Discussion Paper No. 33

Correlation Persistence in Financial
Markets: A Network Theory Approach

Hongwei Chuang
February, 2015

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANCorrelation Persistence in Financial Markets: A Network
Theory Approach
Hongwei Chuang
Graduate School of Economics and Management, Tohoku University27-1 Kawauchi, Aoba Ward, Sendai 980-8576, Japan
hongwei@econ.tohoku.edu.tw

Abstract
I propose a persuasion model to show the correlation of stock price can substantially vary over time in a general network economy in which investors update theirown information following by the Bayesian rules and evolving in the social network
effect. To describe the mutual interconnections and model the financial network,
an empirical strategy, Granger-Causality network, is proposed based on the multivariate vector autoregression (MVAR) models. The empirical study is conductedby examining the unique dataset that consists of all brokers’ daily trading information for a decade in Taiwan. The empirical results show the density of brokers’
financial network can be positively correlated to the market realized volatility. Moreover, the density can also be viewed as an indicator to the systemic risk of market.
The empirical evidence shed new light not only on providing an explanation to thephenomenon of correlation persistence but also measuring the systemic risk of themarket from the perspective of network.
Key words: Financial Network; Realized Volatility; Long Memory; Systemic Risk
JEL classification: C3; G2

1. Introduction
Rational choice theory is widely used for modeling individual behavior in making economic decisions that assumes a person will be able to fully understandthe consequences and take all the time they need for every choice they make.
But, in reality, an individual’s rationality is limited by the information he has,the cognitive limitations of their minds, and the finite amount of time they
have to make a decision. Instead, it is argued that most speculative traders
in stock market do not have any source of private information regarding asset
payoffs, instead, they seem to make trading decisions based on media accounts,
tips from friend, advice from analysts and experts, etc. Therefore, an individual who anticipates a future change in his expected believes should incorporatea wide variety of settings such as persuasion (DeMarzo, Vayanos and Zwiebel(2003)). Since these speculators believe their information are superior than1others, they tend to persuade other investors once the communication was
taken place.
This paper mainly studies the dynamics of volatility that have long been
documented in the literature. Return volatility of individual stocks and markets varies over time and leads to a heavy-tailed unconditional return distribution (Bollerslev and Jubinski (1999); Lobato and Velasco (2000)). Furthermore, the phenomenon of long memory is observed only after applying certaintransformations such as square or absolute value to the series. A common
explanation for such time-varying volatility is lumpy diffusion of information
into the market (Clark (1973); Epps and Epps (1976); Andersen (1996)). In
this study, I propose a persuasion model to study information communicationon the effect of market volatility. In this economy, individuals can exchange
their opinions once word-of-mouth (Hong, Kubik, and Stein (2005)) communication occurs. Investors dynamically update their believes evolving in a waythat weight between their subjective opinions and information of individuals
whom they listen to relative to other individuals. For example, investors iand j each have different information at time t, xti and xtj . Suppose that investors i and j meet. Investor j tries to “persuade” investor i by revealing his
information. If investor i “listens to” investor j, the updating rule for i is:
xt+1= (1 − λi )xti + λi xtj
i(1)

where λi ∈ [0, 1) represents the parameter of social isolation. The weight
depends on the magnitude of investor’s social isolation. If investor is highlysocial isolated, i.e., λi → 0, there would be no information persuasion withhis neighbors so that the believes in determining his invest decisions remains
unchanged.
The equilibrium analysis suggests that the information communication
structure can affect volatility dynamics in the market. And, the price volatility
would vary substantially over time in a general network economy. To implement the empirical study, I collect the dataset which contains all daily trading
information from November 6, 2001 to March 2, 2011 in Taiwan stock market,
covering the period of financial crisis in 2007 with The sample records theinformation of trading stocks, order types (buy or sell), trading amounts, andtrading prices for each broker. During the sample period, there are total 95
brokers and 1330 common stocks that have complete trading records over the2314 trading days.
This study focuses on studying the brokers trading data instead of using
the account level. It is because of preventing tracking those infrequent trading
behaviors in stock market. Since the majority of investors are rarely trade,
identifying their mutual interaction could easily become irrelevant. The brokers trading dataset can also prevent constructing the sparse trading matrix
2and raising the estimation issues. 1 Most importantly, my analysis of the sizeof the brokers financial network would be more stable over time.
The brokers trading data can be viewed as the trading records of a group ofinvestors. To mimic each brokers’ short- and long-term realized trading gains
and losses, I adopt the method developed in Barber, Lee, Liu, and Odean(2009) but focus on brokers’ trades rather than on investor groups. A statistical methodology I proposed is to identify the possible linkages based on
the property of the brokers’ portfolio holdings. To capture the information
diffusion process by a directed network structure, I adopt the multivariate
vector autoregressive (MVAR) models to empirically measure their mutual interconnectedness from a system-wide perspective. 2 The financial networks I
constructed are in the spirit of Granger-Causality according to the parameter
estimations of MVAR(1) model in Section 3.
The empirical results indicate that when the brokers network density increases, the market realized volatility would increase in the following. It makes
the dynamic network density can be a systemic risk indicator, i.e., a probe to
the market status. The remainder of the paper is organized as follows: Section
2 introduces the model. Section 3 describes the data and methodology. The
main empirical results are illustrated in Section 4. Section 5 concludes.
2. ModelAssumes that there are N investors, enumerated by i ∈ N = {1, 2, ..., N } ina T + 1-period economy, t = 0, ..., T. Each investor maximizes his expectedutility of terminal wealth, and has constant absolute risk aversion (CARA)
preferences with homogeneous risk aversion coefficient, γ,
Ui = E[−e−γWi,T ],(2)where Wi,T is investor i’s terminal wealth.
The net supply (liquidity) of the risky asset traded in the economy is L.
The terminal value, v, of the risky asset is assumed asv = v̄ + η.

(3)

The first component of v is a weighted sum of information coming from H pub1

As shown in Gomez-Rodriguez, Leskovec, and Krause (2012), the exact maximum
likelihood estimation is not feasible for large networks.
2 The definition of linkage in this study is different from that of Ozsoylev, Walden,
Yavuz, and Bildik (2013) who define the connection between each pair of investors if
the two agents traded in the same stock in the same direction in a short-term period.
Mine is also different from Pareek (2012), who defined that two fund managers who
allocate 5% or more of their portfolio on the same stock are connected to each other.

3

lic information sources, like financial statements, company reports, macroeconomic announcements, etc that are available publicly known by all investors.
It is denoted by
v̄ =

H
X

wh vh .

(4)h=1

The second component, η, is independent of the vh ’s and follows N (0, σv2 ) whichis unobservable. Therefore, the value of the risky asset is normally distributedwith mean v̄ and variance σv2 . The precision is denoted as τv = σv−2 .
At time 0, each investor receive a noise signal about the asset’s value fromthe H sources,
s0i =

H
X

wi,h vh + i

(5)

h=1

where wi,h is denoted as investor i’s subjective weight on information source hand i ∼ N (0, σ 2 ) is independent across information sources vh and investors.
Investors use their initial signals to form an estimate of the asset’s payoff. At
time T , the true value of the asset, v, is revealed. The precision is denoted asτ = σ −2 .
In this study, I assumed that investors are connected in a simple directed
graph 3 and updated their information via a Bayesian rule which are different
from the setting of Walden (2013).
Investors’ Network
Investors are assumed to be connected in a network, represented by agraph G = (N , E). The relation E ⊂ N × N describes which investors areconnected in the network. Investors gain information from other investors
about the risky payoff via their connection within a network that can bemodeled by vertices (nodes) representing the investors and the directed edges(links) representing the direction of learning. A convenient representation of
the network is by the adjacency matrix A ∈ {0, 1}N ×N with
 1 if investor j is directly linked to i,

Aij =  0 otherwise.

Here, I assume that investors are self-connected. That is, Aii = 1, ∀i. In the mymodel, investor gain information from other investors via persuasion. Whoman investor gains from is determined by a network which can be represented
as following.
3

A graph is “simple” if multiple edges between the same pair of vertices are forbidden. A graph is called “directed” if edges exhibit inherent direction, implying
every relationship presented is asymmetric.

4

Definition 1 For a given network G ∈ G,
D(i) ≡ {k ∈ {1, ..., N } \ {i} : k → i}(6)

is defined to capture investor i’s information structure.
Note that the vector D = (D(1), ..., D(N ))0 fully captures the interaction
among investors impose by the network G ∈ G. I refer k ∈ D(i) as an information source for investor i. In additional, an investor is always an information
source for himself.
Information Updating Rules
The graph G determines how investors share information with each other.
Specifically, at time t + 1, investor i shares all signals he had received up until
to time t with all his neighbors. Let Sit denote the information set that investor
i has received up until t, either directly or via his network. The dynamics ofsti ∈ Sit are given the recursive formulasti = (1 − λi )st−1
+ λi
i

X

Tij stj(7)j∈D(i)

where λi ∈ [0, 1) is a parameter that reflects the extent to which social networkof friends is relevant to investor. It is denoted λ = (λ1 , ..., λN )0 . The term (1 −λi ) is the weight that investor i places on his own public information at timet − 1. Here, the parameter λi can be intuitively denoted as i’s degree of social
interaction. Investor i is said to be socially isolated if i’s time-t information is
not influenced by the social network of friends. Social isolation shuts down thesocial channel through which the provision of information affect i’s believe.
Tij denotes the probability that investor i listen to investor j in i’s network,where Tij > 0 if j ∈ D(i) and Tij = 0 otherwise. And
X

Tij = 1.

(8)j∈D(i)

Definition 2 In most cases, I assume the investors are uniform listening,that is,1
for j ∈ D(i).
(9)
Tij =
#D(i)
An Illustration
Consider a network G with N = 5 investors as shown in the directed graphof Figure 1. The information structures of investors are D(1) = {2, 3}, D(2) =∅, D(3) = {1, 2, 5}, D(4) = {1, 2, 3, 5}, D(5) = {1}. The uniform listening
5

Fig. 1. Illustration of a social network (N=5)

matrix for Figure 1 is:




1 1
0 2 2 0 0
0 0 0 0 0




1
1
1

T =3 3 0 0 3.

1 1 11 
0
4 4 4
4 



(10)

10000

3. Data and Methodology
Taiwan Stock Market
My dataset comes from the Taiwan stock market, which is the world’s
12th largest financial market. During the sample period, the Taiwan Stock Exchange (TSE) opens from 9:00 AM to 1:30 PM. Buy and sell orders interact todetermine the executed price subject to applicable auto-matching rules every
90 seconds. Orders are executed in strict price and time priority. A daily pricelimit of 7% holds in each direction and a trade-by-trade intraday price limit of
two ticks from the previous trade price. The commission fee of TSE is 0.1425%
of the trading value and a transaction tax on stock sales of 0.3%. Capital gains
are not taxed, whereas cash dividends are taxed at ordinary income tax rates
for domestic investors and at 20% for foreign investors. Corporate income istaxed at a maximum rate of 25%, whereas personal income is taxed at a maximum rate of 40%. The accumulated investor account numbers at securities
companies is approximately 16 million in 2011, which is almost 76% of thetotal population.
Brokers Daily Trading Information
I collect all daily trading information of all brokers in Taiwan over the period from November 6, 2001 to March 2, 2011 to carry out my empirical study.
6

I chose the brokers’ dataset as opposed to account-level data because tracking those infrequent investors’ trading behavior and identifying their mutualinteraction could easily make noise and become irrelevant to study investors’
trading behavior via networks. Most importantly, it would also prevent toconstruct the sparse trading matrix. The dataset includes the stocks, ordertypes (buy or sell), trading amounts, and trading prices that are traded byeach broker every day. Therefore, I can investigate the trading behavior of alarge amount of investors within a financial network in a market through thetrading information of brokers. Further, these brokers can also be categorized
into five types as banking (13 brokers), bills finance corporations (1 broker),
specialized brokerage firms (32 brokers), integrated securities firms (31 brokers), and foreign financial institutions (18 brokers). During the sample period,
there are total 95 brokers and 1330 stocks have complete trading information.
The average the trading amount of these brokers is approximately 97% of thedaily whole market’s total trading amount.
Short- and Long-term Trading Gains and Losses
The first step of my empirical work is to construct each broker’s daily buy
and sell portfolios to mimic a group of investors’ daily stocks purchase andsale over a period of time. I use the method similar to Barber, Lee, Liu, andOdean (2009) but focus on brokers’ trades in the short- and long-term trading
period to evaluate a group of investors’ trading gains and losses in each broker,
which are denoted by two vectors Gt = (g1,t , ..., gn,t )0 and Lt = (`1,t , ..., `n,t )0 attime t respectively. Take the trading of broker Yuanta Securities on March 29,
2002 as an example, the mimicking portfolios can be constructed as follows:
if Yuanta Securities buys 900 shares of HTC and sells 700 shares, I add 200
shares of HTC on the buy portfolio, whereas no HTC shares are added to thesell portfolio on that day. The purchase price will be recored as the difference
between the total value of buys and the total value of sells divided by the net
shares. I consider the shares being included in the portfolios for a fixed horizon,
z, with the window length of 5 (short-term) and 20 (long-term) trading days.
zare
The volume-weighted (VW) realized trading gains denoted by gi,ttherefore calculated as:
z
gi,t
=

ni
X

#net shares of stock j purchased× rjt+1 ,
#net
total
sharespurchasedbybrokerij=1(11)

where rjt+1 is the one-day return of the stock j after traded at time t andni denotes the number of stocks that the broker i holds considering z-days
holding period. An analogous calculation occurs for the VW realized trading
losses, `zi,t , which is defined as`zi,t =

ni
X

#net shares of stock j sold× rjt+1 .
#net
total
sharessoldbybroker
i
j=1
7

(12)

z
and `zi,t , that is,
The net trading profit, pzi,t is defined as the difference of gi,tzpzi,t = gi,t− `zi,t .

(13)

In the following study, I use the net trading profit time series, pzt , for eachbroker to fit the multivariate time series model as in the following Section andconstruct the brokers financial network to describe the mutual interconnectedness.
Multivariate Vector Autoregression (MVAR)
To construct the financial networks of 95 brokers, MVAR models can beused to empirically measure their mutual interconnectedness from a systemwide perspective. The network structure could therefore be inferred by using
the Granger-Causality test for investigating whether one time series can provide forecasting power to others and therefore to determine the interconnectedness among these brokers. The causality can therefore reflect both statistic
correlations and economic connections among the multivariate time series ofa group of investors’ net trading profit.
MVAR model is to consider Yt = (y1t , ..., ynt )0 , t = 1, 2, ..., T, which denotes a weakly stationary multivariate time series of dimension n defined ona probability space (Ω, F, P). A p-th order vector autoregressive can be represented in a simple form as
Yt = c +p
XΦk Yt−k + t ,

(14)

k=1

where c denotes an (n × 1) vector of constants and Φk denotes an (n × n)
matrix of autoregressive coefficients for k = 1, 2, ..., p. The t is a vector withΣ symmetric definite matrix. The stationary condition is satisfied if all rootsof |Φ(Z)| = 0 lie outside the unit circle.
For a large VAR model (n > 2), the Granger-Causality test can be used totest whether one variable is influenced only by itself and not by other variablesin the model system. Yt can be arranged and partitioned in subgroups Y1t and
Y2t with dimensions n1 and n2 (n = n1 + n2 ), respectively. VAR model can be
represented as a matrix form:










=

Φ11 (β) Φ12 (β) Y1t 

Φ21 (β) Φ22 (β)

Y2t









+

.

C1 

C2

Σ1 

Σ2

(15)

The Wald statistic to test H0 : Cβ = c, where C is a s × (n2 p + n) matrix ofrank s and c is an s-dimensional vector can be obtained from√d0 −1
T (C β̂ − c)[C(Γ̂−1− χ2 (s).
(16)
p ⊗ Σ)C ] (C β̂ − c) →8

In this study, I set the significance level of α = 0.01.
Granger-Causality Network
Based on the MVAR model, the financial network I construct follows thesimilar concept of Grange causality as in Eichler (2007). I define a network
G = (V, E), where V is a set of elements called nodes (95 brokers in thisstudy) and E is a set of directed edges which belong to the class{j → i|i, j ∈ V, i 6= j} ∈/ E ⇔ Φij (k) = 0 ∀k,(17)where Φ is the autoregressive coefficients matrix in Equation (14). A network
can be represented by an adjacency matrix A ∈ {0, 1}n×n , with

Aij =



 1 if broker j is directly linked to i,(18) 0 otherwise.

Moreover, I assume that a broker is connected with himself, that is, Aii = 1
for all i. This class represents the directed edges corresponding to direct causal
relations between the components of Yt which can be identified by GrangerCausality test as shown in Equation (16). For simplicity, I consider the caseof k = 1 and use a 180-day rolling window to estimate Φ in Equation (14).
4. Empirical Results
All economic networks are heterogeneous with respect to both their agents andinteraction strength which can also strongly vary in time. Previous studies of
efficient and equilibrium networks assumed homogeneity. The most oblivious
stylized fact of realized volatility has shown long range dependence (Andersen,
Bollerslev, Diebold and Labys (2003)), which means the autocorrelation of
the square and absolute returns shows very strong persistence that lasts forextended periods. Schennach (2013) shown that long memory can arise in a
simple linear homogeneous economic subsystems with a short memory areinterconnected to form a network.
Realized Volatility (RV) and Density
For calculating the market realized volatility, I follow the Andersen, Bollerslev, Diebold and Labys (2003) by using the one-minute data of Taiwan Capitalization Weighted Stock Index (TWSI) over the period from October 25,
2002 to March 2, 2011. The time series plot of TWSI RV and short-term density is in Figure 1. The time series plot of TWSI RV and long-term density
is in Figure 2. It can find that when the density of brokers financial network
becomes higher, the following market realized volatility almost rises as well.
Regression Analysis
9Fig. 2. TWSI RV and Short-term Density

Fig. 3. TWSI RV and Long-term Density

Ozsoylev and Walden (2011) predicts that asset volatility increases as the
network centralization increases. To test the prediction, we implement the
regression of
RVt = αz + β z DEN Stz + t ,(19)
where RV is the market realized volatility, DEN S is the density of brokers
financial network in different time of z. The main regression results are shownin Table 1.
From Table 1, it can find that the slopes are all very significant andpositive. One standard deviation increase of brokers financial network density
leads to 0.226% increase in market realized volatility for z = 5. For z = 20,
0.193% increase in market realized volatility.
10Table 1
Regression of Realized Volatility on Network Density (×10−3 )αz

βz

Significance-0.0683

2.26**(0.1229)(0.9040)-0.0203

1.93(0.0991)(0.7390)

z=5

z=20

***

Density and Systemic Risk
Moreover, I also plot the figures to illustrate the relationship between
price and short- and long-term density in Figure 4 and Figure 5. From theboth figures, we can find the time series plots of densities like reflections to
the price pattern. When the density is at the relative low status, the priceoften reaches at the peak. The empirical results shows that the density can be
an indicator to the systemic risk for the market.

Fig. 4. TWSI and Short-term Density

5. Conclusions
When explaining and understanding the social reality in which we exist, if thecomplexity of society rises, precise categorical statements will lose meaning
and meaningful statements will cease to be precise and categorical. Fortunately, with the development of complex network analysis, networks become anatural tool to help obtain more fundamental insights in understanding bothsocial and economic phenomena, such as contagion, globalization, information
diffusion, neighborhood effects, and social learning.
Although networks have been appeared in almost every aspect of science11Fig. 5. TWSI and Long-term Density

and technology, extracting insights from networks of trading behavior withina market remains a less undeveloped area. This study primarily investigates
the relationship of the density of brokers network and the realized volatility ofmarket. My contribution to the network literature is to use a persuasion model
to explain the correlation persistence in the market and provide a multivariate
time series model to measure the mutual interconnectedness among a groupof investors. The empirical results indicate that network density is positively
correlated with the realized volatility of market. When the network density
increases, the following realized volatility of market also becomes large as well.
Acknowledgment
Thanks for seminar participants in National Chung Hsing University, Tohoku University and 2014 Japanese Association of Financial Econometrics and Engineering
Summer Meeting in Seijo University. Thanks for the research found supports from
Institute of Statistical Science and Institute of Sociology in Academia Sinica andthe Data Science and Service Research (DSSR) Center in Tohoku University. I am
solely responsible for any mistakes in these pages.

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