry has a promoting and then restraining effect on the pharmaceutical industry in the short term, and the financial industry has a promoting and then restraining effect on the pharmaceutical industry in the short term. (5) In the sample interval, the consumption industry is mainly affected by itself, while the role of the pharmaceutical industry and the financial industry is very small. The pharmaceutical industry is mainly affected by itself and the consumption industry, while the role of the financial industry is very small. The financial industry is mainly affected by itself and the consumption industry, while the role of the pharmaceutical industry is very small. This situation has consequences for individual investors and institutional investors, since some stock returns can be expected, creating opportunities for arbitrage and for abnormal returns, contrary to the assumptions of random walk and information efficiency. The research on the correlation between asset returns will help to accurately price assets and avoid losses caused by price fluctuations during the epidemic.


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## Body

## 1. Introduction

Since the second half of the last century, the entire international financial environment has changed, and now the financial risks faced have increased dramatically compared with before. All companies and financial institutions are facing more uncertain systemic financial risks. Risk management has become a key and basic technology that all market participants must master. An important aspect of risk management is estimating the correlation between different asset prices.Classical capital theory research is basically based on efficient market theory. Fama summarized that an efficient market is a market in which prices always “fully reflect” all information [1]. There are three sufficient conditions for a specific efficient market: (1) There is no transaction cost in the process of securities trading; (2) market participants do not need to pay for information; and (3) all participants agree with the use of the latest information that affects the current and future price distribution of each security. It embodies an ideal competitive equilibrium, under which the return of assets can be obtained to obey the normal distribution.However, different markets show various forms in different periods, especially in the stock market, where there are many stylized facts [2–4], such as volatility clustering [5–7], fat tails [8–10], and long memories [11, 12], which are difficult to explain by modern financial theory. Therefore, modern financial theory based on efficient market hypothesis and rational expectation theory is not accurate enough in the risk management.Since COVID-19 appeared at the end of 2019 in the city of Wuhan, China, and the World Health Organization (WHO) declared a global epidemic on March 12, 2020. China is the first and most seriously affected country by COVID-19. At the same time, as the world’s second-largest economy, the Chinese economy plays a vital role in global integration. Studying the impact of the epidemic on the Chinese financial market will also provide some reference and inspiration for other countries. The epidemic has had a huge impact on the Chinese real economy, with experts estimating that the consumption, pharmaceutical, and financial industries have been most affected. From the sudden outbreak of COVID-19 in December 2019 and the “closure” of the city of Wuhan to the basic control of the epidemic in April 2020, and then to the gradual resumption of production in all industries in June 2020, the Chinese financial market (especially stock market) was profoundly impacted by the shock of COVID-19. With the gradual control and improvement of the epidemic, it experienced a violent shock process from a sharp decline to a gradual rise.As mentioned above, industries that have been greatly impacted, such as the consumption industry, the pharmaceutical industry, and the financial industry, have attracted much attention. Under the impact of the epidemic, how do the stock prices of these three industries change? Is there a positive correlation or a negative correlation in its stock return? Is the correlation matrix consistent with the stochastic characteristics of the efficient market hypothesis? And whether there is a linkage effect of mutual influence between the returns of the consumption industry, the pharmaceutical industry, and the financial industry? These issues are the hot spots and frontiers of the recent academic fields.In the related research of the financial market, the analysis of the financial data by different methods developed in statistical physics has become a very interesting research area for physicists and economists. It is meaningful and valuable for scholars to analyze the correlation coefficient between stock return time series because this contains a significant amount of information on the nonlinear interactions in the financial market and is a parameter in terms of the Markowitz portfolio theory. The cross-correlation matrix between several stocks, which has unexpected properties due to complex behaviors, such as mispricing, bubbles, market crashes, and so on, is an important parameter to understand the interactions in the financial market. To analyze the actual cross-correlation matrix, the random matrix theory (RMT) proposed by Wigner is a useful and technical tool for eliminating the randomness in the actual cross-correlation matrix [13–16]. Laloux et al. selected 406 stocks in S&P500 and calculated the correlation coefficient matrix for a total of 1309 days from 1991 to 1996, which found that about 94% of the eigenvalues fell within the prediction range and less than 6% of the eigenvalues fell outside the prediction range [17]. Plerou et al. analyzed the correlation matrix of stock price volatility by selecting the 30 minute high-frequency data of 1000 stocks of the largest U.S. companies from 1994 to 1995 and the daily data of 422 stocks from 1962 to 1997, and they finally took an conclusion that most of the eigenvalues of the correlation matrix were within the prediction range, and only a few of the largest eigenvalues fell outside the prediction range [18]. Subsequently, Stanley et al. further analyzed the correlation matrix of the returns of these 1000 stocks and found that there were about 20 maximum eigenvalues outside the range of theoretical prediction values, and the larger eigenvalues are synchronized with the correlation coefficients [19]. Since scholars like Laloux and Plerou applied random matrix theory (RMT) to financial markets, it has been used to study the statistical characteristics of cross-correlation between different financial markets.For example, Wilcox and Gebbie found that the proportion of eigenvalues beyond the prediction range of the South African stock market was larger than that of the U.S. market [20]. Kulkarni and Deo used RMT to study the volatility of the Indian stock market, and the results showed that the maximum eigenvalue of the correlation coefficient and the volatility had positive synchronization; that is, the greater the maximum eigenvalue was, the greater the volatility of the stock market was at that time, and vice versa [21]. Oh et al. took the Korean stock market as the research object, and the empirical results showed that the maximum eigenvalue was 52 times larger than the maximum value among the eigenvalues predicted by the RMT [22]. El Alaoui used random matrix theory to analyze eigenvalues and see if there was a presence of pertinent information by using Marčenko–Pastur distribution [23]. Urama et al. investigated the universal financial dynamics in two stock markets in sub-Saharan Africa through an in-depth analysis of the cross-correlation matrix of price returns in Nigerian Stock Market (NSM) and Johannesburg Stock Exchange (JSE), for the period 2009 to 2013. And the results found that the strength of correlations between stocks was higher in JSE than that of the NSM [24]. Kumar et al. analyzed daily prices of 42 stocks listed in the Nifty50 index of the National Stock Exchange of India from 2006 to 2019 and they found that global and local extreme events affected the correlations among the Nifty50 stocks of the Indian stock market [25]. Taştan and Imamoglu attempted to investigate the cross-correlation between stocks listed under the XU100 index of Borsa Istanbul with several ratios and indices of the stock markets worldwide by using the random matrix theory approach through a correlation matrix, and it was found that XU100 has a distinguishing behavior compared to other indices and rates in terms of eigenvalue and related eigenvector structures [26].In order to separate the non-noise information in the financial correlation matrix, random matrix theoretical denoising methods such as the LCPB denoising method, PG + denoising method, and KR denoising method have been proposed [27–30].Some literatures also analyze the dynamic evolution of stock structure by deviating from eigenvalues [31–33], and some scholars study and compare the characteristics of stock structure in different industries by using the corresponding eigenvector analysis of deviating eigenvalues [34–36]. In addition, partial correlation coefficient analysis can also eliminate some intermediary effects and reveal the correlation characteristics of stocks at a deeper level [37–39].Combined with the literatures, it can be found that most scholars study and determine whether there is a correlation in the stock return matrix. It is divided into two parts: one part is consistent with the characteristics of the random matrix (“noise”); and the other part is the difference part (“real information”). The magnitude of the difference reflects how far the empirical correlation matrix deviates from the random correlation matrix.However, few literatures further analyze whether there is interaction between industries after studying the efficient market hypothesis. The financial market allocates capital resources in different production fields, and the function of the financial market to disperse and transfer risks is the key to the risk allocation function of the market economy. It is also the main research content of portfolio selection theory that using known information to construct a portfolio can effectively disperse investment risk. The core of the optimal decision of portfolio lies in its ability to diversify investments in different assets and reduce various nonsystematic risks of the portfolio so as to obtain the benefits of portfolio investment. The efficiency of risk reduction depends to a large extent on the correlation between the returns of various assets in the portfolio. The random matrix theory can be used to study the real information and noise contained between the returns of assets. This paper has four contributions, as follows:(1)
Focusing on the daily returns of three major industries that were greatly affected by COVID-19 during the epidemic period-the consumption industry, the pharmaceutical industry, and the financial industry. Combined with random matrix theory (RMT), this paper analyzes the statistical characteristics of the return correlation matrix of Chinese stock market during the epidemic. And we also start from the current hot spots, integrate theory with practice, and deeply analyze the internal mechanisms between Chinese stock markets.(2)
Selecting the specific period from the outbreak of the epidemic to the control of the epidemic to study the maximum eigenvalue and the corresponding eigenvector of the correlation matrix, as well as the relationship between the maximum eigenvalue and the correlation coefficient, so as to verify the effectiveness of the entire market after the impact of the epidemic.(3)
For the first time, the three industries of consumption, medicine, and finance are linked to further analyze whether there is a linkage effect between industries during the epidemic, that is, whether one industry can be promoted or restrained by the influence of another industry.(4)
Although this paper focuses on the correlation between the three industries most affected during the epidemic, the methods and ideas can be widely applied to other markets, such as the bond market and futures market, which are conducive to investors’ risk aversion in the face of external environmental shocks in the financial market.The rest of the structure of this paper is arranged as follows: the second part is the research design, which introduces the construction of empirical return correlation matrix, variable description, and data source. The third part is the empirical results, which mainly include the comparison between the eigenvalues and eigenvectors of the empirical correlation matrix and the random matrix, and the linear relationship between the maximum eigenvalue and correlation coefficient. The fourth part is a one-step analysis of the linkage between the consumption industry, the pharmaceutical industry, and the financial industry. The fifth part is the conclusion and enlightenment.


## 2. Research Design

### 2.1. Model Construction

#### 2.1.1. Financial Correlation Coefficient Matrix

Recording the valid trading day price sequence of stockii=1,2,···,N: Pi1,Pi2,···,PiN. Where Ni represents the valid trading days of stock i, Pit represents the closing price of stock i on the valid trading day t, and we define the logarithmic return of stock i as follows:(1)rit=lnPit−lnPit−1.For any two stocksi and j, since their valid trading price series are different, when calculating the correlation, we take out their common valid trading daytime series: u1,u2,···,um, where m represents the total number of common valid trading days of stocks i and j.Thus, the Pearson correlation coefficient between any two stocksi and j is as follows:(2)ρij=rirj>−<ri><rj<ri2>−<ri>2<rj2>−<rj>2,where < ··· > is the statistical average, and the value range of ρij is [−1, 1]. If ρij=−1, it means that there is a completely negative correlation between the two stocks. If ρij=1, it means that there is a completely positive correlation between the two stocks. If ρij=0, it means that the two stocks are not related. Thus, using equations (1) and (2), we can get the correlation coefficient matrix C of the stock market, which has the size of N×N and the element of ρij.


#### 2.1.2. Random Matrix Theory

Considering such a random matrix:R=1/LA×AT, where A is a random matrix of N×L, which is composed of N independent sequences with length L, and each sequence is subject to N0,1 distribution. The statistical characteristics of matrix R are known, especially when N⟶∞, L⟶∞, Q=L/N>1 is fixed, the distribution function Pλ of eigenvalue λ of matrix R has an analytical form:(3)Pλ=Q2πσ2λmax−λλ−λminλ,where λmin⩽λ⩽λmax, and(4)λminmax=σ21+1Q±21Q,and σ2=1.


#### 2.1.3. Principle and Procedure of Empirical Correlation Matrix Analysis Based on RMT

The principle of universality is one of the most important reasons for the success of RMT in multiple research fields; that is, under the average energy level interval scale, the eigenvalue distribution law does not depend on the specific distribution of elements in the system. By using this characteristic, we can obtain the essence of the internal interaction of the intended research system by comparing the similarities and differences between the research system and the random system in the distribution of eigenvalues and eigenvector elements. In other words, the random properties and special nonrandom properties inside the system can be investigated by studying the universality properties of the matrix of the research system and the random system through comparative analysis.To explore the internal structure and dynamic behavior of the economic system, the most important thing is to study the statistical characteristics of price fluctuations and the relationship between different stocks, which can be converted into the statistical characteristics of the empirical cross-correlation matrix of the stock market. Through the theoretical analysis of the information structure of the empirical correlation matrix, the content of the empirical correlation matrix can be divided into “information” and “noise,” so identifying the “information” and “noise” in the empirical correlation matrix has become an important way to study the statistical characteristics of the empirical correlation matrix. Based on the universality of random matrix theory studied above, this paper intends to compare and analyze the statistical characteristics of the empirical correlation matrix and real symmetric random correlation matrix of three typical industries in the Chinese stock market, and then study the nonrandom and random characteristics of the empirical correlation matrix. The former is the “information,” and the latter is the “noise.”Specific analysis procedures are as follows:(1)
Calculate Pearson’s correlation coefficient (PCC)(2)
Construct the empirical correlation matrix and obtain its eigenvalues(3)
Analyze the statistical characteristics of the correlation coefficients, eigenvalues, and eigenvectors


### 2.2. Variable Description and Data Source

This paper selects the closing prices of 98 constituent stocks of the three major indices: Shanghai Stock Exchange Consumption (30 constituent stocks), Shanghai Stock Exchange Medicine (38 constituent stocks), and Shanghai Stock Exchange Finance (30 constituent stocks) from January 2, 2020, to June 3, 2020, to measure the daily returns.The reasons for selecting the data of 98 constituent stocks of the three major Shanghai Composite indexes from January 2, 2020, to June 3, 2020, are as follows:(1)
This paper mainly studies the reaction of the three industries in the stock market in a short time after the impact of COVID-19 and selects all the constituent stocks in the industry indices to be representative.(2)
The constituent stocks of the index will be replaced in June and December every year, and the constituent stocks of the three industry indices will remain unchanged within the research range, which makes the research more accurate.When analyzing the correlation matrix of empirical returns, the datum ofN=98 and L=100 are selected for analysis. When further analyzing the linkage between the three industries of consumption, medicine, and finance, the daily returns of Shanghai Stock Exchange Consumption Index (Consumption), Shanghai Stock Exchange Medicine Index (Medicine), and Shanghai Stock Exchange Finance Index (Finance) are selected for analysis.The specific stock codes are shown in Table1.Table 1
Description of 98 stocks in the consumption, pharmaceutical, and financial industries.
IndustryStock codeConsumption600132, 600298, 600315, 600438, 600519, 600597, 600598, 600600, 600737, 600779, 600809, 600811, 600827, 600872, 600873, 600887, 601933, 603043, 603156, 603198, 603288, 603317, 603369, 603517, 603589, 603605, 603708, 603711, 603866, 603983Pharmaceutical600056, 600062, 600079, 600085, 600161, 600195, 600196, 600201, 600216, 600252, 600267, 600276, 600299, 600332, 600380, 600420, 600436, 600511, 600521, 600535, 600566, 600572, 600664, 600763, 600771, 600867, 600998, 601607, 603222, 603233, 603259, 603658, 603707, 603718, 603858, 603882, 603883, 603939Financial600000, 600015, 600016, 600030, 600036, 600048, 600340, 600606, 600837, 600919, 600999, 601066, 601155, 601166, 601169, 601211, 601229, 601288, 601318, 601319, 601328, 601336, 601398, 601601, 601628, 601688, 601818, 601939, 601988, 601998


## 2.1. Model Construction

### 2.1.1. Financial Correlation Coefficient Matrix

Recording the valid trading day price sequence of stockii=1,2,···,N: Pi1,Pi2,···,PiN. Where Ni represents the valid trading days of stock i, Pit represents the closing price of stock i on the valid trading day t, and we define the logarithmic return of stock i as follows:(1)rit=lnPit−lnPit−1.For any two stocksi and j, since their valid trading price series are different, when calculating the correlation, we take out their common valid trading daytime series: u1,u2,···,um, where m represents the total number of common valid trading days of stocks i and j.Thus, the Pearson correlation coefficient between any two stocksi and j is as follows:(2)ρij=rirj>−<ri><rj<ri2>−<ri>2<rj2>−<rj>2,where < ··· > is the statistical average, and the value range of ρij is [−1, 1]. If ρij=−1, it means that there is a completely negative correlation between the two stocks. If ρij=1, it means that there is a completely positive correlation between the two stocks. If ρij=0, it means that the two stocks are not related. Thus, using equations (1) and (2), we can get the correlation coefficient matrix C of the stock market, which has the size of N×N and the element of ρij.


### 2.1.2. Random Matrix Theory

Considering such a random matrix:R=1/LA×AT, where A is a random matrix of N×L, which is composed of N independent sequences with length L, and each sequence is subject to N0,1 distribution. The statistical characteristics of matrix R are known, especially when N⟶∞, L⟶∞, Q=L/N>1 is fixed, the distribution function Pλ of eigenvalue λ of matrix R has an analytical form:(3)Pλ=Q2πσ2λmax−λλ−λminλ,where λmin⩽λ⩽λmax, and(4)λminmax=σ21+1Q±21Q,and σ2=1.


### 2.1.3. Principle and Procedure of Empirical Correlation Matrix Analysis Based on RMT

The principle of universality is one of the most important reasons for the success of RMT in multiple research fields; that is, under the average energy level interval scale, the eigenvalue distribution law does not depend on the specific distribution of elements in the system. By using this characteristic, we can obtain the essence of the internal interaction of the intended research system by comparing the similarities and differences between the research system and the random system in the distribution of eigenvalues and eigenvector elements. In other words, the random properties and special nonrandom properties inside the system can be investigated by studying the universality properties of the matrix of the research system and the random system through comparative analysis.To explore the internal structure and dynamic behavior of the economic system, the most important thing is to study the statistical characteristics of price fluctuations and the relationship between different stocks, which can be converted into the statistical characteristics of the empirical cross-correlation matrix of the stock market. Through the theoretical analysis of the information structure of the empirical correlation matrix, the content of the empirical correlation matrix can be divided into “information” and “noise,” so identifying the “information” and “noise” in the empirical correlation matrix has become an important way to study the statistical characteristics of the empirical correlation matrix. Based on the universality of random matrix theory studied above, this paper intends to compare and analyze the statistical characteristics of the empirical correlation matrix and real symmetric random correlation matrix of three typical industries in the Chinese stock market, and then study the nonrandom and random characteristics of the empirical correlation matrix. The former is the “information,” and the latter is the “noise.”Specific analysis procedures are as follows:(1)
Calculate Pearson’s correlation coefficient (PCC)(2)
Construct the empirical correlation matrix and obtain its eigenvalues(3)
Analyze the statistical characteristics of the correlation coefficients, eigenvalues, and eigenvectors


## 2.1.1. Financial Correlation Coefficient Matrix

Recording the valid trading day price sequence of stockii=1,2,···,N: Pi1,Pi2,···,PiN. Where Ni represents the valid trading days of stock i, Pit represents the closing price of stock i on the valid trading day t, and we define the logarithmic return of stock i as follows:(1)rit=lnPit−lnPit−1.For any two stocksi and j, since their valid trading price series are different, when calculating the correlation, we take out their common valid trading daytime series: u1,u2,···,um, where m represents the total number of common valid trading days of stocks i and j.Thus, the Pearson correlation coefficient between any two stocksi and j is as follows:(2)ρij=rirj>−<ri><rj<ri2>−<ri>2<rj2>−<rj>2,where < ··· > is the statistical average, and the value range of ρij is [−1, 1]. If ρij=−1, it means that there is a completely negative correlation between the two stocks. If ρij=1, it means that there is a completely positive correlation between the two stocks. If ρij=0, it means that the two stocks are not related. Thus, using equations (1) and (2), we can get the correlation coefficient matrix C of the stock market, which has the size of N×N and the element of ρij.


## 2.1.2. Random Matrix Theory

Considering such a random matrix:R=1/LA×AT, where A is a random matrix of N×L, which is composed of N independent sequences with length L, and each sequence is subject to N0,1 distribution. The statistical characteristics of matrix R are known, especially when N⟶∞, L⟶∞, Q=L/N>1 is fixed, the distribution function Pλ of eigenvalue λ of matrix R has an analytical form:(3)Pλ=Q2πσ2λmax−λλ−λminλ,where λmin⩽λ⩽λmax, and(4)λminmax=σ21+1Q±21Q,and σ2=1.


## 2.1.3. Principle and Procedure of Empirical Correlation Matrix Analysis Based on RMT

The principle of universality is one of the most important reasons for the success of RMT in multiple research fields; that is, under the average energy level interval scale, the eigenvalue distribution law does not depend on the specific distribution of elements in the system. By using this characteristic, we can obtain the essence of the internal interaction of the intended research system by comparing the similarities and differences between the research system and the random system in the distribution of eigenvalues and eigenvector elements. In other words, the random properties and special nonrandom properties inside the system can be investigated by studying the universality properties of the matrix of the research system and the random system through comparative analysis.To explore the internal structure and dynamic behavior of the economic system, the most important thing is to study the statistical characteristics of price fluctuations and the relationship between different stocks, which can be converted into the statistical characteristics of the empirical cross-correlation matrix of the stock market. Through the theoretical analysis of the information structure of the empirical correlation matrix, the content of the empirical correlation matrix can be divided into “information” and “noise,” so identifying the “information” and “noise” in the empirical correlation matrix has become an important way to study the statistical characteristics of the empirical correlation matrix. Based on the universality of random matrix theory studied above, this paper intends to compare and analyze the statistical characteristics of the empirical correlation matrix and real symmetric random correlation matrix of three typical industries in the Chinese stock market, and then study the nonrandom and random characteristics of the empirical correlation matrix. The former is the “information,” and the latter is the “noise.”Specific analysis procedures are as follows:(1)
Calculate Pearson’s correlation coefficient (PCC)(2)
Construct the empirical correlation matrix and obtain its eigenvalues(3)
Analyze the statistical characteristics of the correlation coefficients, eigenvalues, and eigenvectors


## 2.2. Variable Description and Data Source

This paper selects the closing prices of 98 constituent stocks of the three major indices: Shanghai Stock Exchange Consumption (30 constituent stocks), Shanghai Stock Exchange Medicine (38 constituent stocks), and Shanghai Stock Exchange Finance (30 constituent stocks) from January 2, 2020, to June 3, 2020, to measure the daily returns.The reasons for selecting the data of 98 constituent stocks of the three major Shanghai Composite indexes from January 2, 2020, to June 3, 2020, are as follows:(1)
This paper mainly studies the reaction of the three industries in the stock market in a short time after the impact of COVID-19 and selects all the constituent stocks in the industry indices to be representative.(2)
The constituent stocks of the index will be replaced in June and December every year, and the constituent stocks of the three industry indices will remain unchanged within the research range, which makes the research more accurate.When analyzing the correlation matrix of empirical returns, the datum ofN=98 and L=100 are selected for analysis. When further analyzing the linkage between the three industries of consumption, medicine, and finance, the daily returns of Shanghai Stock Exchange Consumption Index (Consumption), Shanghai Stock Exchange Medicine Index (Medicine), and Shanghai Stock Exchange Finance Index (Finance) are selected for analysis.The specific stock codes are shown in Table1.Table 1
Description of 98 stocks in the consumption, pharmaceutical, and financial industries.
IndustryStock codeConsumption600132, 600298, 600315, 600438, 600519, 600597, 600598, 600600, 600737, 600779, 600809, 600811, 600827, 600872, 600873, 600887, 601933, 603043, 603156, 603198, 603288, 603317, 603369, 603517, 603589, 603605, 603708, 603711, 603866, 603983Pharmaceutical600056, 600062, 600079, 600085, 600161, 600195, 600196, 600201, 600216, 600252, 600267, 600276, 600299, 600332, 600380, 600420, 600436, 600511, 600521, 600535, 600566, 600572, 600664, 600763, 600771, 600867, 600998, 601607, 603222, 603233, 603259, 603658, 603707, 603718, 603858, 603882, 603883, 603939Financial600000, 600015, 600016, 600030, 600036, 600048, 600340, 600606, 600837, 600919, 600999, 601066, 601155, 601166, 601169, 601211, 601229, 601288, 601318, 601319, 601328, 601336, 601398, 601601, 601628, 601688, 601818, 601939, 601988, 601998


## 3. Empirical Results

### 3.1. Descriptive Statistics

Figures1(a)–1(c) shows the sequence charts of 100 day daily returns of 98 selected constituent stocks in the consumption industry, the pharmaceutical industry, and the financial industry, while Figure 1(d) is the sequence chart of 100 day daily returns of three industry indices. The sequence charts of three industry indices prices are in accordance with the evidence obtained by Lo and Mackonlay [40] and Mehla and Goyal [41], which showed that the indices prices series are nonstationary. It can be seen from the figures that(1)
On February 3, 2020, the indices of the consumption industry, the pharmaceutical industry, and the financial industry were all affected by COVID-19, and investors sold their stocks in short hands sharply, which caused stock market turbulence. The stock prices of the three industries hit new lows one after another, among which the consumption industry was the most affected, with a return of about −10% on that day; the financial industry was next affected, with a return of about −8% on that day, and the pharmaceutical industry was the least affected, with a return of about −3% on that day. Even from the perspective of individual stocks, the returns of some pharmaceutical constituent stocks rose slightly on that day. The reasons for the sharp decline in the daily returns of the three major industries were that the epidemic had suppressed the consumption demands of residents, the consumption data had been clearly reflected during the Spring Festival, and offline consumption had been seriously damaged. For example, a large number of prepared vegetables in restaurants were sold at low prices, waiters were transferred to logistics work, and new business forms might appear, but the losses of the consumption industry were still serious. The financial industry might suffer from the rise of nonperforming ratio and the decline of securities trading volume in the future. As N98 masks and disposable medical rubber gloves suddenly became necessities during the epidemic period, the pharmaceutical industry was still favored by investors despite the decline of stock prices, and the returns of individual stocks still rose slightly on the same day.(2)
After the sharp fluctuations in the stock market on February 3, 2020, the three major industry indices rose rapidly the next day. The overall return of the pharmaceutical industry was the largest, followed by the consumption industry, and the financial industry was the smallest. The reason for the sharp rise in the daily returns of the three industries the next day was that the central bank and the government rescued the market. The People’s Bank of China invested 1.2 trillion yuan in open market reverse repo operations to ensure sufficient liquidity supply. The overall liquidity of the banking system was 900 billion yuan more than that of the same period last year. In one day on February 3, 2020, alone, about 10 billion insurance funds had been copied into the market. According to sources, many large insurance institutions had also issued internal instructions to equity investment managers: net sales were not allowed on the same day.(3)
The volatility of the return of the financial industry was the smallest, while the volatilities of the returns of the consumption industry and the pharmaceutical industry were larger. The reason was that financial stocks were big blue chips, and most of the short-term fluctuations were caused by short-term speculation. For example, the outbreak of the epidemic made investors pessimistic about their stocks, leading to a large sale of stocks. With the intervention of the central bank and the government and the promulgation of a number of favorable policies, the stock market was gradually stable, and the return of the financial industry fluctuated little.Figure 1
Sequence charts of daily returns of 98 stocks in the three major industries and the three major industry indices. (a) A daily return sequence chart of 30 stocks in the consumption industry, (b) a daily return sequence chart of 38 stocks in the pharmaceutical industry, (c) a daily return sequence chart of 30 stocks in the financial industry, (d) a daily return sequence chart of Shanghai consumption, Shanghai medicine, and Shanghai finance.
(a)(b)(c)(d)The stocks of the consumption industry and the pharmaceutical industry were different from those of the financial industry. The stock price and the scale of listed companies were different, especially in the pharmaceutical industry. With the spread of the epidemic in March, new medicines were put into production, and a large number of people rushed to buy medical supplies, which made investors think that pharmaceutical stocks were profitable in the short term. The investors, therefore, increased the frequency of position adjustments in pharmaceutical stocks, resulting in a large fluctuation in the returns of pharmaceutical stocks during the epidemic. It was the same reason that the returns on consumption stocks fluctuate violently. Although offline consumption was damaged, the epidemic forced people to stay at home, and the online consumption industry began to improve. Investors were bullish on stocks in the consumption industry, and frequently sold and bought their constituent stocks in the short term, resulting in large fluctuations in the returns of consumption stocks during the epidemic.


### 3.2. Analysis of Statistical Characteristics of the Correlation Coefficient

This paper first studies the distribution of the correlation coefficientCiji≠j. Figure 2 shows the distribution histogram of the empirical correlation matrix. It can be seen from the figure that (1) Cij approximately obeys a symmetric normal distribution with a mean value greater than 0 (a mean value equals 0.3564). (2) Most of the correlation coefficients are positive, indicating that there are basically positive correlations between the sample datum. (3) The phenomenon of “fat tail” indicates that the frequency of extreme returns is higher than the prediction of normal distribution.Figure 2
Distribution histogram of the correlation coefficient.Let’s take a look at the correlation of other countries’ stock markets. The average correlation coefficient of the US market from 1964 to 1996 was 0.1, and from 1983 to 1989 it was 0.18, which is also the highest average in history. It was 0.03 in 1994-1995 and 0.06 in 1996-1997 [19]. Therefore, it can be concluded that during the epidemic period, the returns of stock prices show abnormal values, mainly because of the sharp decline on February 3, 2020, and the sharp rise on February 4, 2020. The samples in the major three industry indices taken in this paper show a high positive correlation. At the same time, it also shows that the investment diversification ability of the Chinese stock market and the market efficiency are weak, which is far behind mature markets such as the United States. In other words, the dynamical changes were impacted by the complex behavior of the market crash (the outbreak of COVID-19), unlike the case of random interactions. Our findings confirm that the possible interactions in the Chinese stock market deviated from those for the random interaction [42].


### 3.3. Analysis of Statistical Characteristics of Eigenvalues

#### 3.3.1. Distribution of Eigenvalues

Then, the eigenvalue distribution of matrixC is compared with that of random matrix R. Table 2 shows the comparison of eigenvalues of the empirical correlation matrix, the random matrix, and the theoretical prediction. It can be found from the table that the eigenvalues of the empirical correlation matrix are quite different from the prediction. Most of the eigenvalues fall within the range of prediction values. There are also three larger eigenvalues that deviate from the prediction values. In particular, the maximum eigenvalue of the empirical correlation matrix (λempirial98=40.0301) is about 11.18 times larger than the largest eigenvalue of the RMT (λRMT98=3.5806). The eigenvalues of the random correlation matrix are all in the prediction value of the whole range [1.0101e − 04, 3.9599]. Although the empirical data are different, some scholars have had come to similar conclusions: Utsugi et al. studied the daily prices of 297 stocks in the S&P500 index of the NYSE from January 1991 to July 2001 and found that the maximum eigenvalue (=52.2) was about 29 times larger than the maximum eigenvalue predicted by RMT (=1.79) [43]. Oh et al. studied the statistical characteristics of the cross-correlation of Korean stock markets by using the RMT method and found that the maximum eigenvalue was 52 times of the maximum eigenvalue predicted by RMT [22].Table 2
Comparison of the eigenvalues of the empirical correlation matrix, random correlation matrix, and prediction.
Empirical correlation matrixRandom correlation matrixRMT predictionN100100100L989898Q1.02041.02041.020Min-eigen2.0664e − 041.9001e − 041.0101e − 04Max-eigen40.03013.58063.9599In Figure3, it compares the probability density distribution differences of the eigenvalues of the empirical correlation matrix, and the random matrix under the condition of Q=1.0204. As shown in the figure, the density distribution of the empirical correlation matrix has a relatively smaller difference than that of the random correlation matrix.Figure 3
Comparison of eigenvalue density distributions of empirical correlation matrix and the random correlation matrix.


#### 3.3.2. Eigenvalue Entropy

Through the comparison of the eigenvalue differences between the empirical correlation matrix and the random correlation matrix above, it can be found that there are indeed several large eigenvalues beyond the prediction range. Kenett et al. [44] drew lessons from the concept of entropy in the physical field and defined eigenvalue entropy to measure the amount of information of eigenvalues. It is defined as follows:(5)SE=−1logN∑i=1NΩilogΩi,where Ωi=wi2/∑i=1Nwi2, wi=λi−λi−1, λ is the eigenvalue of the correlation matrix.The characteristics of eigenvalue entropy are as follows:(1)
SE≥0, if and only when one of Ω1,Ω2,···,Ωn is 1, that is, there is only one maximum eigenvalue, and other eigenvalues are all 0, SE=0. In other cases, the eigenvalue entropy is a constant greater than 0.(2)
For a givenN, when Ω1,Ω2,···,Ωn is all equal, that is, when the eigenvalues are equally spaced, the maximum eigenvalue entropy is 1.(3)
The larger the amount of information is, the lower the entropy of the eigenvalue is, and the closer the entropy is to 0. On the contrary, the smaller the amount of information is, the greater the entropy of the eigenvalue is, and the closer the entropy is to 1.As can be seen from Table3,(1)
There is a great difference in the amount of information between the empirical correlation matrix and the random correlation matrix. The eigenvalue entropy of the random correlation matrix is close to 1, indicating that the amount of information is very small and the information has little value, which is in line with the efficient market hypothesis. The eigenvalue entropy of the empirical correlation matrix is close to 0, indicating that the empirical correlation matrix has a large amount of information.(2)
For the empirical correlation matrix, there is a significant difference between the eigenvalue entropy with and without the maximum eigenvalue. However, there is no significant difference in the random correlation matrix. The eigenvalue entropy without the maximum eigenvalue of the empirical correlation matrix is 0.2459, which is significantly higher than the original eigenvalue entropy of 0.0392, indicating that the maximum eigenvalue entropy has a decisive impact on the amount of information of the matrix.(3)
After removing three larger eigenvalues beyond the range, the eigenvalue entropy of the empirical correlation matrix increases, while the eigenvalue entropy of the random correlation matrix decreases. It shows that other large eigenvalues also contain a certain amount of information.Table 3
Comparison of eigenvalue entropies between an empirical correlation matrix and a random matrix under different conditions.
Empirical matrixRandom matrixEigenvalue entropy with the max-eigenvalue0.03920.7592Eigenvalue entropy without the max-eigenvalue0.24590.7444Eigenvalue entropy without three large eigenvalues0.37540.7255Therefore, it can be concluded that the maximum eigenvalue contains most of the real information of the matrix (that is, the stock market), while other larger eigenvalues also contain a small amount of real information.


#### 3.3.3. Eigenvalue and Correlation Coefficient

From the comparison of eigenvalue entropies, it can be found that the real information of the empirical correlation matrix depends on the maximum eigenvalue. This section further analyzes the relationship between the maximum eigenvalue and the correlation coefficient.In the 100-day observation period, with 20 days as the fixed time window and 5 as the sliding unit, 17 subobservation periods are generated. We calculate the correlation matrix of each period, the average correlation matrixCiji≠j and eigenvalue. Then, we study the relationship between them.(6)Ctf,tl=C1+j,20+j,j=0,5,10,15,20,‥‥‥,75,80,where tf and tl are the start time and end time, respectively.It can be seen from Figure4 that there is a strong positive linear relationship between the maximum eigenvalue and the correlation coefficient. The specific regression equation is ACC = 0.0097 ∗ ME−0.00488, the R-square is 0.9674, and adjust the R-square is 0.9653, indicating that the maximum eigenvalue have a significant positive promoting effect on its correlation coefficient.Figure 4
Regression diagram of the maximum eigenvalue and average correlation coefficient.In order to further study the relationship between eigenvalues and correlation coefficients, we observe the distribution histograms of correlation coefficients without the maximum eigenvalues and without the eigenvalues beyond the prediction range.The specific steps are as follows:(1)
The eigenvalues of the correlation matrixC are calculated, and the sequence of all eigenvalues from large to small is reduced to a diagonal matrix D.The eigenvectors corresponding to the eigenvalues are calculated, and each eigenvector is arranged in columns to obtain matrix E.(2)
Then the corresponding eigenvalues are changed to 0 to get two new matrices: the first is to change the maximum eigenvalue of matrixD to 0, the remaining eigenvalues remain unchanged, and to record the new matrix as M; the second is to change the eigenvalue beyond the prediction range to 0, and the remaining unchanged, and to record the new matrix as F.(3)
Finally, the correlation coefficient distribution histogram is obtained according toN1=EME−1 and N2=EFE−1.It can be seen from Figure5 that, after removing the maximum eigenvalue, the coefficient mean is close to 0, but there is still a certain difference from the standard normal distribution. After removing the eigenvalues beyond the prediction range, the coefficient distribution basically conforms to the standard normal distribution. This fully verifies that a part of the empirical correlation matrix C coincides with the random correlation matrix R, that is, the matrix C contains noise.Figure 5
Histograms of correlation coefficients without the maximum eigenvalues and without the eigenvalues beyond the prediction range. (a) Histogram of the distribution of correlation coefficients without the maximum eigenvalue and (b) histogram of the distribution of correlation coefficients without the maximum eigenvalues beyond the prediction range.
(a)(b)


### 3.4. Analysis of Statistical Characteristics of Eigenvectors

The elements of the eigenvectorulk;l=1,···,N of the random matrix R follow a standard normal distribution(7)ρrmu=12πexp−u22.As shown in Figure6, this section compares the eigenvector distribution of the maximum eigenvalue and the minimum eigenvalue of matrix C with the corresponding normal distribution. It is found that the distribution of the eigenvector corresponding to the minimum eigenvalue is basically consistent with the random matrix distribution, and the distribution on both sides of 0 is uniform.Figure 6
Distribution histograms of eigenvectors corresponding to the minimum and maximum eigenvalues. (a) Histogram of the eigenvector corresponding to the minimum eigenvalue, (b) histogram of the eigenvector corresponding to the maximum eigenvalue.
(a)(b)The distribution of the eigenvector corresponding to the maximum eigenvalue is obviously different from the random matrix (normal) distribution. The elements generally tend to the left of 0, and the distribution is uneven. Thus, it is further verified that the empirical correlation matrixC contains the random part (noise) and the maximum eigenvalue, which contains the real market information. For deviating eigenvalues, Zhou et al. [45] also found that the distribution for each stock was very noisy and deviated from Gaussian.


## 3.1. Descriptive Statistics

Figures1(a)–1(c) shows the sequence charts of 100 day daily returns of 98 selected constituent stocks in the consumption industry, the pharmaceutical industry, and the financial industry, while Figure 1(d) is the sequence chart of 100 day daily returns of three industry indices. The sequence charts of three industry indices prices are in accordance with the evidence obtained by Lo and Mackonlay [40] and Mehla and Goyal [41], which showed that the indices prices series are nonstationary. It can be seen from the figures that(1)
On February 3, 2020, the indices of the consumption industry, the pharmaceutical industry, and the financial industry were all affected by COVID-19, and investors sold their stocks in short hands sharply, which caused stock market turbulence. The stock prices of the three industries hit new lows one after another, among which the consumption industry was the most affected, with a return of about −10% on that day; the financial industry was next affected, with a return of about −8% on that day, and the pharmaceutical industry was the least affected, with a return of about −3% on that day. Even from the perspective of individual stocks, the returns of some pharmaceutical constituent stocks rose slightly on that day. The reasons for the sharp decline in the daily returns of the three major industries were that the epidemic had suppressed the consumption demands of residents, the consumption data had been clearly reflected during the Spring Festival, and offline consumption had been seriously damaged. For example, a large number of prepared vegetables in restaurants were sold at low prices, waiters were transferred to logistics work, and new business forms might appear, but the losses of the consumption industry were still serious. The financial industry might suffer from the rise of nonperforming ratio and the decline of securities trading volume in the future. As N98 masks and disposable medical rubber gloves suddenly became necessities during the epidemic period, the pharmaceutical industry was still favored by investors despite the decline of stock prices, and the returns of individual stocks still rose slightly on the same day.(2)
After the sharp fluctuations in the stock market on February 3, 2020, the three major industry indices rose rapidly the next day. The overall return of the pharmaceutical industry was the largest, followed by the consumption industry, and the financial industry was the smallest. The reason for the sharp rise in the daily returns of the three industries the next day was that the central bank and the government rescued the market. The People’s Bank of China invested 1.2 trillion yuan in open market reverse repo operations to ensure sufficient liquidity supply. The overall liquidity of the banking system was 900 billion yuan more than that of the same period last year. In one day on February 3, 2020, alone, about 10 billion insurance funds had been copied into the market. According to sources, many large insurance institutions had also issued internal instructions to equity investment managers: net sales were not allowed on the same day.(3)
The volatility of the return of the financial industry was the smallest, while the volatilities of the returns of the consumption industry and the pharmaceutical industry were larger. The reason was that financial stocks were big blue chips, and most of the short-term fluctuations were caused by short-term speculation. For example, the outbreak of the epidemic made investors pessimistic about their stocks, leading to a large sale of stocks. With the intervention of the central bank and the government and the promulgation of a number of favorable policies, the stock market was gradually stable, and the return of the financial industry fluctuated little.Figure 1
Sequence charts of daily returns of 98 stocks in the three major industries and the three major industry indices. (a) A daily return sequence chart of 30 stocks in the consumption industry, (b) a daily return sequence chart of 38 stocks in the pharmaceutical industry, (c) a daily return sequence chart of 30 stocks in the financial industry, (d) a daily return sequence chart of Shanghai consumption, Shanghai medicine, and Shanghai finance.
(a)(b)(c)(d)The stocks of the consumption industry and the pharmaceutical industry were different from those of the financial industry. The stock price and the scale of listed companies were different, especially in the pharmaceutical industry. With the spread of the epidemic in March, new medicines were put into production, and a large number of people rushed to buy medical supplies, which made investors think that pharmaceutical stocks were profitable in the short term. The investors, therefore, increased the frequency of position adjustments in pharmaceutical stocks, resulting in a large fluctuation in the returns of pharmaceutical stocks during the epidemic. It was the same reason that the returns on consumption stocks fluctuate violently. Although offline consumption was damaged, the epidemic forced people to stay at home, and the online consumption industry began to improve. Investors were bullish on stocks in the consumption industry, and frequently sold and bought their constituent stocks in the short term, resulting in large fluctuations in the returns of consumption stocks during the epidemic.


## 3.2. Analysis of Statistical Characteristics of the Correlation Coefficient

This paper first studies the distribution of the correlation coefficientCiji≠j. Figure 2 shows the distribution histogram of the empirical correlation matrix. It can be seen from the figure that (1) Cij approximately obeys a symmetric normal distribution with a mean value greater than 0 (a mean value equals 0.3564). (2) Most of the correlation coefficients are positive, indicating that there are basically positive correlations between the sample datum. (3) The phenomenon of “fat tail” indicates that the frequency of extreme returns is higher than the prediction of normal distribution.Figure 2
Distribution histogram of the correlation coefficient.Let’s take a look at the correlation of other countries’ stock markets. The average correlation coefficient of the US market from 1964 to 1996 was 0.1, and from 1983 to 1989 it was 0.18, which is also the highest average in history. It was 0.03 in 1994-1995 and 0.06 in 1996-1997 [19]. Therefore, it can be concluded that during the epidemic period, the returns of stock prices show abnormal values, mainly because of the sharp decline on February 3, 2020, and the sharp rise on February 4, 2020. The samples in the major three industry indices taken in this paper show a high positive correlation. At the same time, it also shows that the investment diversification ability of the Chinese stock market and the market efficiency are weak, which is far behind mature markets such as the United States. In other words, the dynamical changes were impacted by the complex behavior of the market crash (the outbreak of COVID-19), unlike the case of random interactions. Our findings confirm that the possible interactions in the Chinese stock market deviated from those for the random interaction [42].


## 3.3. Analysis of Statistical Characteristics of Eigenvalues

### 3.3.1. Distribution of Eigenvalues

Then, the eigenvalue distribution of matrixC is compared with that of random matrix R. Table 2 shows the comparison of eigenvalues of the empirical correlation matrix, the random matrix, and the theoretical prediction. It can be found from the table that the eigenvalues of the empirical correlation matrix are quite different from the prediction. Most of the eigenvalues fall within the range of prediction values. There are also three larger eigenvalues that deviate from the prediction values. In particular, the maximum eigenvalue of the empirical correlation matrix (λempirial98=40.0301) is about 11.18 times larger than the largest eigenvalue of the RMT (λRMT98=3.5806). The eigenvalues of the random correlation matrix are all in the prediction value of the whole range [1.0101e − 04, 3.9599]. Although the empirical data are different, some scholars have had come to similar conclusions: Utsugi et al. studied the daily prices of 297 stocks in the S&P500 index of the NYSE from January 1991 to July 2001 and found that the maximum eigenvalue (=52.2) was about 29 times larger than the maximum eigenvalue predicted by RMT (=1.79) [43]. Oh et al. studied the statistical characteristics of the cross-correlation of Korean stock markets by using the RMT method and found that the maximum eigenvalue was 52 times of the maximum eigenvalue predicted by RMT [22].Table 2
Comparison of the eigenvalues of the empirical correlation matrix, random correlation matrix, and prediction.
Empirical correlation matrixRandom correlation matrixRMT predictionN100100100L989898Q1.02041.02041.020Min-eigen2.0664e − 041.9001e − 041.0101e − 04Max-eigen40.03013.58063.9599In Figure3, it compares the probability density distribution differences of the eigenvalues of the empirical correlation matrix, and the random matrix under the condition of Q=1.0204. As shown in the figure, the density distribution of the empirical correlation matrix has a relatively smaller difference than that of the random correlation matrix.Figure 3
Comparison of eigenvalue density distributions of empirical correlation matrix and the random correlation matrix.


### 3.3.2. Eigenvalue Entropy

Through the comparison of the eigenvalue differences between the empirical correlation matrix and the random correlation matrix above, it can be found that there are indeed several large eigenvalues beyond the prediction range. Kenett et al. [44] drew lessons from the concept of entropy in the physical field and defined eigenvalue entropy to measure the amount of information of eigenvalues. It is defined as follows:(5)SE=−1logN∑i=1NΩilogΩi,where Ωi=wi2/∑i=1Nwi2, wi=λi−λi−1, λ is the eigenvalue of the correlation matrix.The characteristics of eigenvalue entropy are as follows:(1)
SE≥0, if and only when one of Ω1,Ω2,···,Ωn is 1, that is, there is only one maximum eigenvalue, and other eigenvalues are all 0, SE=0. In other cases, the eigenvalue entropy is a constant greater than 0.(2)
For a givenN, when Ω1,Ω2,···,Ωn is all equal, that is, when the eigenvalues are equally spaced, the maximum eigenvalue entropy is 1.(3)
The larger the amount of information is, the lower the entropy of the eigenvalue is, and the closer the entropy is to 0. On the contrary, the smaller the amount of information is, the greater the entropy of the eigenvalue is, and the closer the entropy is to 1.As can be seen from Table3,(1)
There is a great difference in the amount of information between the empirical correlation matrix and the random correlation matrix. The eigenvalue entropy of the random correlation matrix is close to 1, indicating that the amount of information is very small and the information has little value, which is in line with the efficient market hypothesis. The eigenvalue entropy of the empirical correlation matrix is close to 0, indicating that the empirical correlation matrix has a large amount of information.(2)
For the empirical correlation matrix, there is a significant difference between the eigenvalue entropy with and without the maximum eigenvalue. However, there is no significant difference in the random correlation matrix. The eigenvalue entropy without the maximum eigenvalue of the empirical correlation matrix is 0.2459, which is significantly higher than the original eigenvalue entropy of 0.0392, indicating that the maximum eigenvalue entropy has a decisive impact on the amount of information of the matrix.(3)
After removing three larger eigenvalues beyond the range, the eigenvalue entropy of the empirical correlation matrix increases, while the eigenvalue entropy of the random correlation matrix decreases. It shows that other large eigenvalues also contain a certain amount of information.Table 3
Comparison of eigenvalue entropies between an empirical correlation matrix and a random matrix under different conditions.
Empirical matrixRandom matrixEigenvalue entropy with the max-eigenvalue0.03920.7592Eigenvalue entropy without the max-eigenvalue0.24590.7444Eigenvalue entropy without three large eigenvalues0.37540.7255Therefore, it can be concluded that the maximum eigenvalue contains most of the real information of the matrix (that is, the stock market), while other larger eigenvalues also contain a small amount of real information.


### 3.3.3. Eigenvalue and Correlation Coefficient

From the comparison of eigenvalue entropies, it can be found that the real information of the empirical correlation matrix depends on the maximum eigenvalue. This section further analyzes the relationship between the maximum eigenvalue and the correlation coefficient.In the 100-day observation period, with 20 days as the fixed time window and 5 as the sliding unit, 17 subobservation periods are generated. We calculate the correlation matrix of each period, the average correlation matrixCiji≠j and eigenvalue. Then, we study the relationship between them.(6)Ctf,tl=C1+j,20+j,j=0,5,10,15,20,‥‥‥,75,80,where tf and tl are the start time and end time, respectively.It can be seen from Figure4 that there is a strong positive linear relationship between the maximum eigenvalue and the correlation coefficient. The specific regression equation is ACC = 0.0097 ∗ ME−0.00488, the R-square is 0.9674, and adjust the R-square is 0.9653, indicating that the maximum eigenvalue have a significant positive promoting effect on its correlation coefficient.Figure 4
Regression diagram of the maximum eigenvalue and average correlation coefficient.In order to further study the relationship between eigenvalues and correlation coefficients, we observe the distribution histograms of correlation coefficients without the maximum eigenvalues and without the eigenvalues beyond the prediction range.The specific steps are as follows:(1)
The eigenvalues of the correlation matrixC are calculated, and the sequence of all eigenvalues from large to small is reduced to a diagonal matrix D.The eigenvectors corresponding to the eigenvalues are calculated, and each eigenvector is arranged in columns to obtain matrix E.(2)
Then the corresponding eigenvalues are changed to 0 to get two new matrices: the first is to change the maximum eigenvalue of matrixD to 0, the remaining eigenvalues remain unchanged, and to record the new matrix as M; the second is to change the eigenvalue beyond the prediction range to 0, and the remaining unchanged, and to record the new matrix as F.(3)
Finally, the correlation coefficient distribution histogram is obtained according toN1=EME−1 and N2=EFE−1.It can be seen from Figure5 that, after removing the maximum eigenvalue, the coefficient mean is close to 0, but there is still a certain difference from the standard normal distribution. After removing the eigenvalues beyond the prediction range, the coefficient distribution basically conforms to the standard normal distribution. This fully verifies that a part of the empirical correlation matrix C coincides with the random correlation matrix R, that is, the matrix C contains noise.Figure 5
Histograms of correlation coefficients without the maximum eigenvalues and without the eigenvalues beyond the prediction range. (a) Histogram of the distribution of correlation coefficients without the maximum eigenvalue and (b) histogram of the distribution of correlation coefficients without the maximum eigenvalues beyond the prediction range.
(a)(b)


## 3.3.1. Distribution of Eigenvalues

Then, the eigenvalue distribution of matrixC is compared with that of random matrix R. Table 2 shows the comparison of eigenvalues of the empirical correlation matrix, the random matrix, and the theoretical prediction. It can be found from the table that the eigenvalues of the empirical correlation matrix are quite different from the prediction. Most of the eigenvalues fall within the range of prediction values. There are also three larger eigenvalues that deviate from the prediction values. In particular, the maximum eigenvalue of the empirical correlation matrix (λempirial98=40.0301) is about 11.18 times larger than the largest eigenvalue of the RMT (λRMT98=3.5806). The eigenvalues of the random correlation matrix are all in the prediction value of the whole range [1.0101e − 04, 3.9599]. Although the empirical data are different, some scholars have had come to similar conclusions: Utsugi et al. studied the daily prices of 297 stocks in the S&P500 index of the NYSE from January 1991 to July 2001 and found that the maximum eigenvalue (=52.2) was about 29 times larger than the maximum eigenvalue predicted by RMT (=1.79) [43]. Oh et al. studied the statistical characteristics of the cross-correlation of Korean stock markets by using the RMT method and found that the maximum eigenvalue was 52 times of the maximum eigenvalue predicted by RMT [22].Table 2
Comparison of the eigenvalues of the empirical correlation matrix, random correlation matrix, and prediction.
Empirical correlation matrixRandom correlation matrixRMT predictionN100100100L989898Q1.02041.02041.020Min-eigen2.0664e − 041.9001e − 041.0101e − 04Max-eigen40.03013.58063.9599In Figure3, it compares the probability density distribution differences of the eigenvalues of the empirical correlation matrix, and the random matrix under the condition of Q=1.0204. As shown in the figure, the density distribution of the empirical correlation matrix has a relatively smaller difference than that of the random correlation matrix.Figure 3
Comparison of eigenvalue density distributions of empirical correlation matrix and the random correlation matrix.


## 3.3.2. Eigenvalue Entropy

Through the comparison of the eigenvalue differences between the empirical correlation matrix and the random correlation matrix above, it can be found that there are indeed several large eigenvalues beyond the prediction range. Kenett et al. [44] drew lessons from the concept of entropy in the physical field and defined eigenvalue entropy to measure the amount of information of eigenvalues. It is defined as follows:(5)SE=−1logN∑i=1NΩilogΩi,where Ωi=wi2/∑i=1Nwi2, wi=λi−λi−1, λ is the eigenvalue of the correlation matrix.The characteristics of eigenvalue entropy are as follows:(1)
SE≥0, if and only when one of Ω1,Ω2,···,Ωn is 1, that is, there is only one maximum eigenvalue, and other eigenvalues are all 0, SE=0. In other cases, the eigenvalue entropy is a constant greater than 0.(2)
For a givenN, when Ω1,Ω2,···,Ωn is all equal, that is, when the eigenvalues are equally spaced, the maximum eigenvalue entropy is 1.(3)
The larger the amount of information is, the lower the entropy of the eigenvalue is, and the closer the entropy is to 0. On the contrary, the smaller the amount of information is, the greater the entropy of the eigenvalue is, and the closer the entropy is to 1.As can be seen from Table3,(1)
There is a great difference in the amount of information between the empirical correlation matrix and the random correlation matrix. The eigenvalue entropy of the random correlation matrix is close to 1, indicating that the amount of information is very small and the information has little value, which is in line with the efficient market hypothesis. The eigenvalue entropy of the empirical correlation matrix is close to 0, indicating that the empirical correlation matrix has a large amount of information.(2)
For the empirical correlation matrix, there is a significant difference between the eigenvalue entropy with and without the maximum eigenvalue. However, there is no significant difference in the random correlation matrix. The eigenvalue entropy without the maximum eigenvalue of the empirical correlation matrix is 0.2459, which is significantly higher than the original eigenvalue entropy of 0.0392, indicating that the maximum eigenvalue entropy has a decisive impact on the amount of information of the matrix.(3)
After removing three larger eigenvalues beyond the range, the eigenvalue entropy of the empirical correlation matrix increases, while the eigenvalue entropy of the random correlation matrix decreases. It shows that other large eigenvalues also contain a certain amount of information.Table 3
Comparison of eigenvalue entropies between an empirical correlation matrix and a random matrix under different conditions.
Empirical matrixRandom matrixEigenvalue entropy with the max-eigenvalue0.03920.7592Eigenvalue entropy without the max-eigenvalue0.24590.7444Eigenvalue entropy without three large eigenvalues0.37540.7255Therefore, it can be concluded that the maximum eigenvalue contains most of the real information of the matrix (that is, the stock market), while other larger eigenvalues also contain a small amount of real information.


## 3.3.3. Eigenvalue and Correlation Coefficient

From the comparison of eigenvalue entropies, it can be found that the real information of the empirical correlation matrix depends on the maximum eigenvalue. This section further analyzes the relationship between the maximum eigenvalue and the correlation coefficient.In the 100-day observation period, with 20 days as the fixed time window and 5 as the sliding unit, 17 subobservation periods are generated. We calculate the correlation matrix of each period, the average correlation matrixCiji≠j and eigenvalue. Then, we study the relationship between them.(6)Ctf,tl=C1+j,20+j,j=0,5,10,15,20,‥‥‥,75,80,where tf and tl are the start time and end time, respectively.It can be seen from Figure4 that there is a strong positive linear relationship between the maximum eigenvalue and the correlation coefficient. The specific regression equation is ACC = 0.0097 ∗ ME−0.00488, the R-square is 0.9674, and adjust the R-square is 0.9653, indicating that the maximum eigenvalue have a significant positive promoting effect on its correlation coefficient.Figure 4
Regression diagram of the maximum eigenvalue and average correlation coefficient.In order to further study the relationship between eigenvalues and correlation coefficients, we observe the distribution histograms of correlation coefficients without the maximum eigenvalues and without the eigenvalues beyond the prediction range.The specific steps are as follows:(1)
The eigenvalues of the correlation matrixC are calculated, and the sequence of all eigenvalues from large to small is reduced to a diagonal matrix D.The eigenvectors corresponding to the eigenvalues are calculated, and each eigenvector is arranged in columns to obtain matrix E.(2)
Then the corresponding eigenvalues are changed to 0 to get two new matrices: the first is to change the maximum eigenvalue of matrixD to 0, the remaining eigenvalues remain unchanged, and to record the new matrix as M; the second is to change the eigenvalue beyond the prediction range to 0, and the remaining unchanged, and to record the new matrix as F.(3)
Finally, the correlation coefficient distribution histogram is obtained according toN1=EME−1 and N2=EFE−1.It can be seen from Figure5 that, after removing the maximum eigenvalue, the coefficient mean is close to 0, but there is still a certain difference from the standard normal distribution. After removing the eigenvalues beyond the prediction range, the coefficient distribution basically conforms to the standard normal distribution. This fully verifies that a part of the empirical correlation matrix C coincides with the random correlation matrix R, that is, the matrix C contains noise.Figure 5
Histograms of correlation coefficients without the maximum eigenvalues and without the eigenvalues beyond the prediction range. (a) Histogram of the distribution of correlation coefficients without the maximum eigenvalue and (b) histogram of the distribution of correlation coefficients without the maximum eigenvalues beyond the prediction range.
(a)(b)


## 3.4. Analysis of Statistical Characteristics of Eigenvectors

The elements of the eigenvectorulk;l=1,···,N of the random matrix R follow a standard normal distribution(7)ρrmu=12πexp−u22.As shown in Figure6, this section compares the eigenvector distribution of the maximum eigenvalue and the minimum eigenvalue of matrix C with the corresponding normal distribution. It is found that the distribution of the eigenvector corresponding to the minimum eigenvalue is basically consistent with the random matrix distribution, and the distribution on both sides of 0 is uniform.Figure 6
Distribution histograms of eigenvectors corresponding to the minimum and maximum eigenvalues. (a) Histogram of the eigenvector corresponding to the minimum eigenvalue, (b) histogram of the eigenvector corresponding to the maximum eigenvalue.
(a)(b)The distribution of the eigenvector corresponding to the maximum eigenvalue is obviously different from the random matrix (normal) distribution. The elements generally tend to the left of 0, and the distribution is uneven. Thus, it is further verified that the empirical correlation matrixC contains the random part (noise) and the maximum eigenvalue, which contains the real market information. For deviating eigenvalues, Zhou et al. [45] also found that the distribution for each stock was very noisy and deviated from Gaussian.


## 4. Further Analysis

The above analysis shows that there is a positive correlation between the returns of the consumption industry, the pharmaceutical industry and the financial industry. The maximum eigenvalue in the empirical correlation matrix has important information and plays a decisive role in the Chinese stock market. In general, the obtained results are obvious, since we have total rejections of the random walk hypothesis and the information efficiency hypothesis. This section further analyzes the interaction between the consumption industry, the pharmaceutical industry, and the financial industry during the epidemic through impulse response function and variance decomposition.


### 4.1. Stability Test

In order to avoid “pseudo regression,” the unit root test is carried out on each variable to verify its stationarity. As shown in Table4, the returns of consumption, medicine, and finance of the Shanghai Stock Exchange (SSE) pass the significance test at the significance level of 1%. Therefore, it can be concluded that all variables reject the original assumption, and the time series data are stable.Table 4
Unit root test of variables.
VariableADF test(c, t, k)Prob1% level5% level10% levelResultConsumption−10.401(1,t, 1)0.0000−3.511−2.891−2.580StableMedicine−9.490(1,t, 1)0.0000−3.511−2.891−2.580StableFinance−10.366(1,t, 1)0.0000−3.511−2.891−2.580Stable


### 4.2. Impulse Response Function

The impulse response function is used to measure the impact trajectory of a standard deviation shock of random disturbance term on the current and future values of other variables. It can more intuitively describe the dynamic interaction and effects between variables. In order to further analyze the dynamic characteristics of consumption, medicine, and finance, impulse response analysis is carried out.As shown in Figure7, for the consumption industry (Consumption), (1) when it is impacted by a positive standard deviation of itself, the impact is first positive and then negative, but the duration of the positive effect is relatively short, reaching a maximum of 0.0175 in the current period, reaching −0.001 in the first period, and returning to zero after the fifth period. It can be seen that the consumption industry has an obvious promoting and then restraining effect on itself, but the degree of impact is short-term and unsustainable. Due to the support of the national macro-control and the government, the return of the consumption industry shows an increasing trend in the current period, but with the basic control of the epidemic, the return of the consumption industry basically returns to stability. (2) The response of the consumption industry to the impact of the financial industry begins to be negative, suddenly becomes positive in the third period, and then becomes negative in the fourth period. From the beginning of the fifth period, the impact gradually disappears. It shows that the financial industry has a restraining effect on the consumption industry in the short term. The main reason is that during the epidemic, the financial industry mainly lent to help the production of medical devices. In the short term, it may inhibit the lending of funds by the consumption industry, resulting in a decline in the return of the consumption industry. With the stability of the epidemic, the return of the consumption industry will also tend to stabilize. (3) The response of the consumption industry to the impact of the pharmaceutical industry begins to be positive; the third and fourth periods are negative, and gradually return to zero after the fifth period. It shows that the pharmaceutical industry has an obvious promoting and then restraining effect on the consumption industry in the short term. The main reason is that after the outbreak of the epidemic, the public demand for masks, gloves, and other medical devices has increased significantly. However, the public can’t go out and buy. In the short term, it will drive online consumption to promote the consumption industry, but it will gradually produce a marginal diminishing effect to inhibit the consumption industry.Figure 7
Impulse response diagrams of the consumption industry, the pharmaceutical industry, and the financial industry.For the financial industry (Finance), (1) when it is impacted by its own standard deviation, the maximum positive response in the current period is 0.0091, which turns negative in the first period and positive in the third period, but turns negative again in the fifth period, and then gradually returns to zero. It shows that the financial industry has a significant role in promoting itself in the short term. The reason is that due to the outbreak of the epidemic and the central bank’s reverse repurchase funds, the return of the financial industry will rise sharply in the short term. (2) The response of the financial industry to the impact of the consumption industry reaches the maximum positive 0.0102 in the current period. After that, although it fluctuates, most of the periods are positive, and only the fourth period briefly changes into a negative effect, which indicates that the consumption industry has a positive role in promoting the financial industry in the short term. During the epidemic period, the public switches to online consumption, and some people will buy goods through bank loans. Meanwhile, the return of the consumption industry gradually increases, and it will inevitably drive the rise of the return of the financial industry in the short term. (3) The response of the financial industry to the impact of the pharmaceutical industry is that the first three periods are positive, the current period has the greatest effect, the third period becomes negative, the fifth period turns positive, and then gradually returns to 0. It shows that the pharmaceutical industry has a promoting and then refraining effect on the financial industry in the short term. The reason is that with the continuous development of the epidemic, banks have also innovated the scope of financial services. Many banks have provided financial service solutions for customers in the pharmaceutical industry, including health and epidemic prevention, drug procurement, and other aspects, and established a green channel for loan approval so as to do a good job in financial security for the pharmaceutical industry fighting on the front line of epidemic prevention. This leads to the promotion of the pharmaceutical industry to the financial industry in the short term, but because the banking business only focuses on the pharmaceutical industry, other businesses are damaged and begin to have a restraining effect.For the pharmaceutical industry (Medicine), (1) when it is impacted by its own standard deviation, the positive effect is the largest in the current period and then gradually decreases, but it is still positive. The third and fourth periods have a negative effect temporarily, and the fifth period turns positive, and then gradually returns to 0. This shows the role of the pharmaceutical industry in promoting itself in the short term. The reason is that during the epidemic, masks and other medical devices have become daily necessities, which will inevitably promote the development of the pharmaceutical industry in the initial stage. However, with the expansion of production lines, masks and other supplies will gradually return to normal after the problem of short supply is solved. (2) The response of the pharmaceutical industry to the impact of the consumption industry is positive in the current period and then changes to negative. It shows that the consumption industry has a promoting and then refraining effect on the pharmaceutical industry in the short term. The reason is that the rise of the return of the consumption industry in the short term includes the purchase of medical supplies by the public, which is bound to promote the pharmaceutical industry. However, when the medical supplies are no longer in short supply, the rise of the return of the consumption industry will crowd out the return of the pharmaceutical industry, resulting in the decline of the return of the pharmaceutical industry. (3) The response of the pharmaceutical industry to the impact of the financial industry is positive in the first period, negative from the second to the fourth period, positive in the fifth period, and then gradually returns to 0. It shows that the financial industry has the effect of promoting and then refraining from promoting the pharmaceutical industry in the short term. The reason is that the establishment of a green channel for loans by banks for the pharmaceutical industry will promote the pharmaceutical industry, but then with the increase of loans in the pharmaceutical industry, it will have a negative impact on its operating efficiency, thereby inhibiting the return of the pharmaceutical industry.


### 4.3. Variance Decomposition

Although the impulse response function can explain the sign and amplitude of the response of each variable to a specific shock, it cannot compare the response intensity of different shocks to a specific variable. In order to further investigate the volatility of the three variables (Consumption, Medicine, and Finance), the variance decomposition is used to decompose the prediction mean square error of the three variables and calculate the relative importance of the impact of each variable.As shown in Table5, in the development process of the consumption industry, the contribution of the consumption industry to itself has a gradual downward trend but has stabilized at 0.9663 since the tenth period, while the contribution of the pharmaceutical industry and the financial industry to the consumption industry has a gradual upward trend, but the proportion is small. It shows that the consumption industry is mainly affected by itself, while the role of the pharmaceutical industry and the financial industry is very small.Table 5
Variance decompositions of the consumption industry, pharmaceutical industry, and financial industry.
VariablePeriodConsumptionMedicineFinanceConsumption000050.96920.52960.5127100.96630.52890.5109150.96630.52890.5109Medicine000050.02280.45520.0640100.02330.45510.0649150.02330.45510.0649Finance000050.00800.01520.4233100.01040.01600.4242150.01040.01600.4242In the development process of the pharmaceutical industry, the contribution of the pharmaceutical industry to itself is basically 0.4551, while the contribution of the consumption industry to the pharmaceutical industry remains at about 0.5289. Although the contribution of the financial industry to the pharmaceutical industry increases period-by-period, it remains at 0.0160 after ten periods. It shows that the pharmaceutical industry is mainly affected by itself and the consumption industry, while the role of the financial industry is very small.In the development process of the financial industry, the contribution of the financial industry to itself is basically 0.4242, while the contribution of the consumption industry to the financial industry remains at about 0.5109. Although the contribution of the pharmaceutical industry to the financial industry increases period by period, it remains at 0.0649 after ten periods. It shows that the financial industry is mainly affected by itself and the consumption industry, while the role of the pharmaceutical industry is very small.


## 4.1. Stability Test

In order to avoid “pseudo regression,” the unit root test is carried out on each variable to verify its stationarity. As shown in Table4, the returns of consumption, medicine, and finance of the Shanghai Stock Exchange (SSE) pass the significance test at the significance level of 1%. Therefore, it can be concluded that all variables reject the original assumption, and the time series data are stable.Table 4
Unit root test of variables.
VariableADF test(c, t, k)Prob1% level5% level10% levelResultConsumption−10.401(1,t, 1)0.0000−3.511−2.891−2.580StableMedicine−9.490(1,t, 1)0.0000−3.511−2.891−2.580StableFinance−10.366(1,t, 1)0.0000−3.511−2.891−2.580Stable


## 4.2. Impulse Response Function

The impulse response function is used to measure the impact trajectory of a standard deviation shock of random disturbance term on the current and future values of other variables. It can more intuitively describe the dynamic interaction and effects between variables. In order to further analyze the dynamic characteristics of consumption, medicine, and finance, impulse response analysis is carried out.As shown in Figure7, for the consumption industry (Consumption), (1) when it is impacted by a positive standard deviation of itself, the impact is first positive and then negative, but the duration of the positive effect is relatively short, reaching a maximum of 0.0175 in the current period, reaching −0.001 in the first period, and returning to zero after the fifth period. It can be seen that the consumption industry has an obvious promoting and then restraining effect on itself, but the degree of impact is short-term and unsustainable. Due to the support of the national macro-control and the government, the return of the consumption industry shows an increasing trend in the current period, but with the basic control of the epidemic, the return of the consumption industry basically returns to stability. (2) The response of the consumption industry to the impact of the financial industry begins to be negative, suddenly becomes positive in the third period, and then becomes negative in the fourth period. From the beginning of the fifth period, the impact gradually disappears. It shows that the financial industry has a restraining effect on the consumption industry in the short term. The main reason is that during the epidemic, the financial industry mainly lent to help the production of medical devices. In the short term, it may inhibit the lending of funds by the consumption industry, resulting in a decline in the return of the consumption industry. With the stability of the epidemic, the return of the consumption industry will also tend to stabilize. (3) The response of the consumption industry to the impact of the pharmaceutical industry begins to be positive; the third and fourth periods are negative, and gradually return to zero after the fifth period. It shows that the pharmaceutical industry has an obvious promoting and then restraining effect on the consumption industry in the short term. The main reason is that after the outbreak of the epidemic, the public demand for masks, gloves, and other medical devices has increased significantly. However, the public can’t go out and buy. In the short term, it will drive online consumption to promote the consumption industry, but it will gradually produce a marginal diminishing effect to inhibit the consumption industry.Figure 7
Impulse response diagrams of the consumption industry, the pharmaceutical industry, and the financial industry.For the financial industry (Finance), (1) when it is impacted by its own standard deviation, the maximum positive response in the current period is 0.0091, which turns negative in the first period and positive in the third period, but turns negative again in the fifth period, and then gradually returns to zero. It shows that the financial industry has a significant role in promoting itself in the short term. The reason is that due to the outbreak of the epidemic and the central bank’s reverse repurchase funds, the return of the financial industry will rise sharply in the short term. (2) The response of the financial industry to the impact of the consumption industry reaches the maximum positive 0.0102 in the current period. After that, although it fluctuates, most of the periods are positive, and only the fourth period briefly changes into a negative effect, which indicates that the consumption industry has a positive role in promoting the financial industry in the short term. During the epidemic period, the public switches to online consumption, and some people will buy goods through bank loans. Meanwhile, the return of the consumption industry gradually increases, and it will inevitably drive the rise of the return of the financial industry in the short term. (3) The response of the financial industry to the impact of the pharmaceutical industry is that the first three periods are positive, the current period has the greatest effect, the third period becomes negative, the fifth period turns positive, and then gradually returns to 0. It shows that the pharmaceutical industry has a promoting and then refraining effect on the financial industry in the short term. The reason is that with the continuous development of the epidemic, banks have also innovated the scope of financial services. Many banks have provided financial service solutions for customers in the pharmaceutical industry, including health and epidemic prevention, drug procurement, and other aspects, and established a green channel for loan approval so as to do a good job in financial security for the pharmaceutical industry fighting on the front line of epidemic prevention. This leads to the promotion of the pharmaceutical industry to the financial industry in the short term, but because the banking business only focuses on the pharmaceutical industry, other businesses are damaged and begin to have a restraining effect.For the pharmaceutical industry (Medicine), (1) when it is impacted by its own standard deviation, the positive effect is the largest in the current period and then gradually decreases, but it is still positive. The third and fourth periods have a negative effect temporarily, and the fifth period turns positive, and then gradually returns to 0. This shows the role of the pharmaceutical industry in promoting itself in the short term. The reason is that during the epidemic, masks and other medical devices have become daily necessities, which will inevitably promote the development of the pharmaceutical industry in the initial stage. However, with the expansion of production lines, masks and other supplies will gradually return to normal after the problem of short supply is solved. (2) The response of the pharmaceutical industry to the impact of the consumption industry is positive in the current period and then changes to negative. It shows that the consumption industry has a promoting and then refraining effect on the pharmaceutical industry in the short term. The reason is that the rise of the return of the consumption industry in the short term includes the purchase of medical supplies by the public, which is bound to promote the pharmaceutical industry. However, when the medical supplies are no longer in short supply, the rise of the return of the consumption industry will crowd out the return of the pharmaceutical industry, resulting in the decline of the return of the pharmaceutical industry. (3) The response of the pharmaceutical industry to the impact of the financial industry is positive in the first period, negative from the second to the fourth period, positive in the fifth period, and then gradually returns to 0. It shows that the financial industry has the effect of promoting and then refraining from promoting the pharmaceutical industry in the short term. The reason is that the establishment of a green channel for loans by banks for the pharmaceutical industry will promote the pharmaceutical industry, but then with the increase of loans in the pharmaceutical industry, it will have a negative impact on its operating efficiency, thereby inhibiting the return of the pharmaceutical industry.


## 4.3. Variance Decomposition

Although the impulse response function can explain the sign and amplitude of the response of each variable to a specific shock, it cannot compare the response intensity of different shocks to a specific variable. In order to further investigate the volatility of the three variables (Consumption, Medicine, and Finance), the variance decomposition is used to decompose the prediction mean square error of the three variables and calculate the relative importance of the impact of each variable.As shown in Table5, in the development process of the consumption industry, the contribution of the consumption industry to itself has a gradual downward trend but has stabilized at 0.9663 since the tenth period, while the contribution of the pharmaceutical industry and the financial industry to the consumption industry has a gradual upward trend, but the proportion is small. It shows that the consumption industry is mainly affected by itself, while the role of the pharmaceutical industry and the financial industry is very small.Table 5
Variance decompositions of the consumption industry, pharmaceutical industry, and financial industry.
VariablePeriodConsumptionMedicineFinanceConsumption000050.96920.52960.5127100.96630.52890.5109150.96630.52890.5109Medicine000050.02280.45520.0640100.02330.45510.0649150.02330.45510.0649Finance000050.00800.01520.4233100.01040.01600.4242150.01040.01600.4242In the development process of the pharmaceutical industry, the contribution of the pharmaceutical industry to itself is basically 0.4551, while the contribution of the consumption industry to the pharmaceutical industry remains at about 0.5289. Although the contribution of the financial industry to the pharmaceutical industry increases period-by-period, it remains at 0.0160 after ten periods. It shows that the pharmaceutical industry is mainly affected by itself and the consumption industry, while the role of the financial industry is very small.In the development process of the financial industry, the contribution of the financial industry to itself is basically 0.4242, while the contribution of the consumption industry to the financial industry remains at about 0.5109. Although the contribution of the pharmaceutical industry to the financial industry increases period by period, it remains at 0.0649 after ten periods. It shows that the financial industry is mainly affected by itself and the consumption industry, while the role of the pharmaceutical industry is very small.


## 5. Conclusion and Enlightenment

2020 is destined to be an unusual year. The outbreak of COVID-19 has had an impact on global financial markets, and Chinese financial markets are the most representative. Many Chinese industries have been impacted during the epidemic. This paper focuses on three industries that have been greatly impacted by COVID-19, namely, the consumption industry, the pharmaceutical industry, and the financial industry. We choose the returns of 98 stocks in three industries in the 100 trading days from January 2, 2020, to June 3, 2020. Based on the random matrix theory, we first analyze whether the stock market during the epidemic conforms to the efficient market hypothesis. Secondly, we further study the linkage between the three industries.By comparing the empirical correlation matrix with the random correlation matrix, it is found that the empirical correlation matrix does not conform to the characteristics of the random correlation matrix, which verifies that the Chinese stock market did not conform to the efficient market hypothesis during the epidemic. And Dias et al. also confirmed that the random walk hypothesis was rejected in the Chinese stock market during the epidemic [42]. Then, through impulse response function and variance decomposition, it is found that there is a certain interaction between industries. The specific conclusions are as follows:(1)
The correlation coefficient is approximately a normal distribution, but the mean value is greater than 0. And the mean value of the relatively mature markets such as the United States is about 0, which means the Chinese stock market is highly correlated.(2)
There are three eigenvalues greater than the prediction value, of which the maximum eigenvalue is about 11.18 times larger than the largest eigenvalue of the RMT. There is a significant positive relationship between the maximum eigenvalue and the correlation coefficient. The specific market performance is that the stock price fluctuations show a high degree of consistency. In investment, the efficiency of asset allocation will be relatively low, and the ability to spread risks is very limited. Further, the external factors of the market play a leading role in the market.(3)
The financial industry has a restraining effect on the consumption industry in the short term, and the pharmaceutical industry has a promoting and then restraining effect on the consumption industry in the short term. The consumption industry has a promoting effect on the financial industry in the short term, and the pharmaceutical industry has a promoting and then restraining effect on the financial industry in the short term. The consumption industry has a promoting and then restraining effect on the pharmaceutical industry in the short term, and the financial industry has a promoting and then restraining effect on the pharmaceutical industry in the short term.(4)
The consumption industry is mainly affected by itself, while the role of the pharmaceutical industry and the financial industry is very small. The pharmaceutical industry is mainly affected by itself and the consumption industry, while the role of the financial industry is very small. The financial industry is mainly affected by itself and the consumer industry, while the role of the pharmaceutical industry is very small.Under these conditions, the Chinese stock market tends to overreact to information and eventually correct it in the following days, whether it is good news or bad news. The high sensitivity of prices to the arrival of new information could likely be due to the climate of pessimism and uncertainty experienced by investors during the global epidemic of 2020. In addition, this situation is of great importance to investors, since it means that the rates of returns could be partially forecasted, creating possibilities for arbitrage operations and the potential occurrence of extraordinary returns.For institutional investors with the purpose of hedging, the most important work is to determine the hedging ratio, so the correlation between the assets hedged and the assets used for hedging returns has an important impact on the final hedging effect. Clearly, it is crucial to accurately estimate the correlation between the returns of the assets being hedged and the assets being used to hedge. If the correlation between asset returns is time-varying, then the hedge ratio related to it needs to be adjusted accordingly to match the change in the former. Therefore, the research on the correlation of asset returns will help to fully grasp the correlation between asset returns, so as to improve the efficiency of institutional investors in risk management.Further prospects and limitations:(1)
This paper chooses a specific industry to study the return over a specific period of time. The time dimension and the number of stocks collected are still not enough, which will have a certain impact on the results. Next, we can increase the choice of industries and the types of stocks. For example, we can choose B stocks to compare with A shocks.(2)
In this paper, there is no special research on small eigenvalues. In the next step, we can do some research on small eigenvalues to discuss whether there is some useful information about small eigenvalues.



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*Source: 5404229-2023-03-18.xml*