Random matrix approach to the dynamics of stock inventory variations

We study the cross-correlation matrix $C_{ij}$ of inventory variations of the most active individual and institutional investors in an emerging market to understand the dynamics of inventory variations. We find that the distribution of cross-correlation coefficient $C_{ij}$ has a power-law form in the bulk followed by exponential tails and there are more positive coefficients than negative ones. In addition, it is more possible that two individuals or two institutions have stronger inventory variation correlation than one individual and one institution. We find that the largest and the second largest eigenvalues ($\lambda_1$ and $\lambda_2$) of the correlation matrix cannot be explained by the random matrix theory and the projection of inventory variations on the first eigenvector $u(\lambda_1)$ are linearly correlated with stock returns, where individual investors play a dominating role. The investors are classified into three categories based on the cross-correlation coefficients $C_{VR}$ between inventory variations and stock returns. Half individuals are reversing investors who exhibit evident buy and sell herding behaviors, while 6% individuals are trending investors. For institutions, only 10% and 8% investors are trending and reversing investors. A strong Granger causality is unveiled from stock returns to inventory variations, which means that a large proportion of individuals hold the reversing trading strategy and a small part of individuals hold the trending strategy. Comparing with the case of Spanish market, Chinese investors exhibit common and market-specific behaviors. Our empirical findings have scientific significance in the understanding of investors' trading behaviors and in the construction of agent-based models for stock markets.


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individual and one institution. We find that the largest and the second largest eigenvalues (λ 1 and λ 2 ) of the correlation matrix cannot be explained by random matrix theory and the projections of investors' inventory variations on the first eigenvector u(λ 1 ) are linearly correlated with stock returns, where individual investors play a dominating role. The investors are classified into three categories based on the cross-correlation coefficients C V R between inventory variations and stock returns. A strong Granger causality is unveiled from stock returns to inventory variations, which means that a large proportion of individuals hold the reversing trading strategy and a small part of individuals hold the trending strategy. Our empirical findings have scientific significance in the understanding of investors' trading behavior and in the construction of agent-based models for emerging stock markets.
3 eigenvalue of the correlation matrix of inventory variations cannot be explained by random matrix theory (RMT) and its eigenvector contains information on stock price fluctuations. Both buying and selling herding behavior has been observed for trending and reversing firms. In a recent paper, an alternative identification approach of investor clusters has been proposed from their real trading activity [12], which is based on the statistical validation of links in complex networks [13][14][15].
In this work, we perform a similar analysis to that in [5] based on the trading records of Chinese investors in the Shenzhen Stock Exchange. As officially documented by the Spanish Stock Exchange, firms are local and foreign credit entities and investment firms which are members of the stock exchange and are the only firms entitled to trade; that is, a firm is a market member that usually acts as a brokerage house for individual and institutional investors 7 . Therefore, in the Spanish case [5], the inventory variation over a given time interval is a sum of the inventory variations of many individuals and institutions. Different from the Spanish case, our data set contains both individual and institutional investors, which allows us to observe interesting investor behavior. Our analysis starts from the perspective of random matrix theory, which has been extensively used to investigate the cross-correlations of financial returns in different stock markets [16][17][18]. However, very few studies have been conducted on the Chinese stocks [19] and, to our knowledge, there is no research reported on the dynamics of inventory variations of Chinese investors. Alternatively, there are studies on Chinese equities at the transaction and trader level from the complex network perspective [20][21][22][23]. We note that random matrix theory has very wide applications in physics from high-energy physics to statistical physics; see [24] for a brief review and see the papers in this issue.
This paper is organized as follows. Section 2 describes the data and the method for constructing the time series of investors' inventory variations. Section 3 studies the statistical properties of the elements, eigenvalues and eigenvectors of the correlation matrix of inventory variations. Section 4 investigates the contemporaneous and lagged cross-correlation between inventors' inventory variations and stock returns to divide investors into three categories and their herding behavior. Section 5 summarizes our findings.

The data
We analyze 39 stocks actively traded on the Shenzhen Stock Exchange in 2003. The database contains all the information needed for the analysis in this work. For each transaction i of a given stock, the data record the identities of the buyer and seller, the types (individual or institution) of the two traders, the price p i and the size q i of the trade, and the time stamp. Therefore, the trading history of each investor is known. For each stock, we identify active traders who had more than 150 transactions, amounting to about three transactions per week. If the number of active traders of a stock is less than 120, we exclude it from analysis. In this way, we have 15 stocks for analysis.
Following Lillo et al [5], we investigate the dynamics of the inventory variation of the most active investors who executed more than 120 transactions for each stock. Although the trading period of each day consists of call auction and continuous auction, their behavior is different in many aspects and is usually studied separately [25,26]. We stress that all the transactions in both call auction and continuous auction are included in our investigation. The daily inventory Table 1. Basic statistics of the investigated stocks. The first column is the stock code, which is the unique identity of each stock. The second and third columns present investor-averaged total inventory variation t v i and average absolute variation |v| t i . The fourth to eighth columns give the number of investors N , the number of trending investors N tr , the number of reversing investors N re , the number of uncategorized investors N un and the slope of the factor versus stock return k. The variables in the ninth to thirteenth columns are the same as those in the five 'all investors' columns but for individual investors, and the last five columns are for institutional investors. Each value in the last row gives the sum of the numbers in the same column.

All investors Individuals Institutions
Code where + p i (t)q i (t) is the total buy quantity on trading day t and − p i (t)q i (t) is the total sell quantity on the same day. The basic statistics of the 80 most active traders and the resultant inventory variations are given in table 1. The time series of the daily inventory variations of three investors trading stock 000001 are shown in figure 1.

Distributions of cross-correlation coefficients
The empirical correlation matrix C is constructed from the time series of inventory variation v i (t) of the investigated stock, defined as Since the results for individual stocks are quantitatively similar, we put the cross-correlation coefficients of the 15 stocks into one sample. We find that the mean value is C i j = 0.02 for the real data and 0 for the shuffled data. When the types of investors are taken into account, the mean value of the cross-correlation coefficients is C i j = 0.048 (shuffled: −0.001) for both i and j being individuals, C i j = 0.014 (shuffled: 0) for both i and j being institutions and C i j = −0.008 (shuffled: −0.001) for i being individual and j being institution. Figure 2 plots the daily returns of stock 000001 and the sliding average values of the correlation coefficients C i j for comparison. We observe that large values C i j appear during periods of large price fluctuations by and large, which is reminiscent of a similar result for crosscorrelations of financial returns [18]. However, the short time period of our data sample does not allow us to reach a decisive conclusion. There are also less volatile time periods with large C i j . The situation is quite similar for other stocks. Figure 3(a) shows the empirical probability distributions of C i j , which is calculated using daily inventory variation. The four curves with different markers correspond to C i j , C ind,ind , C ind,ins and C ins,ins , respectively. It is found that most coefficients are small and the tails are exponentials: where λ + = 8.8 ± 0.2 and λ − = 11.1 ± 0.3 for C i j , λ + = 8.9 ± 0.2 and λ − = 12.5 ± 0.4 for C ind,ind , λ + = 10.8 ± 0.4 and λ − = 10.5 ± 0.4 for C ind,ins and λ + = 6.6 ± 0.5 and λ − = 8.7 ± 0.5 for C ins,ins , respectively. We find that there are more positive cross-correlation coefficients (λ + < λ − ) when both investors are individuals or institutions. In contrast, the distribution is symmetric (λ + ≈ λ − ) when one investor is an individual while the other is an institution. This finding implies that herding behavior is more likely to occur among the same type of investors, and individuals have larger probability to herd than institutions. We shuffle the original time   series and perform the same analysis. The resulting distributions collapse onto a single curve, which has an exponential form where the parameter λ shuf = 23.3 is determined using robust regression [27,28]. It is not surprising that real data have higher cross-correlations than shuffled data, which is confirmed by λ ± < λ shuf . This exponential distribution is different from the Gaussian distribution for the shuffled data of financial returns [18]. Figure 3(b) plots the distributions in double logarithmic coordinates where the negative parts are reflected to the right with respect to C i j = 0. Nice power laws are observed spanning 7 over three orders of magnitude: where γ + = 0.69 ± 0.01 and γ − = 0.69 ± 0.01 for C i j , γ + = 0.67 ± 0.02 and γ − = 0.62 ± 0.02 for C ind,ind , γ + = 0.67 ± 0.02 and γ − = 0.70 ± 0.02 for C ind,ins and γ + = 0.72 ± 0.02 and γ − = 0.73 ± 0.02 for C ins,ins , respectively. It is found that γ − ≈ γ + and all the power-law exponents are close to each other. An intriguing feature is that the distributions of C i j , C ind,ind and C ind,ins exhibit evident bimodal behavior, which is reminiscent of the distributions of waiting times and inter-event times of human short message communication [29]. Certainly, the underlying mechanisms are different and the factors causing the bimodal distribution of the cross-correlations are unclear.
We have tried to fit the data with the stretched exponential distribution [30,31]. However, we do not observe significant improvement, especially in the tails. The worst fit is obtained for the 'ind-ind' case.
It is natural that we are interested more in large cross-correlations. The preceding discussions focus on cross-correlations not larger than 0.6. As shown in figure 3(a), there are pairs of inventory variation time series that have very large cross-correlations that look like outliers. For better visibility, we plot in figure 3(c) the numbers of occurrences of positive and negative cross-correlations in ten nonoverlapping intervals for the four types of pairs. It is shown that N (C > 0) > N (C < 0) in all intervals for C = C i j , C ind,ind and C ind,ins . In contrast, N (C ind,ins > 0) < N (C ind,ins < 0) when C ind,ins < 0.6 and N (C ind,ins > 0) > N (C ind,ins < 0) when C ind,ins > 0.6. Hence, for larger cross-correlations (|C| > 0.6), there are many more occurrences of positive cross-correlations than negative ones for all four types of pairs. This striking feature can be attributed to two reasons. The first is that a large proportion of investors react to the same external news in the same direction [5]: they buy following good news and sell following bad news. The second is that investors imitate the trading behavior of others of the same type and rarely imitate other investors of different types. The second reason is rational because the friends of individual (or institutional) investors are more likely individual (or institutional) investors.

The eigenvalue spectrum
For the correlation matrix C of each stock, we can calculate its eigenvalues whose density f c (λ) is defined as [16] where n(λ) is the number of eigenvalues of C less than λ. If M is a T × N random matrix with zero mean and unit variance, f c (λ) is self-averaging. In particular, in the limit N → ∞, T → ∞ and Q = T /N 1 fixed, the probability density function f c (λ) of eigenvalues λ of the random correlation matrix M can be described as [16,18,32]  The solid line is the spectral density obtained by shuffling independently the buyers and the sellers in such a way as to maintain the same number of purchases and sales for each investor as that in the real data. The dashed blue line shows the spectral density predicted by random matrix theory using equation (7) with Q = 237/80 = 2.96. The inset shows the largest eigenvalue λ 1 ( ) and the second largest eigenvalue λ 2 ( ) of all 15 investigated data sets from 15 stocks. The solid line indicates the upper thresholds by shuffling experiments, and the dashed line presents the threshold predicted by random matrix theory.
with λ ∈ [λ min , λ max ], where λ max min is given by and σ 2 is equal to the variance of the elements of M [16,32]. The variance σ 2 is equal to 1 in our normalized data. Figure 4 illustrates the probability distribution f c (λ) of the correlation matrix of inventory variation of investors trading stock 000001. The solid line is the spectral density obtained by shuffling independently the buyers and the sellers in such a way as to maintain the same number of purchases and sales for each investor as that in the real data, while the dashed blue line shows the spectral density predicted by random matrix theory using equation (7) with Q = 237/80 = 2.96. We find the largest eigenvalue is well outside the bulk and the second largest eigenvalue also escapes the bulk. The results for the other 14 stocks are quite similar. In the inset of figure 4, we plot the largest eigenvalues λ 1 and the second largest eigenvalues λ 2 for all 15 stocks. We find that all the largest eigenvalues are well above the upper thresholds determined from shuffling experiments and the thresholds λ max in equation (8) predicted by random matrix theory. Moreover, all the second largest eigenvalues are above the two threshold lines and some of them are well above the thresholds. These findings indicate that both the largest and the second largest eigenvalues carry information about the investors, which is different from the results of the Spanish stock market, where only the largest eigenvalue is larger than the up thresholds while the second largest eigenvalue is within the bulk [5]. This discrepancy can be attributed to the difference in the two markets and the fact that our analysis In contrast, at least two classes of Chinese investors can be identified based on λ 2 . However, it is hard to understand what the differences between the two classes of Chinese investors are.

Distribution of eigenvector components
If there is no information contained in an eigenvalue, the normalized components of its associated eigenvector should conform to a Gaussian distribution [16][17][18]: Since the empirical eigenvalue distribution f c (λ) deviates from the theoretic expression (6) from random matrix theory with two large eigenvalues outside the bulk of the distribution, it is expected that the associated eigenvectors also contain certain information.
For correlation matrices of financial returns, the components of an eigenvector with the eigenvalue λ in the bulk of its distribution (λ min < λ < λ max ) are distributed according to equation (9) [16][17][18]. Panels (a) and (b) of figure 5 show the empirical distributions of the eigenvector components u with the eigenvalues in the bulk for two typical stocks. Rather than analyzing the vector for one eigenvector, we normalized the components of each eigenvector and put all the eigenvectors together to gain better statistics, since each eigenvector has only 80 components. We find that the distributions of ten stocks are well consistent with the Gaussian, while the other five stocks exhibit high peaks at the center around u = 0.
For deviating eigenvalues λ 1 and λ 2 , the distribution for each stock is very noisy and deviates from Gaussian. We treat the components of the 15 eigenvectors as a sample to have better statistics. The two distributions obtained are illustrated in figures 5(c) and (d). It is evident that both deviate from the Gaussian distribution and the distribution for λ 1 is more skewed. We have plotted the components of each eigenvector and found that they are remarkably asymmetric with respect to u = 0. The quantile-quantile plots in reference to a normal distribution further confirm that the eigenvectors are not Gaussian. We do not show those figures here.

Information in eigenvectors for deviating eigenvalues
We have shown that the largest and the second largest eigenvalues deviate from the RMT prediction and the distributions of their eigenvector components are not Gaussian. It implies that these eigenvectors carry some information. For u(λ 2 ), it is not clear what kind of information they carry. We find no evident dependence of the magnitude of u i (λ 2 ) on the average absolution inventory variation |v i | , the total variation v i , or the maximal absolute variation max{|v i |}. The same conclusion is obtained for u i (λ 1 ), which differs from the conclusion that the eigenvector components of the return correlation matrix depend on the market capitalization in a logarithmic form [18]. For the correlation matrix of stock returns, the deviating eigenvalues except λ 1 contain information on industrial sectors [18] or stock categories [19]. It is natural to conjecture that λ 2 for inventory variations contains information on investor classification based on their trading behavior. Unfortunately, we are not able to identify clear differences in the trading behavior of investors, although we will show in the next section that the investors can be categorized into three trading types. We find no relation between the trading strategy category and the magnitude of the vector component u(λ 2 ). We thus focus on extracting the information from u(λ 1 ).
For the correlation matrix whose elements are the correlation coefficients of price fluctuations of two stocks, the eigenvector of the largest eigenvalue contains market information [16,17]. The market information indicates the collective behavior of stock price movements [33], which can be unveiled by the projection of the time series on the eigenvector [18]. We follow this approach and calculate the projection G(t) of the time series ]/σ i on the eigenvector u(λ 1 ) corresponding to the first eigenvalue [5]: The projection G can be called the factor associated with the largest eigenvalue [5]. We plot the factor G(t) against the normalized return R(t) for stock 000001 in figure 6(a). There is a nice linear dependence between the two variables and a linear regression gives the slope k = 0.83 ± 0.04. It indicates that these most active investors have a dominating influence on the price fluctuations. This is not surprising because the majority of the investors had very few trading activities. Panels (b) and (c) of figure 6 illustrate the relation between the factor for individuals and institutions and the return of the stock. Linear regression gives k ind = 0.83 ± 0.04 and k ins = 0.41 ± 0.06. Comparing panels (b) and (c) with (a), we find that the influence of individuals matches excellently with the whole sample, which can be quantified by the facts that k ind = k and k ind < k. The results are similar for other stocks. The resulting k ind and k ins are plotted in figure 6 against k for all 15 stocks. For individual investors, we find that k ind = k for 13 stocks and k ind > k for 2 stocks. In contrast, we find that k ins < k except for one stock. An interesting question is to ask the lagged correlation between returns R and dominant mode G. In particular, one would be interested in the potential predictability of G on the returns. The factor G(t) has predictive power if it is significantly correlated with R(t + 1). This test cannot be done using the data in figure 6 because the G(t) time series contains information on the time period investigated. Indeed, the analysis performed above is in a descriptive manner, as done in factor analysis in econometrics [34]. A crude analysis by plotting R(t + 1) against G(t) does not uncover any significant correlations. A proper way is to determine the G(t) time series in moving windows. However, we do not proceed due to the fact that we have only the data over one year. Nevertheless, we argue that this can be done when longer data sets are available and this idea can be adopted in other cases when the records are sufficiently long.

Categorization of investors
Following Lillo et al [5], we divide the investors into three categories according to the crosscorrelation coefficient C V i R between the inventory variation V i and the stock return R. The investor i belongs to the trending or reversing category if its inventory variation is positively or negatively correlated with the return. We use a wieldy significant threshold to categorize the investors: where N T is the number of time records for each time series [5]. We also verify the robustness of equation (11)   As shown in the last row of table 1, the numbers of the three kinds of investors (trending, reversing and uncategorized) are 46, 34 and 373 for institutional investors and 41, 381 and 336 for individual investors, respectively. We find that most institutional investors are uncategorized and there are more trending investors than reversing investors. These results are different from the Spanish case, where only one-third investors are uncategorized and the number of reversing firm investors is about thrice as much as the number of trending firm investors [5]. In contrast, about half of the individuals are reversing investors and only 6% of the individuals are trending investors. The observation that most investors are uncategorized is probably due to the fact that the Chinese market was emerging and its investors are not experienced. Comparing individuals and institutions, we find a larger proportion of individuals exhibiting reversing behavior. This indicates that these individuals buy when the price drops and sell when the price rises on the same day. This finding is very interesting since it explains the worse performance in stock markets [10,11].
The empirical evidence for the significant cross-correlation between inventory variation V i (t) of trending and reversing investors and stock return R(t) leads us to adopt a linear model for the dynamics of inventory variation as a first approximation [5]: where γ i is proportional to the cross-correlation coefficient C V i ,R . It follows immediately that the cross-correlation coefficient between the inventory variations of two investors is If two investors belong to the same category, either trending (γ i > 2σ and γ j > 2σ , significantly) or reversing (γ i < −2σ and γ j < −2σ , significantly), the value of C i j is expected to be significantly positive. In contrast, if two investors belong, respectively, to the trending and reversing categories, the value of C i j is expected to be significantly negative. To show the performance of the model, we plot the contours of the correlation matrix of inventory variation for all investors, for individuals and for institutions, where the investors are sorted according to their cross-correlation coefficients C V R of the inventory variation with the price return.

Causality
In section 4.1, we have shown that the inventory variation V i (t) and the stock return R(t) have significant positive or negative correlation for part of the investors. It is interesting to investigate the lead-lag structure between these two variables. For the largest majority of reversing and trending firms in the Spanish stock market, it is found that returns Granger cause inventory variation but not vice versa at the day or intraday level, and the Granger causality disappears over longer time intervals [5]. Here, we aim to study the same topic for both individual and institutional investors. We first investigate the autocorrelation function C V (t)V (t+τ ) of the inventory variation time series sampled in 15 min time intervals. Figure 8(a) shows the three autocorrelation functions for all the trending, reversing and uncategorized investors. Each autocorrelation function is obtained by averaging the autocorrelation functions of the investors in the same category to have better statistics. It is found that the inventory variation is long-term correlated and the correlation is significant over dozens of minutes and the correlation is stronger for trending investors than reversing investors, which can be partly explained by the order splitting behavior of large investors [5,[35][36][37][38]. We also find that uncategorized investors have stronger autocorrelations than trending and reversing investors, which is different from the Spanish case [5]. However, it  is not clear why uncategorized investors have stronger autocorrelations than trending investors. Figures 8(b) and (c) illustrate the results for individuals and institutions. We observe that institutions have stronger long memory than individuals. This implies that institutions are more persistent in adopting their trading strategies while individuals are more likely to change their strategies.
Panels (d)-(f) of figure 8 illustrate the averaged lagged cross-correlation functions C V (t)R(t+τ ) between inventory variations and returns. The results in the three panels are qualitatively the same. For uncategorized investors, no significant cross-correlations are found between inventory variations and returns, which is trivial due to the 'definition' of this category, as shown in figures 7(a)-(c). For trending and reversing investors, it is evident that the returns lead the inventory variations by dozens of minutes (τ < 0), where the cross-correlation C V (t)R(t+τ ) is significantly nonzero. When the price drops, trending investors will sell stock shares to reduce their inventory in a few minutes, while reversing investors will buy shares to increase their inventory. When the price rises, trending investors will buy shares to increase their inventory in a few minutes, while reversing investors will sell shares to reduce their inventory. In the meanwhile, we also observe nonzero cross-correlations for τ > 0 in shorter time periods, which means that the inventory variations lead returns.
To further explore the lead-lag structure between inventory variations and returns, we perform Granger causality analysis. We define an indicator I (X → Y ), whose value is 1 if X Granger causes Y and 0 otherwise [5].  Figure 8 shows that there are more investors with I (R → V ) = 1 than investors with I (V → R) = 1. On average, bidirectional Granger causality is observed at the intraday time scales and the Granger causality disappears at the weekly level or longer. Moreover, individual investors are more probable to be influenced by the intraday price fluctuations than institutions, because the I (R → V ) values of individuals are greater than those of institutions at the same time scale level.
We then investigate the impact of investor category on the causality indicator. The results for T = 4 (i.e. 1 h) are depicted in figures 8(j)-(l). The middle parts bounded by two vertical lines at C V R = ±σ correspond to uncategorized investors. The left parts (C V R < −σ ) correspond to reversing investors and the right parts C V R > ±σ correspond to trending investors. It is found that an investor adjusts his inventory following price fluctuations with very large probabilities when his |C V R | value is large. This conclusion holds for both individual and institutional investors. The strong Granger causality from inventory variations to returns and the weak but significant causality from returns to inventory variations cannot be attributed to the non-Gaussianity in the distributions of the variables, as verified by bootstrapping analysis. The effect that inventory variations Granger cause return is stronger for institutional investors. Qualitatively similar results are obtained for other T values. The observation that inventory variations seem to also Granger cause returns for both trending and reversing firms is not surprising. It reflects that executed orders causing inventory variations have temporary or permanent price impacts [39][40][41]. We note that the Granger causality from inventory variation to return is much weaker for the Spanish market [5]. This can be explained  by the fact that the Chinese stock market was most inefficient as a young emerging market and the investors were less skilled and more irrational.

Herding behavior
Herding and positive feedbacks are essential for the boom of bubbles [42,43]. These topics have been studied extensively to understand the price formation process [44][45][46][47]. Herding is a phenomenon where a group of investors are trading in the same direction over a period of time.
Here, we try to investigate possible herding behavior in different groups of investors. We study possible buy and sell herding behavior among groups of the same investors. Investors are classified into different groups based on their types (individual or institution) and their categories (reversing, trending or uncategorized). We define a herding index as follows [5]: where N + is the number of buying investors and N − is the number of selling investors in the same group over a given time horizon. When the herding index h is smaller than 5% under a binomial null hypothesis, we assess that herding is present. In our analysis, we fix the time horizon to one day and determine the number of days that herding was present for different groups of investors. The results are depicted in table 2. According to table 2, there are no buying and selling herding days observed for trending institutions. For trending individuals and reversing institutions, herding is observed in only one stock on very few days. For categorized investors, we see slightly more herding days in a few stocks. For reversing investors, the number of herding days is greater than that for other investors and we observe comparable buying and selling herding days. Our findings are consistent with those for the Spanish stock market, especially in the sense that reversing investors are more likely to herd [5]. Our analysis also allows us to conclude that individuals are more likely to herd than institutions in 2003.

Summary
In summary, we have studied the dynamics of investors' inventory variations. Our data set contains 15 stocks actively traded on the Shenzhen Stock Exchange in 2003 and the investors can be identified as either individuals or institutions.
We studied the cross-correlation matrix C i j of inventory variations of the most active individual and institutional investors. It is found that the distribution of cross-correlation coefficient C i j is asymmetric and has a power-law form in the bulk and exponential tails. The inventory variations exhibit stronger correlation when both investors are either individuals or institutions, which indicates that the trading behavior is more similar within investors of the same type. The eigenvalue spectrum shows that the largest and the second largest eigenvalues of the correlation matrix cannot be explained by the random matrix theory and the components of the first eigenvector u(λ 1 ) carry information about stock price fluctuation. In this respect, the behavior differs for individual and institutional investors.
Based on the contemporaneous cross-correlation coefficients C V R between inventory variations and stock returns, we classified investors into three categories: trending investors who buy (sell) when stock price rises (falls), reversing investors who sell (buy) when stock price rises (falls), and uncategorized investors. We also observed that stock returns predict inventory variations. It is interesting to find that about 56% of individuals hold trending or reversing strategies and only 18% of institutions hold strategies. Moreover, there are far more reversing individuals (50%) than trending individuals (6%). In contrast, there are slightly more trending institutions (10%) than reversing institutions (8%). Hence, Chinese individual investors are prone to selling winning stocks and buying losing stocks, which provides supporting evidence that trading is hazardous to the wealth of individuals [9,10].
In this work, we have focused on the zero-lag cross-correlations among inventory variations. It is possible to perform further investigations. One possibility is to study the correlation matrix constructed from time-lagged cross-correlations of the inventory variations as in [48][49][50]. In the Granger causality analysis, the lagged cross-correlations between inventory variation and return have already been considered. For inventory variation and return and their magnitudes, when longer time series become available, we can perform detrended crosscorrelation analysis [51] and multifractal cross-correlation analysis [52,53] and determine their statistical significance [54]. Another possibility is to research the absolute values of the inventory variations, which is a natural idea as the pair of return and volatility.