# Stock fluctuations are correlated and amplified across networks of interlocking directorates

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## Abstract

Traded corporations are required by law to have a majority of outside directors on their board. This requirement allows the existence of directors who sit on the board of two or more corporations at the same time, generating what is commonly known as interlocking directorates. While research has shown that networks of interlocking directorates facilitate the transmission of information between corporations, little is known about the extent to which such interlocking networks can explain the fluctuations of stock price returns. Yet, this is a special concern since the risk of amplifying stock fluctuations is latent. To answer this question, here we analyze the board composition, traders’ perception, and stock performance of more than 1,500 US traded corporations from 2007-2011. First, we find that the fewer degrees of separation between two corporations in the interlocking network, the stronger the temporal correlation between their stock price returns. Second, we find that the centrality of traded corporations in the interlocking network correlates with the frequency at which financial traders talk about such corporations, and this frequency is in turn proportional to the corresponding traded volume. Third, we show that the centrality of corporations was negatively associated with their stock performance in 2008, the year of the big financial crash. These results suggest that the strategic decisions made by interlocking directorates are strongly followed by stock analysts and have the potential to correlate and amplify the movement of stock prices during financial crashes. These results may have relevant implications for scholars, investors, and regulators.

## Keywords

stock market corporate governance interlocking networks information transmission financial traders## 1 Introduction

According to corporate governance standards in the US [1], traded corporations are required to have a minimum of three directors in their board and a majority of outside directors [2], [3]. Typically, these outside directors either have their primary affiliation with a different corporation, are self-employed, or retired [2]. This allows a director to sit on the board of two or more corporations at the same time - including or not their own affiliation board, generating what is commonly known as interlocking directorates [2]–[5]. In turn, interlocking networks emerge as the result of many interconnected interlocking directorates [2], [3]. Indeed, interlocking networks have been the focus of numerous research analyzing their role in the performance of corporations, organizational failure, economic downturns, hegemony, CEO pay, the sale price of a corporation, stock synchronicity, and prices for a corporation’s services, among others [2], [3], [6]–[10]. Yet, the strongest consensus so far is that interlocking networks mainly favor the transmission of information among corporations [3], [5], [11]. For instance, this information advantage has been observed when central corporations of interlocking networks can adopt new market strategies quicker via information they gather from their interlocking directorates [12], [13].

A major question that remains to be answered is whether interlocking networks can affect market processes [3]. Board members are typically major stockholders and report financial strategies to their investors [2], [3]. Indeed, previous work has suggested that interlocking directorates may play an important role in the information gathered by stock analysts, especially when there is substantial media coverage [14]. This is a special concern of regulators since the risk of amplifying stock fluctuations is latent [2], [3]. In fact, this question has motivated important actions in the US Congress dating back to the early 1900’s [2]. Section 8 of the Clayton Act explicitly prohibited interlocks if the linked corporations would violate antitrust laws if combined into a single corporation [2], [4]. Over the years, these actions have promoted a decrease in the number of interlocking directorates, especially within the financial sector [3]. However, even with these limitations, it has been shown that financial shocks can be transmitted across the interlocking network regardless of the degrees of separation between corporations [2], [3], [15]–[17].

To shed new light on the above question, we study the extent to which interlocking networks can explain stock fluctuations. We analyze data on the board composition, traders’ perception, and stock performance of more than 1,500 US traded corporations (see the Appendix for further details). To capture the behavior of interlocking networks across a variety of financial periods, we focus our analysis on the period covered by the big 2008 financial crash and its pre and post financial periods. We use the data extracted from *RiskMetrics*[18], which is a yearly compilation from 2007-2011 of the board composition of over 1,500 large US traded corporations. Daily closing stock prices for these corporations are extracted from *WRDS* database [19], and stocks are categorized in a market sector according to *Yahoo Finance* criteria [20]. We analyze traders’ perception on these traded corporations using trading data and electronic communication among traders from one US trading firm for the period 2007-2009 [21], [22]. Here, we explore the extent to which interlocking networks can explain stock correlations, the plausible mechanisms linking interlocking networks and stock markets, and the potential effect of such associations.

## 2 Results

### 2.1 Network characterization

We constructed interlocking networks for each year from 2007-2011 formed only by the traded corporations observed in a particular year. A link between two corporations is established if they share at least one board member in the same year. Consistent with previous studies and regulations [2], [3], we find that the observed boards in these interlocking networks have a median size of 9 directors, of which the majority are outside directors (>50%). The 99% of the observed interlocking directorates (i.e., links between two corporations) share only one board member. In total, there is approximately 20% of corporations who do not share any single director with another corporation, whereas the other 80% are on average 4.6 interlocking directorates apart from each other. Interestingly, 85% of all interlocking directorates are formed between corporations that belong to different market sectors, and we find no significant connectivity differences across market sectors. These results are also in agreement with the characterization of interlocking networks found in previous studies [16], [17].

Whereas the characterization of the interlocking network remains fairly constant across the observation period, the financial market exhibited important fluctuations from year to year. In 2007, 46% of the 1,422 observed traded corporations increased their stock price by the end of the year. In contrast, in 2008, the year of the big financial crash, only 12% of the 1,453 observed traded corporations increased their stock price. This was followed by a recovery period in 2009 and 2010 were 81% of 1,357 and 1,413 observed corporations increased their stock price, respectively. Finally, in 2011, the market had a relatively bad period with 41% of 1,447 observed corporations increasing their stock price. Therefore, these data bring us the opportunity to investigate the association between interlocking networks and stock fluctuations across different financial periods.

### 2.2 Interlocking networks and stock correlations

**D**of size $N\times N$, where

*N*is the number of observed traded corporations in a year, ${D}_{ij}=1/({d}_{ij})$, and ${d}_{ij}$ is the degree of separation (number of links) between corporation

*i*and

*j*in the interlocking network. Note that the degree of separation of a corporation to itself is ${d}_{ii}=0$, and unconnected corporations have an infinite degree of separation (${d}_{ij}=\mathrm{\infty}$). The greater ${D}_{ij}$, the higher the proximity between the two corporations in the interlocking network. Note that our matrix of network proximity takes into account all pairs of traded corporations.

Additionally, we define a matrix of market similarity $\mathbf{S}=\frac{1}{M-1}R{R}^{T}$ of size $N\times N$, where *M* is the total number of trading days in a year, and ${R}_{ik}=({z}_{ik}-\u3008{z}_{ik}\u3009)/{\sigma}_{{z}_{ik}}$ is the standard deviation normalized daily log returns of stock *i* in day *k* such that ${z}_{ik}=log({p}_{i}(k)/{p}_{i}(k-1))$ and ${p}_{i}(k)$ is the closing stock price of corporation *i* in day *k*. Therefore, the matrix of market similarity **S** corresponds to the correlation matrix ${\mathrm{\Sigma}}_{R}$ of *R*[23]. The higher ${S}_{ij}$, the higher the similarity or temporal correlation of the stock price movements of corporation *i* and *j* in the year.

We calculate partial Mantel correlations [24] (both Spearman and Pearson correlations) between the matrices of network proximity and market similarity while controlling for other proximity matrices given by market sector, geographic distance, board size, fraction of directors with financial expertise, and average stock price in the year (see the Appendix for further details). Following previous studies [25]–[29], we use the average stock price in the year as surrogate of firm size given its relevance and undisputed effect on the analysis of stock fluctuations. We also divide our analysis into the biggest market sectors (Basic Materials, Consumer Goods, Financial, Health Care, Industrial Goods, Services, and Technology), where sector matrices are simply sub-matrices of the full proximity matrices composed of corporations from a single sector.

To further support the validity of the correlations between network proximity and market similarity, we test whether corporations that increase or decrease their network proximity from one year to the next one also increase or decrease their market similarity accordingly. In specific, for each year from 2008-2011, we calculate a new matrix of network proximity given by the difference between the network proximity between two corporations in a given year ${D}_{ij}(t)$ and their network proximity in the previous year ${D}_{ij}(t-1)$ (only taking into account those corporations that are present in both years). Similarly, we generate a new matrix of market similarity for each year from 2008-2011, in which the new elements of each of these matrices are given by the difference between the market similarity between two corporations in a given year ${S}_{ij}(t)$ and their market similarity in the previous year ${S}_{ij}(t-1)$. Finally, for each year, we measure the correlation between the newly generated matrices of network proximity and market similarity.

Figure 2 (green/Y region) shows that the new correlations between proximity changes are, in fact, equivalent to the positive correlations in each year. Overall, these findings reveal that changes in the degree of separation between two corporations in the interlocking network are correlated with changes in their stock correlations. If the degree of separation decreases, the stock correlation increases, and vice versa.

### 2.3 Interlocking networks and stock markets

To unveil potential mechanisms explaining the previous association between stock correlations and interlocking networks, we study the behavior of one the main financial actors towards interlocking directorates. In specific, we analyze the perception and information gathered by stock analysts on traded corporations and whether this is associated with their trading activity. As proxy for the information collected by stock analysts, we use trading data and the electronic communication among a group of financial traders from a US trading company from 2007-2009 (see the Appendix). Previous work has revealed that traders are constantly tracking business press coverage of traded corporations and exchanging this information among their peers [21]. An illustrative example of this information concerning corporate directorates is the following message between two traders on July 7th 2008: ‘Microsoft willing to enter talks if Yahoo elects new board.’ In fact, it has been shown that traders’ communications can signal their understanding of market volatility [21], [22]. Therefore, this suggests that the more strategic decisions made by interlocking directorates, the higher the chances that there is a media coverage of relevance to traders, and the higher the potential that this correlates with their trading activity.

To test the above hypothesis, for each year from 2007-2009, we investigate the association of the frequency at which traders talk about traded corporations with the centrality in the interlocking network and the traded volume of such corporations. The frequency is calculated by the total number of times a ticker (e.g. GOOG as for Google) is mentioned over the entire year by this group of traders. We measure a corporation’s centrality by $\u3008{D}_{i}\u3009=\frac{1}{N-1}{\sum}_{j}{D}_{ij}$, where ${D}_{ij}$ is the network proximity between two corporations, as mentioned above. We find that similar results are obtained if we replace $\u3008{D}_{i}\u3009$ by other centrality measures, such as the total number of interlocking directors or the community participation coefficient [31]. The traded volume is calculated by the total amount of US dollars traded by this group of traders over the entire year.

### 2.4 Exposure to market fluctuations in interlocking networks

Finally, to test the extent to which interlocking networks can amplify the exposure of traded corporations to market fluctuations, we compare the centrality of corporations in the interlocking network with their stock performance over the year. The centrality of a corporations is again measured by $\u3008{D}_{i}\u3009$. We measure a corporation’s short-term and long-term stock performance by its beta (${\beta}_{i}$) and its yearly stock price return (${r}_{i}$), respectively.

The commonly known beta of a stock is given by ${\beta}_{i}=Cov({z}_{i},{z}_{b})/Var({z}_{b})$, where ${z}_{i}$ and ${z}_{b}$ are the daily log returns of stocks *i* and the benchmark return, respectively (see the Appendix). The higher (lower) the beta, the more the stock moves in the same (opposite) direction and farther apart from the benchmark return. This means that during good or bad market periods, the short-term stock performance of a traded corporation increases the higher or the lower the beta of the stock, respectively. The yearly stock price return is given by ${r}_{i}=log({p}_{i}({t}_{f})/{p}_{i}({t}_{o}))$, where ${p}_{i}({t}_{o})$ and ${p}_{i}({t}_{f})$ are the daily closing stock prices of corporation *i* at the beginning and at the end of the calendar year, respectively. The higher the yearly return, the better the long-term stock performance of a traded corporation. For each year from 2007-2011, we measure the effect of centrality on the short-term and long-term stock performance using a multivariate linear regression model controlling for the corporations’ individual characteristics (see the Appendix).

## 3 Discussion

US governance standards favor the participation of outside directors in corporate boards, which in turn allows the creation of interlocking directorates. While direct competitors are prohibited to create interlocking directorates, these corporations have on average only four degrees of separation. The financial sector is a good example of this market environment. There is almost no interlocking directorate between financial corporations, but they are indirectly linked by their interlocking directors sitting on the board of a third corporation belonging to a different market sector. Indeed, as we mentioned before, 85% of interlocking directorates are formed between corporations belonging to different market sectors. While interlocking networks facilitate the transmission of information, it has been unclear whether they can also explain stock fluctuations in the financial market.

Our findings have shown that interlocking directorates seem to be one of the factors favoring the existence of stock correlations. We acknowledge that other confounding factors can explain stock correlations. Interestingly, a potential mechanism explaining the link between interlocking networks and stock markets appears to be the enhanced attention of stock analysts to central corporations. It remains to be seen whether other measures of centrality can provide better insights about these associations. Importantly, because interlocking networks may amplify market fluctuations during financial crashes, future work should also explore the impact of the different dynamics found across market sectors. Importantly, it does not escape our notice that these results provide valuable non-opinionated insight for scholars, regulators, investors, and board members themselves.

## Electronic Supplementary Material

Below are the links to the electronic supplementary material.

## Appendix

### 5.1 Board composition data

We use the data extracted from *RiskMetrics*[18], which is a yearly compilation from 2007-2011 of the board composition of over 1,500 large US traded corporations. We use only those traded corporation on which we could find stock price information in our stock price data. Board members were manually disambiguated using gender, age, and affiliation data provided by *RiskMetrics*. These data also have information of whether a board member is consider a financial expert by the traded corporation. In some cases, the same board member is classified as a financial member in one corporations but a non-financial expert in a second corporation. We decided to classify board members as financial experts if they are considered to be experts at least in one corporation. The board composition and their financial expertise for each year is provided in Additional file 1.

### 5.2 Stock price data

Daily closing stock prices from 2007-2011 for these corporations are extracted from *WRDS* database [19], and stocks are categorized in a market sector according to *Yahoo Finance* criteria [20]. The stock data and market classification for each year is provided in Additional file 1.

### 5.3 Traders data

Our data includes the full population of more than 3 million instant messages sent/received and more than 1 million of trading decisions of day traders at a typical small-to-medium sized US trading company from 2007-2009. All trading related data was automatically captured by the company’s trading system, which is specially designed for accuracy in recording, and used by most other companies in the industry. The study conforms to Institutional Review Board (IRB) criteria. There was no subject interaction, all data was 100% archival, and the company and the subjects were anonymized. Legally, all data used in the study is owned by the company. All traders at the company know the company owns the data and that their communications and trading behavior is recorded by law. We received written permission from the company to use these data for research purposes and publishing contingent on identifying characteristics of the company and its traders remaining confidential and anonymous. For further details see Refs. [21], [22]. We use only those tickers that we were able to disambiguate across traders’ communications. Tickers’ mentions and traded volume are normalized (to preserve the company’s confidentiality) and provided in Additional file 1.

### 5.4 Control proximity matrices

Partial Mantel correlations between network proximity and market similarity are controlled by proximity in market sector **F**, average stock price in the year **T**, board size **B**, fraction of directors with financial expertise **E**, and geographic distance **G**. The proximity matrix **F** corresponds to ${F}_{i,j}=1$ if traded corporation *i* and *j* belong to the same sector, 0 otherwise. In the rest of the control proximity matrices, the elements ($i,j$) of these matrices correspond to the inverse of the normalized absolute differences between the value measured in corporation *i* and *j*, such that higher values always represent corporations with higher proximity. For proximity matrix **T**, ${T}_{ij}=(max\{k1\}-k{1}_{ij})/(max\{k1\}-min\{k1\})$ where $k{1}_{ij}=|\u3008log({p}_{i})\u3009-\u3008log({p}_{j})\u3009|$ and $\u3008log({p}_{i})\u3009$ is the average log price of stock *i* over the year. For proximity matrix **B**, ${B}_{ij}=(max\{k2\}-k{2}_{ij})/(max\{k2\}-min\{k2\})$ where $k{2}_{ij}=|{b}_{i}-{b}_{j}|$ and ${b}_{i}$ is the total number of board members of traded corporation *i* over the year. For proximity matrix **E**, ${E}_{ij}=(max\{k3\}-k{3}_{ij})/(max\{k3\}-min\{k3\})$ where $k{3}_{ij}=|{e}_{i}-{e}_{j}|$ and ${e}_{i}$ is the fraction of board members with financial expertise of traded corporation *i* over the year. For proximity matrix **G**, ${G}_{ij}=(max\{k4\}-k{4}_{ij})/(max\{k4\}-min\{k4\})$ where $k{4}_{ij}$ is the geographic distance between corporation *i* and *j* as function of their geographic coordinates latitude and longitude extracted from their zip codes. Zip codes were collected from *Yahoo Finance* and from *HighBeam.Com*. These data are provided in Additional file 1. We find a majority of negligible associations of market similarity with the control proximity matrices of firm size, board size, fraction of financial experts in the board, and geographic distance (see Figures S1-S4 in Additional file 2).

### 5.5 Robustness of correlations between network proximity and market similarity

We find that each of the positive correlations shown in Figure 2 is also higher than the expected correlation between network proximity and market similarity when we randomly generate new matrices of network proximity (see Figure S5 in Additional file 2). These new matrices are generated by bootstrapping elements with replacement from the original matrices of network proximity. Additionally, the observed correlations between network proximity and market similarity shown in Figure 2 cannot be reproduced if we disconnect boards (i.e., ${d}_{ij}=\mathrm{\infty}$) when a certain degree of separation has been exceed (see Figure S6 in Additional file 2).

### 5.6 Benchmark return

To calculate the benchmark return, we use the average daily log returns ${z}_{b}=\u3008{z}_{i}\u3009$ of all the observed corporations. For individual market sectors, we did not find statistical differences if we use $\u3008{z}_{i}\u3009$ or the average daily stock returns only of corporations from individual market sectors. The distribution of betas and yearly stock returns are shown in Figure S7 in Additional file 2.

### 5.7 Multivariate linear regression model

The model is defined by ${\mathit{performance}}_{i}\sim \u3008{D}_{i}\u3009+\u3008log({p}_{i})\u3009+{b}_{i}+{f}_{i}+{\u03f5}_{i}$. The response variable *performance* can take either the beta of the stock ${\beta}_{i}$ or the yearly stock return ${r}_{i}$. We control for the average price of the stocks in the year ($\u3008log({p}_{i})\u3009$), the size of the boards (${b}_{i}$), the fraction of directors with financial expertise in each board (${f}_{i}$), and the Gaussian noise (${\u03f5}_{i}$). We scale all predictor variables to be able to compare their effect. The model has a goodness-of-fit of ${\mathrm{R}}^{2}\sim 0.3$ for all years. Interestingly, the control variables board size and fraction of directors with financial expertise show a majority of negligible effects, while firm size yields positive and negative effects for short-term and long-term stock performance (see Figures S8-S10 in Additional file 2).

### 5.8 Mantel correlation

*n*objects represented by their similarity matrices ${X}_{ij}$ and ${Y}_{ij}$, i.e., dyadic data. For example in our case, network proximity and market similarity. The Pearson Mantel correlation coefficient is computed as

where $\overline{X}=\frac{2}{n(n-1)}{\sum}_{i>j}{X}_{ij}$ is the average $Xij$ value (similarly for $\overline{Y}$). Note that the sum is only made on the strictly upper triangular part of the matrices. This is so because similarity matrices are symmetric and have their diagonal elements equal to one. For the Spearman rank Mantel correlation, we first substitute the elements of the matrices ${X}_{ij}$ and ${Y}_{ij}$ by their respective rank. In a similar way we can compute partial Mantel correlations [32]. To compute the 95% confidence interval of the correlation coefficient, we use a bootstrap procedure [33]. We re-sample simultaneously the rows and the columns of the similarity matrices, and then for each re-sampling we compute again the Mantel correlation coefficient. The re-sampling procedure results in the empirical distribution of the Mantel correlation coefficient, from which we extract the 95% confidence interval.

## Notes

### Acknowledgements

Funding was provided by the European Research Council through an Advanced Grant (JB), the Spanish Ministry of Education through a FPU PhD Fellowship (LJG), FP7-REGPOT-2010-1 program under project 264125 EcoGenes (RPR), and the Kellogg School of Management (MS).

## Supplementary material

## References

- 1.(2013), [http://nysemanual.nyse.com/lcm/]
- 2.Mizruchi MS: What do interlocks do? An analysis, critique, and assessment of research on interlocking directorates.
*Annu Rev Sociol*1996, 22: 271–298. 10.1146/annurev.soc.22.1.271CrossRefGoogle Scholar - 3.Burt RS (2006) Interlocking directorates behind the S&P indices. School of Business, Chicago., [http://faculty.chicagobooth.edu/ronald.burt/research/IDS&P.pdf]Google Scholar
- 4.Dooley PC: The interlocking directorate.
*Am Econ Rev*1969, 59: 314–323.Google Scholar - 5.Carpenter MA, Westphal JD: The strategic context of external network ties: examining the impact of director appointments on board involvement in strategic decision making.
*Acad Manag J*2001, 4: 639–660. 10.2307/3069408CrossRefGoogle Scholar - 6.Pennings JM:
*Interlocking directorates*. Jossey-Bass, San Francisco; 1980.Google Scholar - 7.Palmer D, Barber BM, Zhou X: The finance conception of control: the theory that ate New York? Reply to Fligstein.
*Am Sociol Rev*1995, 60: 504–508. 10.2307/2096290CrossRefGoogle Scholar - 8.Fligstein N, Brantley P: Bank control, owner control, or organizational dynamics: who controls the large modern corporation?
*Am J Sociol*1992, 98: 280–307. 10.1086/230009CrossRefGoogle Scholar - 9.Fligstein N: Networks of power of the finance conception of control? Comment on Palmer, Barber, Zhou, and Soysal.
*Am Sociol Rev*1995, 60: 500–503. 10.2307/2096289CrossRefGoogle Scholar - 10.Khanna T, Thomas C: Synchronicity and firm interlocks in an emerging market.
*J Financ Econ*2009, 92: 182–204. 10.1016/j.jfineco.2008.03.005CrossRefGoogle Scholar - 11.Borgatti SP, Cross R: A relational view of information seeking and learning in social networks.
*Manag Sci*2003, 49: 432–445. 10.1287/mnsc.49.4.432.14428CrossRefGoogle Scholar - 12.Connelly BL, Johnson JL, Tihanyi L, Ellstrand AE: More than adopters: competing influences in the interlocking directorate.
*Organ Sci*2011, 22: 688–703. 10.1287/orsc.1100.0555CrossRefGoogle Scholar - 13.Davis GF: Agents without principles? The spread of the poison pill through the intercorporate network.
*Adm Sci Q*1991, 36: 583–613. 10.2307/2393275CrossRefGoogle Scholar - 14.Haunschild PR, Beckman CM: When do interlocks matter?: Alternate sources of information and interlock influence.
*Adm Sci Q*1998, 43: 815–844. 10.2307/2393617CrossRefGoogle Scholar - 15.Vedres B: The constellation of economic power: the position of political actors, banks, and large corporations in the network of directorate interlock in Hungary.
*Int Netw Soc Netw Anal*1997, 23: 44–59.Google Scholar - 16.Kogut B, Walker G: The small world of Germany and the durability of national networks.
*Am Sociol Rev*2001, 66: 317–335. 10.2307/3088882CrossRefGoogle Scholar - 17.Battiston S, Bonaneau E, Weisbuch G: Decision making dynamics in corporate boards.
*Physica A*2003., 322: 10.1016/S0378-4371(02)01930-1Google Scholar - 18.(September 2013), [http://wrds-web.wharton.upenn.edu/wrds/ds/riskmetrics/rmdirectors/index.cfm]
- 19.(September 2013), [http://wrds-web.wharton.upenn.edu/wrds/]
- 20.(September 2013), [http://biz.yahoo.com/p/s_conamed.html]
- 21.Saavedra S, Duch J, Uzzi B: Tracking traders’ understanding of the market using e-communication data.
*PLoS ONE*2011., 6: 10.1371/journal.pone.0026705Google Scholar - 22.Saavedra S, Malmgren RD, Switanek N, Uzzi B: Foraging under conditions of short-term exploitative competition: the case of stock traders.
*Proc R Soc Lond B*2013., 280: 10.1098/rspb.2012.2901Google Scholar - 23.Fenn DJ, Porter MA, Williams S, McDonald M, Johnson NF, Jones NS: Temporal evolution of financial market correlations.
*Phys Rev E*2011., 84: 10.1103/PhysRevE.84.026109Google Scholar - 24.Mantel N: The detection of disease clustering and a generalized regression approach.
*Cancer Res*1967, 27: 209–220.Google Scholar - 25.Banz RW: The relationship between return and market value of common stocks.
*J Financ Econ*1981, 9: 3–18. 10.1016/0304-405X(81)90018-0CrossRefGoogle Scholar - 26.Schwert GW: Size and stock returns, and other empirical regularities.
*J Financ Econ*1983, 12: 3–12. 10.1016/0304-405X(83)90024-7CrossRefGoogle Scholar - 27.Cheung YW, Ng LK: Stock price dynamics and firm size: an empirical investigation.
*J Finance*1992, 47: 1985–1997. 10.1111/j.1540-6261.1992.tb04693.xCrossRefGoogle Scholar - 28.Davis LJ: The cross-section of realized stock returns: the pre-COMPUSTAT evidence.
*J Finance*1994, 49: 1579–1593. 10.1111/j.1540-6261.1994.tb04773.xCrossRefGoogle Scholar - 29.Berk JB: A critique of size-related anomalies.
*Rev Financ Stud*1995, 8: 275–286. 10.1093/rfs/8.2.275CrossRefGoogle Scholar - 30.Moro E, Vicente J, Moyano LG, Gerig A, Farmer JD, Vaglica G, Lillo F, Mantegna RN: Market impact and trading profile of hidden orders in stock markets.
*Phys Rev E*2009., 80: 10.1103/PhysRevE.80.066102Google Scholar - 31.Guimerá R, Amaral LAN: Functional cartography of complex metabolic networks.
*Nature*2005, 433: 895–900. 10.1038/nature03288CrossRefGoogle Scholar - 32.Legendre P: Comparison of permutation methods for the partial correlation and partial mantel test.
*J Stat Comput Simul*2000, 67: 37–73. 10.1080/00949650008812035MathSciNetCrossRefGoogle Scholar - 33.Bradley E, Tibshirani RJ:
*An introduction to the bootstrap*. Chapman & Hall/CRC, Boca Raton; 1994.Google Scholar

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