آنالیز تئوری ماتریس تصادفی همبستگی متقابل در بازار سهام ایالات متحده: شواهد حاصل از ضریب همبستگی پیرسون و ضریب همبستگی دیتند

In this study, we first build two empirical cross-correlation matrices in the US stock market by two different methods, namely the Pearson’s correlation coefficient and the detrended cross-correlation coefficient (DCCA coefficient). Then, combining the two matrices with the method of random matrix theory (RMT), we mainly investigate the statistical properties of cross-correlations in the US stock market. We choose the daily closing prices of 462 constituent stocks of S&P 500 index as the research objects and select the sample data from January 3, 2005 to August 31, 2012. In the empirical analysis, we examine the statistical properties of cross-correlation coefficients, the distribution of eigenvalues, the distribution of eigenvector components, and the inverse participation ratio. From the two methods, we find some new results of the cross-correlations in the US stock market in our study, which are different from the conclusions reached by previous studies. The empirical cross-correlation matrices constructed by the DCCA coefficient show several interesting properties at different time scales in the US stock market, which are useful to the risk management and optimal portfolio selection, especially to the diversity of the asset portfolio. It will be an interesting and meaningful work to find the theoretical eigenvalue distribution of a completely random matrix R for the DCCA coefficient because it does not obey the Marčenko–Pastur distribution.

Random matrix theory (RMT) is a popular technical tool for investigating the cross-correlations in financial markets [1], [2] and [3]. RMT, which was first proposed by Wigner [4] in 1951, can be used to describe the statistical properties of energy levels in complex quantum systems. In 1999, Laloux et al. [5] and Plerou et al. [6] applied the RMT approach to analyze the cross-correlations in the US stock market. The former chose the daily prices of 406 assets of the Standard & Poor (S&P) 500 index from 1991 to 1996 as the empirical data [5], and the research object of the latter was the 30-min high-frequency data of the largest 1000 US companies during the years 1994–1995 [6]. Although the empirical data of Refs. [5] and [6] are different, Laloux et al. [5] and Plerou et al. [6] came to a similar conclusion: the largest eigenvalue corresponding to the financial market (i.e., the US stock market), is roughly 25 times greater than the maximum eigenvalue (i.e., λ+λ+) predicted by RMT. The same results were drawn by Plerou et al. [7] in 2002 who studied three different empirical datasets: the first one is the same as Ref. [6], the second one is 30-min high-frequency data of 881 US stocks in the period of 1996–1997, and the third one is 1-day data of 422 US stocks over the 35 years period 1962–1996. However, Utsugi et al. [8] analyzed the daily prices of 297 stocks of S&P 500 index from January 1991 to July 2001, and found that the largest eigenvalue (=52.2) is approximately 29 times larger than the maximum eigenvalue (λ+=1.79λ+=1.79). Here, we may ask what happened in the US stock market in recent years which burst the US sub-prime crisis and the European debt crisis. Since RMT was introduced in financial markets by Laloux et al. [5] and Plerou et al. [6], it has been widely used to study the statistical properties of cross-correlations in different financial markets [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] and [28], such as the Warsaw stock market [13] and [22], the South African stock market [15], the Istanbul stock market [16], the foreign exchange market [19], and the Indian stock market [20] and [21]. More recently, Eom et al. [23] examined the topological properties of stock networks by using the minimal spanning tree (MST) method and RMT in financial data. They stated that the largest eigenvalue evidently has an effect on the construction of stock networks. Namaki et al. [25] studied the stability of correlation matrices of both the Tehran stock market (as an emerging market) and the Dow Jones Industrial Average (DJIA, as a mature market) via the mean values of the distribution of correlation coefficients and RMT under the global perturbation. Their results showed that DJIA is more sensitive to perturbations than the Tehran stock market. Oh et al. [26] analyzed the statistical properties of the cross-correlations in the Korean stock market through RMT and found that the largest eigenvalue is 52 times greater than the maximum eigenvalue predicted by RMT. Kumar et al. [27] applied RMT to investigate the cross-correlation properties of 20 financial indices before and during the US sub-prime crisis. They observed that few of the largest eigenvalues deviate dramatically from the prediction by RMT and the deviation has changed during the US sub-prime crisis. The time-lag RMT (TLRMT) was also introduced in financial markets to investigate long-range collective movements [29] and [30]. For instance, Podobnik et al. [29] examined 88 constituent stocks of S&P 500 index in 2009 during the period 1983–2009 and revealed pronounced peaks in times of crisis by applying the TLRMT. Accurately, they showed singular values versus year and found marked peaks during the largest market shocks and financial crises, such as the Black Monday, the Dot-com Bubble, and the US sub-prime crisis. Wang et al. [30] investigated 48 world financial indices and employed the TLRMT to demonstrate the decay of cross-correlations with time lags. They reported long-range power-law cross-correlations in the volatilities (i.e., the absolute returns) which can quantify risk, and indicated that cross-correlations between the volatilities decay much more slowly than cross-correlations between the returns. In the aforementioned previous works, the cross-correlation matrices are constructed by Pearson’s correlation coefficient (PCC) CijCij, which is defined by between stock ii and stock View the MathML sourcej,Cij=〈rirj〉−〈ri〉〈rj〉σiσj, where riri and rjrj are returns of stocks ii and jj respectively, View the MathML sourceσi=〈ri2〉−〈ri〉2 is the standard deviation of riri and 〈⋯〉〈⋯〉 represents the time average over the period studied. PCC can describe the linear correlation between two time series which are both assumed to be stationary [31]. However, in the real world, especially in the financial markets, the time series are often non-stationary and heterogeneous [32] and [33]. In other words, PCC may lose effectiveness if the time series are non-stationary or non-Gaussian distributions. Zebende [34] also pointed out PCC is not robust and can be misleading if outliers are present because the real world recordings are characterized by a high level of heterogeneity. To address the drawbacks of PCC, based on the detrended fluctuation analysis (DFA) method [35] and the detrended cross-correlation analysis (DCCA) method [36] and [37], Zebende [34] recently developed a novel detrended cross-correlation coefficient, i.e., DCCA coefficient ρij(s)ρij(s), to quantify the level of cross-correlation between non-stationary time series, where ss is the time scale. The outstanding advantage of this nonlinear cross-correlation coefficient is that it can investigate the cross-correlations at different time scales [38] and [39]. Vassoler and Zebende [40] adopted the DCCA coefficient to analyze and quantify cross-correlations between time series of air temperature and relative humidity. Podobnik et al. [41] studied the statistical significance of ρij(s)ρij(s), and showed that the tendency of the Chinese stock market to follow the US stock market is extremely weak by using the coefficient ρij(s)ρij(s). Taking into account what has been discussed above, we hereby put forward three questions as follows: (i) Can the DCCA coefficient ρij(s)ρij(s) efficiently measure the cross-correlations of financial time series by combining RMT? (ii) If the cross-correlations are calculated by PCC and ρij(s)ρij(s) respectively, what are the differences between the two coefficients? (iii) What properties do the cross-correlations in financial markets have at different time scales, where the cross-correlations are measured by ρij(s)ρij(s) and RMT? Therefore, in this study, we aim to answer the above-mentioned questions. In the empirical process, we first choose the daily closing prices of 462 constituent stocks of S&P 500 index from January 3, 2005 to August 31, 2012 as the research data. Next, we respectively use PCC and the DCCA coefficient ρij(s)ρij(s) to construct the empirical cross-correlation matrices of the 462 stocks and combine them with RMT to analyze the statistical properties of the cross-correlations in the US stock market. Then, we present the statistical results from the two coefficients. At the same time, we also show some statistical properties of the cross-correlations at different scales, in which the cross-correlations are calculated by the coefficient ρij(s)ρij(s). The rest of this paper is organized as follows. In the next section, we present the empirical data and describe the methodologies of the DCCA coefficient and RMT. In Section 3, we show the main empirical results and discussion. Finally, we draw some conclusions in Section 4.

In this paper, we use two different approaches, namely PCC and the DCCA coefficient, to construct the empirical cross-correlation matrices, and combine them with the method of RMT to analyze the statistical properties of cross-correlations in the US stock market. We choose the returns of 462 constituent stocks of S&P 500 index during the period of January 3, 2005–August 31, 2012 as the sample data. In the empirical process, we investigate the statistical properties of cross-correlation coefficients, the distribution of eigenvalues, the distribution of eigenvector components, and the inverse participation ratio. We also present some results of the above-mentioned properties by the DCCA coefficient at different time scales. The basic findings of the cross-correlations in the US stock market in our study, which are different from the results reached by previous studies, can be summarized as follows: (i) The largest eigenvalue corresponding to the financial market is roughly 80 times for CijCij greater than the maximum eigenvalue predicted by RMT. After removing the market factor, the largest eigenvalue is still approximately 11 times larger than the maximum eigenvalue λ+λ+ of RMT; (ii) In our study, the eigenvalues that fall in the bulk predicted by RMT are much less than the results of Refs. [5] and [6] investigated the earlier years (e.g., 1994–1995). This phenomenon still happened after filtering out the market effect; and (iii) The two estimated slopes of Eq. (17) by the two method are roughly equal to 0.99, which are greater than 0.85 reported by Plerou et al. [7]. There are two possible origins to explain the differences between our outcomes and the earlier results in Refs. [5], [6] and [7] as follows: on the one hand, along with the rapid growth of the economic, science and technology, the relationships and co-operations among different companies are more and more tightened, which leads to the cross-correlations in the US stock market more and more remarkable. The higher the cross-correlations in the market are, the higher the largest eigenvalue of the cross-correlation matrix is. So the portfolio of the returns on the eigenvector corresponding to the largest eigenvalue carries more market information compared to the earlier years. On the other hand, many literatures [29], [33] and [46] reported that the largest market shocks and economic crises often lead to a large co-movement effect and interaction effect in the financial markets. In our study, there are two financial crises (i.e., the US sub-prime crisis and European debt crisis) in the analyzed period, which trend to cause the increase of the cross-correlations among the companies in the American stock market. Based on (i) and (ii), we also confirm that the market factor strongly influences the eigenvalue distribution and the bulk of the original empirical cross-correlation matrix is not pure noise. The above-mentioned three different results compared to the earlier studies are the main similitudes of the two coefficients, while hereby we try to show their differences and answer the three questions which were put forward in the introduction as follows: (i) Although the DCCA coefficient method has some good properties (e.g., the nonlinear property and the scale resolution), the theoretical eigenvalue distribution of a completely random matrix View the MathML sourceR for the method of the DCCA coefficient is not the Marčenko–Pastur distribution. Therefore, care should be taken when using the method of the DCCA coefficient and RMT to investigate the cross-correlations. (ii) We find that the DCCA coefficient method has similar results and properties with PCC, such as the properties of the largest eigenvalue and the corresponding eigenvector, whereas the main difference between the two methods is just as the aforementioned feature that the eigenvalue distribution of View the MathML sourceR for the DCCA coefficient method does not obey the Marčenko–Pastur distribution. Also, the distribution of eigenvalues of the empirical cross-correlation matrix constructed by the DCCA coefficient is different from that of PCC. (iii) The empirical cross-correlation matrices built by the DCCA coefficient at different scales are different and present different properties, such as the distribution of eigenvalues and the inverse participation ratio, which are useful to the risk management and optimal portfolio selection [51], especially to the diversity of the asset portfolio. So it is an interesting and meaningful future work to find the theoretical eigenvalue distribution of View the MathML sourceR at different time scales.