Chinese Stock So of i

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Nonlinear Interdependence of the Chinese Stock

Markets

Abdol S. SOOFI a, Zhe LI b, Xiaofeng HUI b

a University of Wisconsin-Platteville, Department of Economics

b School of Management, Harbin Institute of Technology

Abstract

The methodologies and assumptions in financial integration studies are

problematic and may lead to spurious empirical results. Using surrogate

data analysis and mutual prediction method of testing for nonlinear in-

terdependence, it is feasible for an analyst, with a scant knowledge of the

underlying dynamics of two dynamical systems, to show whether the sys-

tems are interdependent. This study applies these techniques in testing for

synchronization of three Chinese stock markets: Shanghai, Shenzhen, and

Hong Kong.

The empirical results of the present study indicate that the stock mar-

kets series are nonlinear and that the Chinese stock exchanges are non-

linearly interdependent. Specifically, the evidence indicates that Shanghai

and Shenzhen markets are bi-directionally interdependent, while Shanghaiand Hong Kong as well as Shenzhen and Hong Kong markets are unidirec-

tionally interdependent, with the direction of interdependence going from

the mainlands markets to the Hong Kong market.

Classification Code : C14, C22, F36Keywords: Chinese stock markets, financial integration, mutual prediction, non-linear interdependence

We are grateful to Lianqyue Cao and anonymous referees for their helpful comments on theearlier drafts of this paper. This study was partially supported by a grant from The NationalNatural Science Foundation of China (70773028).

Corresponding author: Abdol S. SOOFI.Address:University of Wisconsin-Platteville, Department of Economics, Platteville WI 53818,U.S.A. Tel.:608-342-1834;Fax: 608-342-1036. E-mail addresses: [email protected]

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1 Introduction

Measuring integration of financial markets within a country or across nationalborders has been a subject of keen interest in financial economics. However,economists rarely agree on definition of the term or the method of measurementof financial integration. In this study, we give the term a more precise meaning bystating that the market for a financial instrument is fully integrated if all potentialparticipants in the market with the same characteristics face a single set of rules,have equal access to the financial market, and are treated equally when they areactive in the market(Baele,et al., 2004). This definition is qualitative in naturewhich is of little value in measuring the degree of integration of the financialmarkets. Below we will provide an exact definition of financial integration and useterms such as financial integration, markets synchronization, and interdependenceof financial markets synonymously.

In measuring financial integration a number of methodological approaches

have been utilized in the past. Many of these earlier studies that are based on in-ternational asset pricing models (IAPMs) use dichotomous integration-segmentationhypothesis, emphasizing that financial markets are either fully integrated or to-tally segmented (Errunza, Losq, Padmanabham, 1992; Stulz, 1981). In additionto the problematic nature of the dichotomous assumption, (IAPMs) are based onrather restrictive assumption of purchasing power parity, a theorem that is basedon holding of the law of one price.

A newer methodology used in more recent capital markets integration stud-ies is less restrictive, and proposes to measure financial integration between twocountries by quantifying the co-movements of innovations in future expected stockreturns. Nevertheless, the method assumes that asset returns are conditionally

multivariate normal concluding that capital asset pricing model holds (Henry,2000; Phylakis and Ravazzolo, 2002). Moreover, these methods assume the un-derlying data generating processes of financial data are linear and stochastic,while tests indicate that data generating processes of many financial time se-ries are nonlinear (e.g. Scheinkman and LeBaron, 1989; Hsieh, 1991; Soofi andCao, 2002; Soofi and Galka, 2003), making the empirical results of such modelsquestionable.

Another strand of integration studies considers price-based, news-based, andquantity-based measures of integration that hinge on working of the law of oneprice.

The price-based measures concentrate on discrepancies between assets of thesame class created by geographic origin of the assets. While the news-based mea-sures are based on the arguments that in a well-integrated financial market newsof local character would have little or no effect on the asset prices. However, newsof global nature and significance would impact the integrated financial markets.The quantity-based measures focuses on cross-border holding of financial assetsby the institutional investors (see Baele, et. al., 2004, for details).

To address some of the methodological difficulties of the linear correlationand international asset pricing models with the latter method employing factormodels, a number of approaches were adopted by researchers in the subsequent

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studies. These methods include multivariate extreme value theory (Longgin andSolnik, 2001), asymmetric multivariate GARCH-M models ( Bekaert and Wu,2000), Poisson jumps (Das and Uppal, 1999), regime-switching models (Ang andBekaert, 2000).

Recently, new approaches in measuring interdependence of financial markets,mostly in the context of cross-country transmissions of financial shocks and con-tagion of financial crisis, have emerged (see for example, Pesaran and Pick, 2007;Dungey,et al., 2005; Bae, Karolyi, and Stulz, 2003). In contrast with the earliercontagion studies that focused on correlation of asset returns (see references tothese earlier works in Bae, et al., 2003), the work by Bae et al., do away withcorrelation as a linear measure of association altogether, and adopt multinominallogistic regression in assessing probability of joint occurrences of large absolutevalue daily returns.

It is important to point out that like many other researchers, Pesaran and Pick(2007), and Dungey al., 2005, distinguish interdependence of financial markets

during the normal times and during periods of high market volatility. Theseauthors specify econometric models that are supposed to distinguish the twoperiods. However, as it is discussed below, specification and estimation of thesemodels are wrought with a great deal of difficulties.

In this paper, we aim to introduce a new approach in testing for nonlineardynamical interdependence between financial markets. This new approach isreadily applicable for testing for cross-country as well as domestic integrationof financial markets. We adopt mutual prediction method in measuring stockmarket integration (markets synchronization) and apply the method in testingfor nonlinear interdependence of the Chinese stock markets: Shanghai, Shenzhen,and Hong Kong. We choose Chinese stock markets as a good example of a fullyintegrated financial system, as the term is defined by Baele, et al. (2004) above.

We would like to emphasis that the method we use, as it stands currently,does not differentiate between contagion and regular financial interdependence.However, for completeness of discussion of the topic of financial integration, wefind it useful to discuss contagion as a special form of financial interdependencein the present study.

This paper is organized as follows. In section 2 we discuss a canonical modelof interdependence (contagion) and review the problems that are associated inestimating the model (Pesaran and Pick, 2007). In section 3 we discuss surro-gate data analysis and method of mutual prediction in detection of nonlinear

interdependence between two dynamical systems (synchronization of oscillatingsystems). Section 4 deals with implementation of the methods discussed in sec-tion 3 on Chinese stock market data and presents the empirical results. Section5 summarizes and concludes the paper.

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2 Methods of testing for interdependence of the

financial markets

For completeness of discussion of financial integration, it is instructive to dis-

tinguish between the good financial interdependence and the bad financialinterdependence (contagion), when the latter occurs during financial crisis. Mas-son (1999) distinguishes normal interdependence of markets and financial con-tagion by identifying three categories of cross-country transmissions of shocks:Monsoonal effect, spill-over effect, and pure contagion. Monsoonal effect occursbecause correlation of economic fundamentals for two or more countries. Thespill-over effect occurs because of external relationship such as trade betweencountries. Finally, pure contagion occurs because of presence of multi-equilibriaof markets and movements of the markets from an optimal equilibrium to a sub-optimal one. The first two categories are considered normal interdependence,while the last one is contagion, a state of market(economy) that experienceshigher correlation between the fundamentals. Accordingly, interdependence offinancial systems is the necessary but not the sufficient condition for presence ofcontagion.

2.1 A canonical model of interdependence of financial mar-

kets

To rigorously define the terms financial interdependence and contagion and dif-ferentiate between the two, we present the model by Pesaran and Pick (2007)developed a two-country canonical model, as follows :

y1t =

1zt+

1x1t+1I(y2tc22,t1) +u1t (1)

y2t =

2zt+

2x2t+2I(y1tc11,t1) +u2t, (2)

whereyit,i = 1, 2;t= 1, , Tis a performance indicator that could be an assetprice, a market, or a country; u1t and u2t are serially uncorrelated errors withzero means, conditional variances 2u1,t1 and

2u2,t1

, and a non-zero correlationcoefficient. xit are (ki1) country-specific observed common factors, are pre-determined, and are distributed independently ofujt for all i andj. The (s1)vector zt contains pre-determined observed common factors such as petroleumprices in international markets. I(.) is an indicator function and is defined as

follows:

I(yitcii,t1) =

1 if (yitcii,t1)> 00 otherwise

where2i,t1= V ar(yit|t1), and t1 is the information set at time t1.Using the simultaneous equations system (1) and (2) interdependence of the

markets is measured by non-zero values of, and contagion is measured by non-zero values of i. These equations are nonlinear in the endogenous variablesyt = (y1t, y2t)

, and linear for the parameters under the simplifying assumption

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that the threshold values ofc1 and c2 are known. Furthermore, consistent, effi-cient estimation of the parameters of the model requires additional simplifyingassumptions such as existence of one country-specific regressors in each equation.Furthermore, it is assumed that the regressors are stationary, strictly exogenous,

and are distributed independently of the error terms u1tandu2t. These are highlyrestrictive assumptions that may lead to dubious empirical results in practice.Also, see, Pesaran and Pick (2007) for the problems associated with estimationof the model.

Nevertheless, in spite of the above mentioned restrictive assumptions, Pesaranand Pick (2007), provide a formal framework for testing for financial integra-tion (interdependence) of two markets. This framework is consistent with theassumptions of linearity and normality that commonly appears in the financialeconometric literature. However, we provide another formal definition of financialintegration (interdependence) in the framework of nonlinear dynamical systemstheories in section (3.1) below, which is non-parametric and does not pose any

assumptions or restrictions on the data. The method works equally well on sta-tionary and non-stationary series.

2.2 Volatility, financial interdependence, and coupled non-

linear oscillators

We consider financial markets as dynamical systems with varying degrees ofvolatility or oscillation. These oscillating systems could be nonlinear and insome instances chaotic. Nonlinear oscillators do not necessarily mean chaoticoscillators. Nonlinearity is the necessary condition for a chaotic system. A linear

system cannot be chaotic. Study of coupled oscillators is an important, activefield of research in physics, which has many applications to communications andcontrol. To have a synchronized set of nonlinear oscillators they must be coupled.

We do know that financial markets of developed, open economies are coupled.The coupling of these markets is due to free flow of news, money, and financialassets across borders. What is not clear, however, is whether these coupledfinancial markets with varying degrees of volatility are synchronized or becomesynchronized after passage of adequate time.

The methodological problems in assessing the degree of financial integrationin one country or the degree of cross-country interdependence of the financialmarkets, as stated in the previous section, lead us to search for powerful tools

from nonlinear dynamical systems theories that were developed to test for theextent of synchronization of nonlinear systems with little a priori knowledge oftheir underlying dynamics.

It turns out that a major strand of research on nonlinear dynamical systemsof recent years deals with the issue of interdependence of dynamical systems,and asks the following important question: Given the time series data for twodynamical systems, and without a knowledge of the exact nature of their under-lying dynamics, to what extent, if at all, does one system influence the behaviorof the second system?

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In answering this question, Schiffet al. (1996) and Pecora et al. (1996) de-veloped tests for dynamical interdependence in bivariate time-series data. Thesetechniques have found many applications in natural and physical sciences, buttheir use in financial economics is rare or nonexistent.

In present study, we consider two stock markets as coupled nonlinear oscil-lators, and proceed to test for synchronization of the markets using the methodproposed by Schiffet al. (1996), which is called mutual prediction method.

The study of dynamical interdependence of nonlinear oscillators, commonlyknown as synchronization in physics literature, has its origin in the works of Fu-

jisaka and Yamada (1983), Afrainmovich, Verichev, and Rabinovich (1986), andPecora and Carrol (1990). A variety of approaches to synchronization studies, in-cluding system-subsystem synchronization, synchronization in unidirectional andbidirectional coupled systems, anti-phase synchronization, partial synchroniza-tion, pulse-coupled synchronization, and generalized synchronization have beendeveloped.

The theoretical framework for generalized synchronization of coupled nonlin-ear oscillators where the form of coupling is unrestricted exists. However, theuse of the model in practical applications involves considerable difficulties (SeeRulkov, et al., 1995). In present study, we assumeforced synchronization, wherethe full coupled system consists of one market as an autonomous driving systemand the second one as a response system.

In communications or control systems, selection of an oscillator as autonomousdriving system and the second one a response system may have material conse-quences; however, in this study it makes no difference which market is a driveror response, since we test for forced synchronization by considering each mar-ket as autonomous (hence the second one as a response) system. By doing so,we cover all possibilities. In short, we are testing for presence of unidirectionalsynchronization between otherwise independent systems. This implies that iden-tifying a system as endogenous or exogenous is not an issue here. Moreover, to beconsistent with the terminology of financial literature, instead of using synchro-nization of coupled oscillators, we use terminologies of nonlinear interdependenceand financial integration interchangeably.

3 Mutual prediction and surrogate data anal-

ysis for detection of dynamical interdepen-

dence

Before applying the methods and algorithms from nonlinear dynamical systemstheories to the stock market data in this study, one should establish that the timeseries are nonlinear. We use the surrogate data technique (Theiler,et al., 1992)in testing for nonlinearity of the stock market indices. We give a more detaileddiscussion of this method in section 4.2 below; however, in the next section, wedescribe the algorithms for detection of dynamical interdependence.

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3.1 Nonlinear mutual prediction and dynamical interde-

pendence

Dynamical interdependence of two or more financial markets implies that theobserved time series originate from the different parts of the same dynamicalsystem, nevertheless, in general, two or more coupled, completely independentsystems could become synchronized also. For cross-country financial integrationstudies the equity markets are part of the global economic system. In the presentcontext, the stock markets are considered subsystems of Chinese financial system.Moreover, presence of dynamical interdependence among the subsystems (theindividual equity markets) implies that:

1. The subsystems communicate, that is, they are coupled together and infor-mation flows between them (news arrival in the financial markets), and/or

2. They are coupled to a common driver, where in the case of the stock markets

the driving force is profit motive.

For coupled (interdependent) dynamical systems, it is possible that theirtemporal evolutions might become synchronized as one adjusts the couplingstrength between them, even though their temporal evolution might not be iden-tical.

Dynamical interdependence, as described in Rulkov, Sushichik, Tsimring, andAbarbanel (1995), which adopts a generalized synchronization approach, impliespredictability of the responsesystems behavior by the driving system. This isthe starting point for testing for interdependence of two systems which assumesexistence of function that projects values from the trajectories of the driving

systemD space into the trajectories in the response system R space. In practice,however, when the degrees and directions of the coupling between the systems areunknown, one aims to reconstruct the dynamics of the two systems by time-delayembedding method, and then estimates statistics for testing for dynamical inter-dependence between the reconstructed systems. This is the basis for the mutualprediction method for testing for interdependence of two dynamical systems.

The method of mutual prediction tests for synchronization of completely inde-pendent, but coupled oscillating systems (stock markets in our case). Examplesof synchronization of completely independent, yet coupled, oscillating systemsfrom biological and physical realms include synchronized intermittent emissions

of light by tens of thousand fireflies to random openings of ion channels in cellmembranes, to organ pipes, just to name a few. In short, synchronization is in-teraction of different systems or subsystems with each other. This means thatthese coupled, different, and independent systems or subsystems adjust the timescales of their oscillations due to the interaction.

We consider three Chinese stock markets as nonlinear dynamical oscillatingsystems. We further consider two indexes at a time for testing and take X(t) asthe driver system and Y(t) as the response system.

It is important to point out that in forced synchronizationone may have one-way synchronization. This means that in a forced synchronization where one

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oscillator is being influenced by another one in a unidirectional manner withoutinfluencing the influencer. One can have bidirectional synchronization where bothsystems are mutually interacting and influencing each other. Hence, in the forcedsynchronization case, ifX is not influencingY, it does not necessarily mean thatY

is not influencingX

. (For excellent discussions of synchronization, see Balanov,A. N. Janson, D. Postnov, and O. Sosnovtsev, 2009) .We search for evidence of coupling between these markets by considering their

dynamics that are represented by the following differential equations:

dX

dt = f(X(t)) (3)

dY

dt = g(Y(t), hc(X(t), Y(t))) (4)

where functionsf andg generate local dynamics, functionh transmits the influ-ence ofX(t) to Y(t), and constant c measures the strength of coupling.

Let X and Y be two potentially coupled dynamical systems with the timeseries observations of xi and yi(i = 1, , N), respectively. Often, in practice,the state variables are not directly observable, and one has no a prioriknowledgeof their individual dynamics or their dynamical interdependence. Instead, theirevolutions are measured by the scalar variables

xi(t) = k(X(t)) +1(t) (5)

yi(t) = k(Y(t)) +2(t) (6)

wherek is the measurement function (possibly nonlinear), and1 and2 are theerror terms representing noise in the data.

Because the time series data used in this study have values in a wide range(see table 2, below), we standardize each series by the following transformations:

xi = xi x

x(7)

yi = yi y

y(8)

where x, y, x, and y are the mean and standard deviation of the xi and yiseries, respectively.

3.2 Time delay embedding and embedding dimension

Using time delay embedding (Takens,1981; Sauer, et al.,1991), reconstruct thephase spaces for xi in an embedding space X using the standardized time series{xi} with a suitable embedding dimension d and a time delay , such that

xi= (xi,xi+, ,xi+(d1)). (9)

Similarly, reconstruct the phase space yi in an embedding space Y:

yi= (yi,yi+, ,yi+(d1)). (10)

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We select time delay by using mutual information approach (Fraser andSwinney,1986), see below.

In practice, optimal determination of embedding dimension and time delay in-volves considerable difficulties, and a large number of methods has been proposed

for calculation of these parameters (see Soofi and Cao, 2002). In this paper wechoose the embedding dimension,d, and time delay,by Cao method (Cao,1997)which is discussed below.

3.3 Determining time delay and embedding dimension

Cao method of determining the embedding dimension is based on the conceptof false neighbors. For a given dimensiond, find the nearest neighbor of eachreconstructed time-delay embedding vector yi(d) and denote it yn(i,d)(d). Notethat the n(i, d)th vector, yn(i,d)(d), is the closest vector to the i

th vector, yi(d).Also note that notation n(i, d) implies that the nearest neighbor of any vector

depends on the dimension d and the ith

vector itself. Now increase d by 1, andthen measure how the distance between these two nearest neighbors changes.This is quantified by

a(i, d) =yi(d+ 1)yn(i,d)(d+ 1)

yi(d)yn(i,d)(d) .

If a pair of nearest neighbors are true neighbors, a(i, d) should not be largerthan some threshold value, otherwise, they are false neighbors. As there are manyreconstructed vectors, a quantity measuring the average changes of all pairs ofnearest neighbors is defined asE(d) = 1

NdNdi=1 a(i, d) which depends only on

the dimension d, given the time delay . Clearly, given different values maylead to different embedding dimensions. See below for a discussion of a methodof simultaneous determination ofd and.

Cao further definedE1(d) =E(d+1)/E(d) for determination of the minimumembedding dimension. Increased from 1 until E1(d) stops changing at some d0.Then d0+ 1 is the minimum embedding dimension according to Cao method.

The algorithm of computing time delaywith mutual information techniqueis based on Shannons entropy, and consists of first constructing a histogram forthe probability distribution of the data. Let denote the probability that thesignal assumes a value inside the ith bin of the histogram. Moreover, let pij() bethe probability that x(t) is in the ith bin and x(t+) is in the j

th bin. Then themutual information for time delay is

I() =i,j

pij() lnpij()2i

pilnpi (11)

The first minimum of I() marks the time lag where x(t+) adds maximuminformation to x(t).

For an unfolding of a time series into a representative state space of a dynam-ical system, optimal embedding dimensiond and time delayare required. Themethods of computing embedding dimension and time delay presuppose prior

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knowledge of one parameter before estimation of the other. Accordingly, calcu-lating one parameter requires exogenous determination of the other.

In this study, we adopt the method of simultaneous estimation of embed-ding dimensions and time delays and use these parameters in prediction of the

original series. We select that combination of the embedding dimension and timedelay that would lead to the minimum prediction error using nonlinear predictionmethod (see Soofi and Cao, 2002 for a discussion of the method)1.

Specifically, let i = f(dj , k, i), [i = 1, , N;j = k = 1, , M], where i,dj, k, and i are the ith prediction error, the jth embedding dimension, the kth

time delay, and the ith nearest neighbors, respectively.We usedM=N= 30 on the stock market indexes and present the estimated

parameters in table 1. We experimented with values of M = N > 30, but theprediction accuracy diminished with larger values ofMandN. We also searchedfor the optimal embedding dimension and time delay for the standardized valuesand find that the optimal dimension and time delay parameters for the series

at level and standardized values are the same. Therefore, we only report theembedding dimension and time delay for the series at level.

Shanghai Shenzhen Hong Kong

d 23 16 18 15 15 26

Table 1: Optimal time delays and embedding dimensions for Chinese Stock mar-ket indices.

We use d and for each stock market that is taken as a driver system for

projection into the trajectories of the second stock market as the response system.For example, if we wish to unfold the Shanghai stock market index and projectvalues from this index into the trajectory of the Shenzhen market, we used = 23and= 15 for unfolding of both indexes, even though d = 16 and = 15 for theShenzhen market.

We select the larger embedding dimensions to unfold the time series into thestate spaces to avoid the problem of false nearest neighbors. Kennelet al. (1992)showed that if a time series is unfolded in a too small phase space, then the nearestneighbors falsely appear to be closer because of the small embedding space.

3.4 Algorithm for mutual prediction methodWe write a possible functional relationship between Xand Y as

Y ?=(X) (12)

and aim at empirically verifying existence of the functional relationshipbetweenthe two reconstructed systems X and Y. If such a relationship exists, then two

1We are grateful to Dr. Liangyue Cao for suggesting this method and also for providing ushis computer codes in simultaneous calculation ofd and

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close states in the phase space of the Xsystem correspond to two close states inthe phase space of the Y system.

As was stated earlier, it does not matter which state variable we choose as au-tonomous or response variable. For measuring nonlinear interdependence what

counts is hc function, and coupling strength coefficient c. Existence of a con-tinuous, differentiable map , where in presence of synchronization creates aone-to-one correspondence between the orbits of X onto the orbits of Y in caseofY= (X), and maps Y onto Xin case ofX= (Y) is the important consid-eration.

Select an arbitrary point x0 in the X space. Suppose the nearest neighborof x0 has a time index of nnnd. Then if function exists, that is, if the twosystems are coupled, then point y0 in the Yspace will have point ynnnd as a closeneighbor also. This means that the nearest neighbors of both points x0 and y0share the same time indexes2. For example, if the nearest neighbor of point x0 isa three-dimensional vector with time indexes (1, 5, 8), then the vector that is the

nearest neighbor of point y0 has the same time indexes (1, 5, 8).In implementing the mutual prediction method of testing for nonlinear interde-

pendence of Chinese stock markets, we follow the method discussed by Breakpsearand Terry (2002) which is a modified, improved version of Schiffet al.(1996) asdiscussed below.

Construct in X a simplex around an arbitrary selected point x(ti) in timet = ti with 2d

x1 vertices each consisting of another vector in X. d

x1 is the

embedding dimension ofX.

Choose these embedding vectors (vertices) such that the size of the simplex

is minimized3

. Denote the points satisfying the criteria of being a vertex in the minimized

simplex as xj(tij), j = 1, . . . , 2dx1 . Also denote the time indices of thevertices as tij, j= 1, . . . , 2dx1 .

Use the time indicestij ofxj(tij) to construct a simplex in the state spaceY with verticesy(tij), j = 1, . . . , 2d

x1 .

Take the weighted average of the vertices in y(tij) to locate the vectory(tij)that was predicted by the vector x(ti)

ypred.(ti) =2dx1

k=1iky(tik)2dx1

k=1ik(13)

where the weighting factors ik, are determined by the distances of thevertices in X from x(ti), giving

ik= (|x(tik)x(ti)|)1. (14)

2Note that we have unfolded the time series into d-dimensional space.3A detailed discussion of how this is done is beyond the scope of this paper. Contact the

corresponding author for the algorithm.

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In this study, we choose the embedding dimension d of the driver systemD.

To calculate the mutual prediction error, take the difference of the predictedvector and the actual vector

y(x)= |ypred(ti)y(ti)|. (15)

To compare the prediction error y(x) with a prediction error based on arandomly selected element of the time series observations calculate

rand= |yrandy(ti)|, (16)

where yrand is calculated using the same procedure used in prediction ofypred.(ti), except that the simplex in X is a random combination of pointson the orbit X weighted with respect to another randomly selected point.

This corresponds to the null hypothesis of no interdependence between themarkets.

The normalized predicted y , y(x), as predicted byx, is calculated by

y(x)= < y(x)>rms< rand >rms

(17)

whererms is the root mean square.

y(x)=1 implies no interdependence (no synchronization).y(x) = 0 impliescomplete synchronization.

Calculate the vertices of simplex in Y as above and then iterate themH-step ahead on their respective orbits to obtain the vertices y(tij +H),

j = 1, . . . , 2dyi

Compare the weighted predicted vectorypred.(ti + H), j = 1, . . . , 2yi to the

actual forward iterate y(ti+H) to obtain future prediction errors.

Normalize the H-step ahead prediction errors by a vector generated fromrandom vertices in X to yield the normalized future prediction error:

H

y(x)=

< Hy(x)>rms

< rand >rms (18)

Hy(x) = 1 implies no interdependence between the systems at H-step pre-diction.

Note that in presence of generalized synchronization the error grows at a ratedetermined by the Lyapunov exponents(Wolf, Swift, Swinney and Vastano,1985),and is less than one for some time steps into the future.

After generating a number of surrogates, which share the spectral density func-tions with the original time series use one-step ahead mutual prediction method

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described above, and conduct Hforecasts of the original time series and the sur-rogate time series separately. If the one-step ahead nonlinear prediction errorsof the original series are smaller than those for any of the surrogates, predictionsare significant.

A plot ofHprediction errors as well as prediction interval for the original andsurrogate series based on the above mentioned algorithm would aid in determiningnonlinear interdependence of the markets.

The deterministic interdependence is detected if the graph of the cross-predictionerrors of the original series is below the graphs of cross-prediction errors for thesurrogate sets, but above the lower bound of the 95% confidence interval.

4 Empirical results

4.1 The data

In this study, we use 3 Chinese stock indices to test for nonlinear interdepen-dence among the markets. These are the major indexes for the stock markets ofHong Kong (Hang Seng Index, HSI), Shanghai (Stock Exchange Composite In-dex, SSI), and Shenzhen Stock Exchange (Component Index, SZI). The samplingtime period for these series are the daily observations from 2 Jan 2000 to 6 June2008. The log normalized graph of the series appear in figure 1. Table 2 showsthe descriptive statistics for the stock markets series.

200 400 600 800 1000 1200 1400 1600 1800 2010

0.5

0

0.5

1

1.5

Time series of SSI,SZI and HSI

observations

series(logform)

SSISZIHSI

Figure 1: Normalized stock market series.

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SSI SZI HSI

mean 2025.3 5363.9 14999variance 1.0e+006 * 1.139 1.0e+006 * 1.6635 1.0e+006 * 2.2079skewness 2.0153 2.1338 1.1253

max 6092.1 19531 31638min 1011.5 2627 8409

Table 2: Descriptive statistics for the stock markets series.

4.2 Test for nonlinearity of the stock market time series

using surrogate data

Before testing for nonlinear interdependence of the three time series, evidence ofnonlinearity for single series should be found.

We use the surrogate data technique (Theiler, et al., 1992) in testing for

nonlinearity of the stock market indices. By using surrogate data method oftesting for nonlinearity of the series we aim to reject the null hypothesis that thedata involve only temporal correlations and are random otherwise.

It is known that noise and limited sample observations may point to nonlin-earity of a stochastic time series when such nonlinearity does not exist(see forexample, Osborne and Provencale, 1989). To exclude such misleading signals,surrogate data method is often used for testing nonlinearity of a series. Themethod generates a number of surrogates for the original series by preserving allthe linear properties of the original data while destroying any nonlinear structurethat may exist in the original series by randomizing the Fourier components.

The strategy in surrogate analysis is finding an inadequate process that mayhave generated the original data and then using a statistical test in showing thatthe observed data are highly unlikely to have been generated by such inadequateprocess.

The null hypothesis that the surrogate series are linear is tested. If the nullis true, then surrogate procedure will not affect measures of nonlinear structure.However, if the measure of nonlinear structure is significantly changed by thesurrogate procedure, then null of linearity of the series is rejected.

The hypothesis testing requires a test statistics, and a variety of test statisticshave been used in surrogate data analysis. The most popular of these statisticsare correlation dimension and some measure of predictability. In this study we

use one step ahead prediction error as the statistics to test for nonlinearity of theseries.

We find motivating the use of surrogate data analysis instead of traditionalboostrap method that is commonly used in economic literature useful at this

juncture.In determining the unknown probability distribution of measures of nonlin-

earity Monte Carlo re-sampling technique is used. The parametric bootstrapmethods (Efron, 1982) use explicit models that must be extracted from the data.The validity of this approach hinges on successful extraction of the model from

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the data. The main shortcoming of parametric bootstrap method is that onecannot be sure about the true underlying process by fitting the data to anymodel. The surrogate data method, which could be considered constrained real-izationmethod, overcomes the weakness of parametric bootstrap method, which

istypical realization

method, by directly imposing the desired structure onto therandomized time series.To avoid spurious results it is essential that the correct structure (suitable

null-hypothesis formulation) is imposed on the original series. One approach inensuring validity of statistical test is determining the most likely linear modelthat might have generated the data, fitting the model, and then test for the nullhypothesis that the data have been generated by the specified model. (Schreiber,1999, pp:42-43).

Surrogate data analysis is the method of choice in physics and nonlinear dy-namical systems analysis. Hence, the mutual prediction method of test for non-linear interdependence uses this approach also.

In this study we generate 35 surrogate data series and compute the predictionerrors q1, q2, . . . q 35, as the discriminating statistics for each series. The choiceof 35 surrogates is arbitrary but adequate for hypothesis testing. The surrogateseries are consistent with the null hypothesis of a linear process.

One may reject the null hypothesis either by rank ordering or hypothesistesting. The rank ordering involves deciding whether q0 of the original seriesappears as the first or last item in the sorted list ofq0, q1, q2, . . . q 35.

If the qs are fairly normally distributed we may use significance test. Underthis method rejection of the null using the significant test requires a t value ofabout 2, at the 95% confidence level, where t is defined as:

t= |q0< q >|

q(19)

where < q > and q are the mean and standard deviation, respectively, of theseries q1, q2, . . . q 35 (For an in-depth discussion of surrogate data analysis seeKugiumtzis, 2002).

Note that we use two surrogate data generating programs, fftsurr(fast Fouriertransform surrogates) and ampsurr(amplitude adjusted surrogates) by Kaplan(2004). First, we generated 35 phase-randomized surrogate data using fftsurr.The surrogate data generated by this program have the spectral density functionsas the original time series. Next, we used the amplitude-adjusted surrogatesamp-surr, where the amplitudes of the surrogates and the original data are identical.Note that the amplitude-adjusted surrogate method re-shuffles the original seriesin a way such that the power spectrum of the surrogates and the original seriesare almost identical, however, the actual values of the original and surrogates aredifferent.

Applying FFT to data with non-Gaussian distribution may result in spuriousrejection of the null hypothesis. This result is due to difference between thedistributions of the surrogates and the original series. To remedy this problemone should distort the original data so that it is transformed to a series with

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Gaussian distribution. Then using the distorted original series, now a Gaussianseries, with surrogate data generation using a method such as FFT, generatea new set of surrogates. Finally, the surrogates are transformed back to thesame non-Gaussian distribution. This method is called amplitude-adjusted phase

randomization (see Galka, 2000, chapter 11).It appears that the Chinese stock market data are non-Guassian, because ofluptokurtic nature of the distribution. Therefore, we choose the results based onamplitude-adjusted phase randomization method.

The root mean square prediction errors in the test are out-of-sample one stepahead prediction errors for the last 700 observations in the original and surrogatetime series. In these prediction exercises, we divided the time series into twoparts, with the first half used as the training set and the last 700 observationswere used as the test set. We used the training set to construct the phase space,then predicted one step ahead. We then used 701 observations as the trainingset, and predicted one step ahead, and continued the process by increasing the

test set by one observation until the data is totally used.The null hypothesis is0 , where0 is the prediction error of the original

series, while is the prediction error of the surrogate data.If t t(n 1) , the null hypothesis is rejected, which means the mean

prediction error of the original time series is significantly less than that of thesurrogate data series.

We calculated nonlinear prediction errors for 35 surrogate data we generatedfor each index. The mean square prediction errors, t-statistics, and the results ofsignificance tests for the series at levels are presented in Table 3.

RMSE Mean RMSE(fft) t-statistics Mean RMSE(amp) t-statistics

SSI 0.0854 0.0747 4.5780 0.1487 (2.4048)

SZI 0.1302 0.1190 1.3560 0.2083 (2.0671)

HSI 0.1334 0.1289 0.7236 0.2059 (2.2993)

Table 3: Testing for nonlinearity using two surrogate data generating methods.Mean RMSE(fft) and RMSE(amp) means the mean RMSE of the surrogate datagenerated by fast Fourier transform and amplitude adjusted methods, respec-tively. RMSE means root mean square prediction errors. t-statistics appear inthe parentheses. * implies significant at 95% confidence level.

According to the t-statistics in Table 3, the prediction errors of the originalseries are smaller than the prediction errors of the amplitude adjusted surrogatedata and are statistically significant. Although, the prediction errors of the otheroriginal series are not significantly less than those of the fft surrogate data.

For the fft surrogate method the original data dont show nonlinearity, whilefor the amplitude surrogate method, the original data show significant nonlinear-ity.

The results point to nonlinearity of all three stock market indices: HSI , SSI,and SZI. Accordingly, we may apply the mutual prediction method for testing fornonlinear interdependence of the stock index time series.

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4.3 Test for nonlinear interdependence of the Chinese stock

markets

After the nonlinearity test of the single stock index series, we tested the nonlinearinterdependence of the Chinese stock markets by the mutual prediction methodthat was discussed in section 3.4.

In this part, we constructed 19 bivariate surrogate data with the same ampli-tude distribution, auto correlation function, and cross-spectral density functionas the original data. However, non-linear structure contained within and betweenthe surrogate series are destroyed. Thus the surrogate algorithm allows testingof the null hypothesis that the time series are produced by a cross-correlatedstochastic system.

Examples of the growth of the non-linear mutual prediction errors from Chi-nese stock indices appear in Figure 2 to Figure 4.

Fig. 2 shows the test results of the standardized mutual prediction errors

of SZI and HSI predicted by SSI. Values for the mutual prediction errors of theoriginal series are shown as solid lines with circles. The entries on the verticalaxis are the normalized predicted stock index ofx, based on another stock indexyusing prediction errors based on a randomly selected point in the original series.For example, the termSZI(SSI)means that we have used Shanghai stock marketindex (SSI) in predicting Shenzhen stock market index (SZI), and then comparedthe root mean square errors (RMSE) of these predictions with the RMSE of thepredictions based on a randomly selected initial point in SZI. Furthermore, theterm implies that we used the time indices of the nearest neighbors of this randompoint in the H-step prediction of Shanghai stock market series (SSI):

HSZI(SSI)=HSZI(SSI)rms

Hrandrms. (20)

The dotted lines show the growth of the prediction errors calculated from the19 surrogate data sets. In this study we calculated the lower bound by 1.96s,where as stated before, is the prediction error of the surrogate data, and s isthe standard deviation of the surrogate errors.The line with x shows the lowerbound of prediction interval at the 95% confidence level.

In Fig. 2, we see that the prediction errors of the original series are sig-nificantly lower than the lower bound of the surrogates, which means the nullhypothesis that the time series are produced by a cross-correlated stochastic sys-

tems is rejected. So the results show that the original series have nonlinear mutualpredictability.

The results in Fig. 2 and Fig. 3 indicate that there is nonlinear mutual(bidirectional) predictability between SSI and SZI. Moreover, there exists uni-directional predictability from SSI to HSI and from SZI to HSI. However, theresults in Fig. 4 dont provide statistically significant evidence that Hong Kongmarket predicts the stock markets in mainland China.

See figure 5 for a schematic view of the direction of predictability, therefore,interdependence between the markets. Note that the direction of arrows shows

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direction of predictability. For example, the arrow emerging from Shanghai andending in Hong Kong indicates that Shanghai stock market predicts Hong Kongmarket, implying a unidirectional dependence of Shanghai market on Hong Kongmarket.

Since the series are non-stationary, standardized series may distort the results.We tested the original data without data transformation, and discovered that thefundamental results concerning the market interdependence were intact, althoughthe values of the statistics were changed. For economy of space we do not reportresults from the raw data.

The presence of nonlinear interdependence is manifested by a more gradualincrease in the cross-prediction error in the original series compared to the surro-gate sets. Four of the six tests remain outside of the confidence intervals for all20 future iterates, corresponding to strong interdependence.

We used both Schiffet al. method as well as Breakspear and Terry simplexmethod on the data. The prediction errors based on Schiff method are smaller

than the prediction errors based on simplex method. The results in Fig. 2 to Fig.4 are based on Schiff technique.

4.4 Comparing the results with the results based on a

traditional linear method

We compared the results obtained in this study with the results from a linearmethod of testing for integration of financial markets. Zhu, et al. (2003) have usedcointegration, fractional cointegration, and Granger causality methods in testingfor integration of Chinese stock markets. The tests in Zhu, et al. (2003) show

no evidence of cointegration (either integrated or fractionally integrated) amongthe stock markets. They could not find any evidence for presence of causalityamong the markets either. Hence, the mutual prediction method of testing forinterdependence of Chinese stock markets data shows completely different resultsfrom those obtained by the traditional linear stochastic methods used in Zhu, elal.(2003) study. Since we have provided ample evidence that the stock marketdata are nonlinear series, it is reasonable to conclude that the linear models havefailed to detect interdependence, while the mutual prediction method succeededin finding the evidence of dynamical interdependence between the markets.

5 Summary and ConclusionsThe methodological problems in assessing the degree of financial integration ofmarkets within a country or the degree of cross-country interdependence of thefinancial markets point to the need for new tools of analysis. To test for pres-ence and strength of interdependence of nonlinear systems with little a prioriknowledge of their underlying dynamics, one could use methods from nonlineardynamical systems theories that are readily available.

We tested nonlinearity and nonlinear interdependence of Chinese stock mar-kets using surrogate data analysis and mutual prediction method. The use of

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Chinese stock markets data instead of data from stock markets of different coun-tries in this study is motivated by two factors. First, the structure of Hong Kongeconomy and financial markets are characteristically different from the economyand financial markets of the mainland. Second, because of calendar effects, bothtrading days effects

(number of business days in a week) andholiday effects

(dif-ferent holidays) between China and other countries, a uniform set of time seriesmeasurements for Chinese and other stock markets data does not exist. Eventhough, statistical methods for dealing with calendar effects exist (see, for ex-ample, Cleveland and Devlin, 1982), testing the data with calendar effects forsynchronization would add one extra layer of complexity to an already complexproblem. We used Chinese stock markets data to side step this issue.

We used two surrogate data generating programs, fftsurrand ampsurrin test-ing for nonlinearity of the series. First, we generated phase-randomized surrogatedata using fftsurr. The surrogate data generated by this program have the spec-tral density functions as the original time series. Next, we used the amplitude-

adjusted surrogates ampsurr, where the amplitudes of the surrogates and theoriginal data are identical, but the actual values of the original and surrogatesare different.

The method of mutual prediction is based on the notion that if two nonlinearsystems are interdependent (synchronized), then there exists a function that mapsvalues from the trajectories of one system into the trajectories of another one.

Using the methods of surrogate data analysis and mutual prediction on Shang-hai, Shenzhen, and Hong Kong stock market data, we showed that these seriesare nonlinear, and are nonlinearly dependent on each other. See figure 5 for aschematic view of the interdependencies.

These results give additional credence to nonlinear methodologies that areused in financial integration studies. However, it should be emphasized that themutual prediction method, as it currently stands, cannot measure a change incoupling strength; therefore, it cannot test for contagion, which is often asso-ciated with a rise in cross-country interdependence during the financial crisis.Nevertheless, we consider presence of nonlinear interdependence between two fi-nancial systems as the necessary condition for presence of contagion. Accordingly,to test for contagion, one must first establish nonlinear interdependence betweentwo dynamical systems. The mutual prediction method is a suitable, efficienttechnique for testing for existence of the necessary condition for contagion.

The method of mutual prediction that we use in this study does not address

multi, m : n synchronization, where m and n are two arbitrary integers. Nev-ertheless, at it was discussed above, we are dealing with forced synchronization(unidirectional synchronization), and the test can determine whether one marketis unidirectionally synchronized with another one. Therefore, it is immaterialwhether the influencing market is synchronized with or is influenced by the othermarkets4.

Currently we are not aware of a time series method that would test for syn-chronization of multi systems. This does not mean that such a method does not

4We are grateful to an anonymous referee for raising this important, fine point.

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exist. Perhaps testing for an m : n synchronization could be the subject of an-other research paper if a methodology to test for simultaneous synchronizationof a multi-system is attainable.

Another important feature of the mutual prediction method is that in addition

to demonstrating presence of integrated markets, it can show the direction ofinfluence by identifying a market as driver or a response system.In addition to extending the study for testing synchronization of an m : n

system, it could be extended in two more directions. First, one could use thismethod in testing for nonlinear interdependence of cross-country equity or moneymarkets, by first adjusting the calendar effects in the data. Secondly, researchis required to modify the mutual prediction method so that it can measure thestrength of coupling. The increase in coupling coefficient, then would be anindicator of rise in coupling strength during the period of financial crisis.

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Figure 5: Flow chart of the interdependencies of Chinas stock markets. Thedirection of arrows shows direction of predictability.

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