Multivariate causality tests with simulation and application

Multivariate Causality Tests with Simulation and

Application

Zhidong Bai

School of Mathematics and Statistics, Northeast Normal University

Department of Statistics and Applied Probability,

National University of Singapore

Heng Li

Department of Mathematics

Hong Kong Baptist University

Wing-Keung Wong

Department of Economics

Hong Kong Baptist University

Bingzhi Zhang

Department of BioStatistics

Columbia University

Corresponding author: Wing-Keung Wong, Department of Economics, Hong Kong Bap-

tist University, Kowloon Tong, Hong Kong. Tel: (852)-3411-7542, Fax: (852)-3411-5580,

Email: [email protected]

Acknowledgments The third author would like to thank Professors Robert B. Miller

and Howard E. Thompson for their continuous guidance and encouragement. This re-

search is partially supported by Northeast Normal University, the National University of

Singapore, Hong Kong Baptist University, and Columbia University.

Multivariate Causality Tests with Simulation and

Application

Abstract The traditional linear Granger causality test has been widely used to ex-

amine the linear causality among several time series in bivariate settings as well as mul-

tivariate settings. Hiemstra and Jones (1994) develop a nonlinear Granger causality test

in a bivariate setting to investigate the nonlinear causality between stock prices and trad-

ing volume. In this paper, we first discuss linear causality tests in multivariate settings

and thereafter develop a nonlinear causality test in multivariate settings. A Monte Carlo

simulation is conducted to demonstrate the superiority of our proposed multivariate test

over its bivariate counterpart. In addition, we illustrate the applicability of our proposed

test to analyze the relationships among different Chinese stock market indices.

Keywords: linear Granger causality, nonlinear Granger causality, U-statistics, simu-

lation, stock markets.

JEL Classification:

C01, C12, G10

Linear and Nonlinear Causality Tests 1

1 Introduction

It is an important issue to detect the causal relation among several time series and it starts

with two series, see, for example, Chiang, et al (2010) and Qiao, et al (2008, 2009) and

the references therein for more discussion. To examine whether past information of one

series could contribute to the prediction of another series, linear Granger causality test

(Granger, 1969) is developed to examine whether lag terms of one variable significantly

explain another variable in a vector autoregressive regression model.

Linear Granger causality test can be used to detect the causal relation between two

time series.1 However, the linear Granger causality test does not perform well in detecting

nonlinear causal relationships. To circumvent the limitation, Baek and Brock (1992)

develop a bivariate nonlinear Granger causality test to examine the remaining nonlinear

predictive power of a residual series of a variable on the residual of another variable

obtained from a linear model. Hiemstra and Jones (1994) have further modified the test.

Nevertheless, the multivariate causal relationships are important but it has not been well-

studied, especially for nonlinear causality relationship. Thus, it is important to extend

the Granger causality test to nonlinear causality test in the multivariate settings.

In this paper, we first discuss linear causality tests in multivariate settings and there-

after extend the theory by developing a nonlinear causality test in the multivariate setting.

For any n variables involved in the causality test, we discuss a n-equation vector autore-

gressive regression (VAR) model to conduct the linear Granger causality test, and test

for the significance of relevant coefficients across equations using likelihood ratio test. If

those coefficients are significantly different from zero, the linear causality relationship is

identified. We then extend the nonlinear Granger causality test from bivariate setting

to multivariate setting. We notice that the bivariate nonlinear Granger causality test

is developed by mainly applying the properties of U-statistic developed by Denker and

1See, for example, Wong, et al (2006), Qiao, et al (2007), Chen, et al (2008), and Foo, et al (2008).


Keller (1983, 1986). Central limit theorem can be applied to the U-statistic whose argu-

ments are strictly stationary, weakly dependent and satisfy mixing conditions of Denker

and Keller (1983, 1986). When we extend the test to the multivariate settings, we find

that the properties of the U-statistic for the bivariate settings could also be used in the

development of our proposed test statistic in the multivariate settings, which is also a

function of U-statistic.

The paper is organized as follows. We begin in next section by introducing definitions

and notations and stating some basic properties for the the linear and nonlinear Granger

causality tests to test for linear and nonlinear Granger causality relationships between two

series. In Section 3, we first discuss the linear Granger causality tests in the multivariate

settings and thereafter develop the nonlinear Granger causality tests in the multivariate

settings. In Section 4, we conduct simulation to demonstrate the superiority of our pro-

posed multivariate test over its bivariate counterpart in the performance of both size and

power. In Section 5, we illustrate the applicability of our proposed test to analyze the

relationships among different Chinese stock market indices. Section 6 gives a summary

of our paper.

2 Bivariate Granger Causality Test

In this section, we will review the definitions of linear and nonlinear causality and discuss

the linear and nonlinear Granger causality tests to identify the causality relationships

between two variables.

2.1 Bivariate Linear Granger Causality Test

The linear Granger causality is conducted based on the following two-equation model:


Definition 2.1. In a two-equation model:

xt = a1 +

p∑i=1

αixt−i +

p∑i=1

βiyt−i + ε1t (1a)

and

yt = a2 +

p∑i=1

γixt−i +

p∑i=1

δiyt−i + ε2t , (1b)

where all {xt} and {yt} are stationary variables, p is the optimal lag in the system, and

ε1t and ε2t are the disturbances satisfying the regularity assumptions of the classical linear

normal regression model. The variable {yt} is said not to Granger cause {xt} if βi = 0

in (1a), for any i = 1, · · · , p. In other words, the past values of {yt} do not provide any

additional information on the performance of {xt}. Similarly, {xt} does not Granger

cause {yt} if γi = 0 in (1b), for any i = 1, · · · , p.

It is well-known that one can test for linear causal relations between {xt} and {yt} by

testing the following null hypotheses separately:

H10 : β1 = · · · = βp = 0, and H2

0 : γ1 = · · · = γp = 0.

From testing these hypotheses, we have four possible testing results:

(a) If both Hypotheses H10 and H2

0 are accepted, there is no linear causal relationship

between {xt} and {yt}.

(b) If Hypothesis H10 is accepted but Hypothesis H2

0 is rejected, then there exists linear

causality running unidirectionally from {xt} to {yt} .

(c) If Hypothesis H10 is rejected but Hypothesis H2

0 is accepted, then there exists linear

causality running unidirectionally from {yt} and {xt} .

(d) If both Hypotheses H10 and H2

0 are rejected, then there exist feedback linear causal

relationships between {xt} and {yt}.


There are several statistics could be used to test the above hypotheses. One of the

most commonly used statistics is the standard F -test. To test the hypothesis H10 : β1 =

· · · = βp = 0 in (1a), the sum of squares of the residuals from both the full regression,

SSRF , and the restricted regression, SSRR, are computed in the equation (1a) and the

F -test follows

F =(SSRR − SSRF )/p

SSRF /(n− 2p− 1), (1)

where p is the optimal number of lag terms of yt in the regression equation on xt and n is

the number of observations. If {yt} does not Granger cause {xt}, F in (1) is distributed

as F(p,n−2p−1). For any given significance level α, we reject the null hypothesis H10 if

F exceeds the critical value Fα (p,n−2p−1). Similarly, one could test for the second null

hypothesis H20 : γ1 = · · · = γp = 0, and identify the linear causal relationship from {xt}

to {yt}.

2.2 Bivariate Nonlinear Granger Causality Test

The general test for nonlinear Granger causality is first developed by Baek and Brock

(1992) and, later on, modified by Hiemstra and Jones (1994). As the linear Granger

causality test is inefficient in detecting any nonlinear causal relationship, to examine the

nonlinear Granger causality relationship between a pair of series, say {xt} and {yt}, one

has to first apply the linear models in (1a) and (1b) to {xt} and {yt} for identifying

their linear causal relationships and obtain their corresponding residuals, {ε1t} and {ε2t}.Thereafter, one has to apply a nonlinear Granger causality test to the residual series, {ε1t}and {ε2t}, of the two variables, {xt} and {yt}, being examined to identify the remaining

nonlinear causal relationships between their residuals.

We first state the definition of nonlinear Granger causality as follows:

Definition 2.2. For any two strictly stationary and weakly dependent series {Xt} and


{Yt}, the m-length lead vector of Xt is given by

Xmt ≡ (

Xt, Xt+1, · · · , Xt+m−1

), m = 1, 2, · · · , t = 1, 2, · · ·

and Lx-length lag vector of Xt is defined as

XLxt−Lx

≡ (Xt−Lx , Xt−Lx , · · · , Xt−1

), Lx = 1, 2, · · · , t = Lx + 1, Lx + 2, · · · .

The m-length lead vector, Y mt and the Ly-length lag vector, Y

Ly

t−Ly, of Yt can be defined sim-

ilarly. Series {Yt} does not strictly Granger cause another series {Xt} nonlinearly

if and only if:

Pr(‖Xm

t −Xms ‖ < e

∣∣∣‖ XLxt−Lx

−XLxs−Lx

‖< e, ‖ YLy

t−Ly− Y

Ly

s−Ly‖< e

)

=Pr(‖Xm

t −Xms ‖ < e

∣∣‖ XLxt−Lx

−XLxs−Lx

‖< e)

,

where Pr(· | · ) denotes conditional probability and ‖ · ‖ denotes the maximum norm which

is defined as

‖X − Y ‖ = max(|x1 − y1|, |x2 − y2|, · · · , |xn − yn|

),

for any two vectors X =(x1, · · · , xn

)and Y =

(y1, · · · , yn

).

Under Definition 2.2, the nonlinear Granger causality test statistic is given by

√n

(C1

(m + Lx, Ly, e, n

)

C2

(Lx, Ly, e, n

) − C3

(m + Lx, e, n

)

C4

(Lx, e, n

))

, (2)

where


C1

(m + Lx, Ly, e, n

) ≡ 2

n(n− 1)

∑∑t<s

I(xm+Lx

t−Lx, xm+Lx

s−Lx, e

) · I(y

Ly

t−Ly, y

Ly

s−Ly, e

),

C2

(Lx, Ly, e, n

) ≡ 2

n(n− 1)

∑∑t<s

I(xLx

t−Lx, xLx

s−Lx, e

) · I(y

Ly

t−Ly, y

Ly

s−Ly, e

),

C3

(m + Lx, e, n

) ≡ 2

n(n− 1)

∑∑t<s

I(xm+Lx

t−Lx, xm+Lx

s−Lx, e

),

C4

(Lx, e, n

) ≡ 2

n(n− 1)

∑∑t<s

I(xLx

t−Lx, xLx

s−Lx, e

), and

I(x, y, e) =

0, if ‖x− y‖ > e

1, if ‖x− y‖ ≤ e.

The test statistic, see Hiemstra and Jones (1994), possesses the following property:

Theorem 2.1. For given values of m, Lx, Ly and e > 0 defined in Definition 2.2,

under the assumptions that {Xt}, {Yt} are strictly stationary, weakly dependent, and sat-

isfy the conditions stated in Denker and Keller (1983), if {Yt} does not strictly Granger

cause {Xt}, then the test statistic defined in (2) is distributed as N(0, σ2(m,Lx, Ly, e)

)

asymptotically, and the estimator of the variance σ2(m,Lx, Ly, e) is given by

σ2(m,Lx, Ly, e

)= dT · Σ · d ,

where

d =

[1

C2

(Lx, Ly, e, n

) ,−C1

(m + Lx, Ly, e, n

)

C22

(Lx, Ly, e, n

) ,− 1

C4

(Lx, e, n

) ,C3

(m + Lx, e, n

)

C24

(Lx, e, n

)]T

and Σ is a matrix containing elements

Σi,j = 4 ·K(n)∑

k=1

ωk(n)

[1

2(n− k + 1)

∑t

(Ai,t(n) · Aj,t−k+1(n) + Ai,t−k+1(n) · Aj,t(n)

)],

in which K(n) = [n1/4], [x] is the integer part of x,


ωk(n) =

1, if k = 1

2(1− [(k − 1)/K(n)]

), otherwise

;

A1,t =1

n− 1

(∑

s 6=t

I(Xm+Lx

t−Lx, Xm+Lx

i,s−Lx, e

) · I(Y

Ly

t−Ly, Y

Ly

s−Ly, e

))− C1

(m + Lx, Ly, e, n

),

A2,t =1

n− 1

(∑

s 6=t

I(XLx

t−Lx, XLx

s−Lx, e

) · I(Y

Ly

t−Ly, Y

Ly

s−Ly, e

))− C2

(Lx, Ly, e, n

),

A3,t =1

n− 1

(∑

s 6=t

I(Xm+Lx

t−Lx, Xm+Lx

s−Lx, e

))− C3

(m + Lx, e, n

),

A4,t =1

n− 1

(∑

s 6=t

I(XLx

t−Lx, XLx

s−Lx, e

))− C4

(Lx, e, n

), and

t, s = max(Lx, Ly

)+ 1, · · · , T −m + 1 .

3 Multivariate Granger Causality Test

In this section, we first discuss the linear Granger causality tests in the multivariate

settings and, thereafter, develop the nonlinear Granger causality test from the bivariate

settings to the multivariate settings.

3.1 Multivariate Linear Granger Causality Test

We first discuss the linear Granger causality test in the multivariate settings.


3.1.1 Vector Autoregressive Regression

The linear Granger causality test is applied in the vector autoregressive regression (VAR)

scheme. For t = 1, · · · , T , the n-variable VAR model is represented as:

y1t

y2t

...

ynt

=

A10

A20

...

An0

+

A11(L) A12(L) . . . A1n(L)

A21(L) A22(L) . . . A2n(L)...

.... . .

...

An1(L) An2(L) . . . Ann(L)

y1, t−1

y2, t−1

...

yn, t−1

+

e1t

e2t

...

ent

, (3)

where (y1t, · · · , ynt) is the vector of n stationary time series at time t, L is the backward

operator in which Lxt = xt−1, Ai0 are intercept parameters, Aij(L) are polynomials in

the lag operator L such that

Aij(L) = aij(1) + aij(2)L + · · ·+ aij(p)Lp−1

and et = (e1t, · · · , ent)′ is the disturbance vector satisfying the regularity assumption of

the classical linear normal regression model.

In practice, it is common to set all the equations in VAR to possess the same lag length

for each variable. So a uniform order p will be chosen for all the lag polynomials Aij(L) in

the VAR model (3) according to a certain criteria such as Akaike’s information criterion

(AIC) or Schwarz criterion (SC). Along with the Gauss-Markov assumptions satisfied for

the error terms, ordinary least square estimation (OLSE) is appropriate to be used to

estimate the model as it is consistent and efficient. However, long lag length for each

variable will consume large number of degrees of freedom. For example, in the model

stated in equation (3), there will be n(np + 1) coefficients including n intercept terms, n

variances and n(n− 1)/2 covariances to be estimated. When the available sample size T

is not large enough, including too many regressors will make the estimation inefficient,

and thus, cause the test unreliable. To circumvent this problem, one could adopt a Near-

VAR model and seemingly unrelated regressions estimation technique instead of applying

OLSE to estimate the equations simultaneously. We skip the discussion of the Near-VAR


model and seemingly unrelated regressions estimation in this paper and, for simplicity,

we only use OLSE to estimate the parameters in the VAR model to identify the causality

relationship among vectors of different time series.

3.1.2 Multiple Linear Granger Causality Hypothesis and Likelihood Ratio

Test

To test the linear causality relationship between two vectors of different stationary time

series, xt = (x1,t, · · · , xn1,t)′ and yt = (y1,t, · · · , yn2,t)

′, where there are n1 + n2 = n series

in total, one could construct the following vector autoregressive regression (VAR) model:

xt

yt

=

Ax[n1×1]

Ay[n2×1]

+

Axx(L)[n1×n1] Axy(L)[n1×n2]

Ayx(L)[n2×n1] Ayy(L)[n2×n2]

xt−1

yt−1

+

ext

eyt

, (4)

where Ax[n1×1] and Ay[n2×1] are two vectors of intercept terms, Axx(L)[n1×n1], Axy(L)[n1×n2],

Ayx(L)[n2×n1], and Ayy(L)[n2×n2] are matrices of lag polynomials, ext and eyt are the cor-

responding error terms.

Similar to the bivariate case, there are four different situations for the existence of

linear causality relationships between two vectors of time series xt and yt in (4):

(a) There exists a unidirectional causality from yt to xt if Axy(L) is significantly different

from zero2 and, at the same time, Ayx(L) is not significantly different from zero;

(b) there exists a unidirectional causality from xt to yt if Ayx(L) is significantly different

from zero and, at the same time, Axy(L) is not significantly different from zero;

(c) there exist feedback relations when both Axy(L) and Ayx(L) are significantly different

from zero;

(d) xt and yt are not rejected to be independent when both Axy(L) and Ayx(L) is not

significantly different from zero.

2We said Axy(L) is significantly different from zero if there exists any term in Axy(L) which is signif-icantly different from zero.


We note that one could consider one more situation as follows:

(e) xt and yt are rejected to be independent when either Axy(L) and Ayx(L) is signif-

icantly different from zero. This is the same situation as either (a), (b) or (c) is

true.

To test the above statements is equivalent to test the following null hypotheses:

(a) H10 : Axy(L) = 0,

(b) H20 : Ayx(L) = 0, and

(c) both H10 and H2

0 : Axy(L) = 0 and Ayx(L) = 0.

One may first obtain the residual covariance matrix Σ from the full model in (4) by

using OLSE for each equation without imposing any restriction on the parameters, and

compute the residual covariance matrix Σ0 from the restricted model in (4) by using OLSE

for each equation with the restriction on the parameters imposed by the null hypothesis,

H10 , H2

0 , or both H10 and H2

0 . Thereafter, besides using the F -test in (1), one could use a

similar approach as in Sims (1980) to obtain the following likelihood ratio statistic:

(T − c)( log|Σ0| − log|Σ| ) (5)

where T is the number of usable observations, c is the number of parameters estimated

in each equation of the unrestricted system, and log|Σ0| and log|Σ| are the natural log-

arithms of the determinants of restricted and unrestricted residual covariance matrices,

respectively. When the null hypothesis is true, this test statistic has an asymptotic χ2

distribution with the degree of freedom equal to the number of restrictions on the coeffi-

cients in the system. For example, when we test H0 : Axy(L) = 0, one should let c equal

to np + 1, and there are n2 × p restrictions on the coefficients in the first n1 equations

of the model. Hence, the corresponding test statistic (T − (np + 1))( log|Σ0| − log|Σ| )

asymptotically follows χ2 with n1 × n2 × p degrees of freedom. The conventional linear

bivariate causality test is a special case of the linear multivariate causality test when


n1 = n2 = 1. Therefore,besides using the F -test stated in (1), one could also use the

likelihood ratio test in (5) to identify the linear causality relationship for two variables in

the bivariate settings.

3.1.3 ECM-VAR model

Consider (Y1t, · · · , Ynt) to be a vector of n non-stationary time series with cointegration.

Let yit = ∆Yit for i = 1, · · · , n be the corresponding stationary differencing series. In

this situation, one should not use the VAR model as stated in (3), but impose the error-

correction mechanism (ECM) on the VAR to test for Granger causality between these

variables. The ECM-VAR framework is:

y1t

y2t

...

ynt

=

A10

A20

...

An0

+

A11(L) A12(L) . . . A1n(L)

A21(L) A22(L) . . . A2n(L)...

......

An1(L) An2(L) . . . Ann(L)

y1, t−1

y2, t−1

...

yn, t−1

+

α1

α2

...

αn

· ecmt−1 +

e1t

e2t

...

ent

, (6)

where ecmt−1 is the error correction term. In particular, in this paper, we consider to

test the causality relationship between two vectors of non-stationary time series, Xt =

(X1,t, · · · , Xn1,t)′ and Yt = (Y1,t, · · · , Yn2,t)

′, we let xit = ∆Xit and yit = ∆Yit be the

corresponding stationary differencing series, where there are n1 + n2 = n series in total.

If Xt and Yt are cointegrated with residual vector vecmt, then, instead of using the VAR

in (4), one should adopt the following ECM-VAR model:


xt

yt

=

Ax[n1×1]

Ay[n2×1]

+

Axx(L)[n1×n1] Axy(L)[n1×n2]

Ayx(L)[n2×n1] Ayy(L)[n2×n2]

xt−1

yt−1

+

αx[n1×1]

αy[n2×1]

· ecmt−1 +

ext

eyt

(7)

where Ax[n1×1] and Ay[n2×1] are two vectors of intercept terms, Axx(L)[n1×n1], Axy(L)[n1×n2],

Ayx(L)[n2×n1], Ayy(L)[n2×n2] are matrices of lag polynomials, αx[n1×1] and αy[n2×1] are the

coefficient vectors for the error correction term ecmt−1. Thereafter, one should test the

null hypothesis H0 : Axy(L) = 0 and/or H0 : Ayx(L) = 0 to identify strict causality

relation using the LR test as discussed in Section 3.1.2.

3.2 Multivariate Nonlinear Causality Test

In this section, we will extend the nonlinear causality test for a bivariate setting developed

by Hiemstra and Jones (1994) to a mulitvariate setting.

3.2.1 Multivariate Nonlinear Causality Hypothesis

As discussed in Section 2.2, to identify any nonlinear Granger causality relationship from

any two series, say {xt} and {yt} in a bivariate setting, one has to first apply the linear

models in (1a) and (1b) to {xt} and {yt} to identify their linear causal relationships

and obtain their corresponding residuals, {ε1t} and {ε2t}. Thereafter, one has to apply

a nonlinear Granger causality test to the residual series, {ε1t} and {ε2t}, of the two

variables being examined to identify the remaining nonlinear causal relationships between

their residuals. This is also true if one would like to identify existence of any nonlinear

Granger causality relations between two vectors of time series, say xt = (x1,t, · · · , xn1,t)′

and yt = (y1,t, · · · , yn2,t)′ in a multivariate setting. One has to apply the VAR model in (4)

or the ECM-VAR model in (7) to the series to identify their linear causal relationships and


obtain their corresponding residuals. Thereafter, one has to apply a nonlinear Granger

causality test to the residual series instead of the original time series. For simplicity, in

this section we will denote Xt = (X1,t, · · · , Xn1,t)′ and Yt = (Y1,t, · · · , Yn2,t)

′ to be the

corresponding residuals of any two vectors of variables being examined.

We first define the lead vector and lag vector of a time series, say Xi,t, similar to the

terms defined in Definition 2.2 as follows. For Xi,t, i = 1, · · · , n1, the mxi-length lead

vector and the Lxi-length lag vector of Xi,t are defined, respectively, as

Xmxii,t ≡ (Xi,t, Xi,t+1, · · · , Xi, t+mxi−1),mxi

= 1, 2, · · · , t = 1, 2, · · · ,

XLxii, t−Lxi

≡ (Xi, t−Lxi, Xi, t−Lxi+1, · · · , Xi, t−1), Lxi

= 1, 2, · · · , t = Lxi+ 1, Lxi

+ 2, · · · .

We denote Mx = (mx1 , · · · ,mxn1), Lx = (Lx1 , · · · , Lxn1

), mx = max(mx1 , · · · ,mxn1),

and lx = max(Lx1 , · · · , Lxn1). The myi

-length lead vector, Ymyii,t , and the Lyi-length lag

vector, YLyii,t−Lyi

, of Yi, t, My, Ly, my, and ly can be defined similarly.

Given mx, my, Lx, Ly, e , we define the following four events:

(a){‖XMx

t −XMxs ‖ < e

} ≡ {‖XMx1i,t −X

mx1i,s ‖ < e, for any i = 1, · · · , n1

};

(b){ ‖ XLx

t−Lx−XLx

s−Lx‖< e

} ≡ {‖XLxii,t−Lxi

−XLxii,s−Lxi

‖ < e, for any i = 1, · · · , n1

};

(c){‖Y My

t − YMys ‖ < e

} ≡ {‖Y myii,t − Y

myii,s ‖ < e, for any i = 1, · · · , n2

}; and

(d){ ‖ Y

Ly

t−Ly− Y

Ly

s−Ly‖< e

} ≡ {‖Y Lyii,t−Lyi

− YLyii,s−Lyi

‖ < e, for any i = 1, · · · , n2

},

where ‖ · ‖ denotes the maximum norm defined in Definition 2.2.

The vector series {Yt} is said not to strictly Granger cause another vector series {Xt}if:

Pr(‖XMx

t −XMxs ‖ < e

∣∣ ‖ XLxt−Lx

−XLxs−Lx

‖< e, ‖ YLy

t−Ly− Y

Ly

s−Ly‖< e,

)

= Pr(‖XMx

t −XMxs ‖ < e,

∣∣ ‖ XLxt−Lx

−XLxs−Lx

‖< e)

,


where Pr(· | · ) denotes conditional probability.

3.2.2 Test Statistic and It’s Asymptotic Distribution

Similar to the bivariate case, the test statistic for testing non-existence of nonlinear

Granger causality can be obtained as follows:

√n

(C1

(Mx + Lx, Ly, e, n

)

C2

(Lx, Ly, e, n

) − C3

(Mx + Lx, e, n

)

C4

(Lx, e, n

))

(8)

where

C1

(Mx + Lx, Ly, e, n

) ≡ 2

n(n− 1)

∑∑t<s

n1∏i=1

I(x

mxi+Lxii,t−Lxi

, xmxi+Lxii,s−Lx

, e) ·

n2∏i=1

I(y

Lyii,t−Lyi

, yLyii,s−Lyi

, e) ,

C2

(Lx, Ly, e, n

) ≡ 2

n(n− 1)

∑∑t<s

n1∏i=1

I(x

Lxii,t−Lxi

, xLxii,s−Lx

, e) ·

n2∏i=1

I(y

Lyii,t−Lyi

, yLyii,s−Ly

, e),

C3

(Mx + Lx, e, n

) ≡ 2

n(n− 1)

∑∑t<s

n1∏i=1

I(x

mxi+Lxii,t−Lx

, xmxi+Lxii,s−Lxi

, e),

C4

(Lx, e, n

) ≡ 2

n(n− 1)

∑∑t<s

n1∏i=1

I(x

Lxii,t−Lxi

, xLxii,s−Lxi

, e), and

t, s = max(Lx, Ly

)+ 1, · · · , T −mx + 1, n = T + 1−mx −max

(Lx, Ly

).

Theorem 3.1. To test the null hypothesis, H0, that {Y1,t, · · · , Yn2,t} does not strictly

Granger cause {X1,t, · · · , Xn1,t}, under the assumptions that the time series {X1,t, · · · , Xn1,t}and {Y1,t, · · · , Yn2,t} are strictly stationary, weakly dependent, and satisfy the mixing con-

ditions stated in Denker and Keller(1983), if the null hypothesis, H0, is true, the test

statistic defined in (8) is distributed as N(0, σ2(Mx, Lx, Ly, e)

). When the test statistic

in (8) is too far away from zero, we reject the null hypothesis. A consistent estimator of

σ2(Mx, Lx, Ly, e) follows:

σ2(Mx, Lx, Ly, e) = ∇f(θ)T · Σ · ∇f(θ)


in which each component Σi,j (i, j = 1, · · · , 4), of the covariance matrix Σ is given by:

Σi,j = 4 ·∑

k≥1

ωkE(Ai,t · Aj,t+k−1),

ωk =

1 if k = 1

2, otherwise,

A1,t = h11

(xMx+Lx

t−Lx, y

Ly

t−Ly, e

)− C1(Mx + Lx, Ly, e) ,

A2,t = h12

(xLx

t−Lx, y

Ly

t−Ly, e

)− C2(Lx, Ly, e) ,

A3,t = h13

(xMx+Lx

t−Lx, e

)− C3(Mx + Lx, e) , and

A4,t = h14

(xLx

t−Lx, e

)− C4(Lx, e) ,

where h1i(zt), i = 1, · · · , 4, is the conditional expectation of hi(zt, zs) given the value of

zt as follows:

h11

(xMx+Lx

t−Lx, y

Ly

t−Ly, e

)= E

(h1

∣∣ xMx+Lxt−Lx

, yLy

t−Ly

),

h12

(xLx

t−Lx, y

Ly

t−Ly, e

)= E

(h2

∣∣ xLxt−Lx

, yLy

t−Ly

),

h13

(xMx+Lx

t−Lx, e

)= E

(h3

∣∣ xMx+Lxt−Lx

), and

h14

(xLx

t−Lx, e

)= E

(h4

∣∣ xLxt−Lx

).

A consistent estimator of Σi,j elements is given by:

Σi,j = 4 ·K(n)∑

k=1

ωk(n)

[1

2(n− k + 1)

∑t

(Ai,t(n) · Aj,t−k+1(n) + Ai,t−k+1(n) · Aj,t(n)

)],

K(n) = [n1/4] , ωk(n) =

1 if k = 1

2(1− [(k − 1)/K(n)]) otherwise,


A1,t =1

n− 1

(∑

s 6=t

n1∏i=1

I(X

mxi+Lxii,t−Lxi

, Xmxi+Lxii,s−Lx

, e) ·

n2∏i=1

I(Y

Lyii,t−Lyi

, YLyii,s−Lyi

, e))

− C1

(Mx + Lx, Ly, e, n

),

A2,t =1

n− 1

(∑

s 6=t

n1∏i=1

I(X

Lxii,t−Lxi

, XLxii,s−Lx

, e) ·

n2∏i=1

I(Y

Lyii,t−Lyi

, YLyii,s−Ly

, e))

− C2

(Lx, Ly, e, n

),

A3,t =1

n− 1

(∑

s 6=t

n1∏i=1

I(X

mxi+Lxii,t−Lx

, Xmxi+Lxii,s−Lxi

, e))− C3

(m + Lx, e, n

),

A4,t =1

n− 1

(∑

s 6=t

n1∏i=1

I(X

Lxii,t−Lxi

, XLxii,s−Lxi

, e))− C4

(Lx, e, n

),

t, s = max(Lx, Ly

), · · · , n and n = T −mx −max(Lx + Ly) + 1 ,

and a consistent estimator of ∇f(θ) is:

∇f(θ) =

[1

θ2

, − θ1

θ22

, − 1

θ4

,θ3

θ24

]T

=

[1

C2

(Lx, Ly, e, n

) , −C1

(m + Lx, Ly, e, n

)

C22

(Lx, Ly, e, n

) , − 1

C4

(Lx, e, n

) ,C3

(Mx + Lx, e, n

)

C24

(Lx, e, n

)]T

.

4 Monte Carlo Simulation

In this section, we present the Monte Carlo simulation3 to demonstrate the superiority of

our proposed multivariate nonlinear Granger causality test over its bivariate counterpart

in the performance of both size and power when the underlying series possess multivariate

nonlinear Granger causality nature.

We have conducted simulations for a variety of time series possessing different multi-

variate nonlinear Granger causality relationships. All simulations show that our proposed

3Readers may refer to Tiku et al (1998, 1999a, 1999b, 2000) and Lean et al (2008) for more informationon simulation.


multivariate nonlinear Granger causality test performs better in both size and power. For

simplicity, we only present the results of the following equation:

Xt = β Yt−1Zt−1 + εt (9)

where {Yt} and {Zt} are i.i.d. and mutually independent random variables generated from

N(0, 1), {εt} is Gaussian white noise generated from N(0, 0.1). Under the model in (9),

the variables {Yt, Zt} nonlinear Granger cause {Xt} if β 6= 0 and there is no Granger

causality relationship if β = 0. The bivariate nonlinear Granger causality test could

detect the bivariate causality relationships well but it may not be able to examine the

causality relationships under multivariate settings including the one set in (9). Thus,

we expect that our proposed multivariate test could perform better than its bivariate

counterpart in this model setting. To justify our claim, we conduct a simulation with

1,000 Monte Carlo runs based on sample size of 50 and 100 observations for each β value.

We set lead length m = 1 and the common lag length Lx = Ly = Lz for all the cases

being examined. A common scale parameter of e = 1.5σ is used where σ = 1 denotes the

standard deviation of standardized series. In the simulation of each replication, the values

of the test statistics for different common lag lengths are compared with their asymptotic

critical values at the 0.05 nominal significance level. The percentage of rejecting the null

of β = 0 is reported in Tables 1 and 2 for sample size of 50 and 100, respectively.

Table 1 displays the simulation results of sample size 50 with the value of β varying

from −0.5 to 0.5 and the common lag length being 1, 2, and 3 for all the cases under

examination. When β = 0, that is, {Xt} is “independent” with both {Yt} and {Zt} imply-

ing that the null hypothesis is true, both bivariate and multivariate tests are conservative

when the common lag length = 1, 2. When the common lag length is equal to 3, both

tests have empirical sizes similar to the nominated significance level of 0.05. In short,

Table 1 exhibits that (a) the average of the simulated size of multivariate test is closer to

the nominated significance level of 0.05. When β is nonzero, our simulation shows that

(b) the powers of both bivariate and multivariate tests perform better when lag length =


1 and their powers reduce when lag length increases, and (c) the power of our proposed

multivariate test is much higher than that of its bivariate counterpart for any lag length

being examined in our paper. We note that (b) is reasonable because only lag one of

both Y and Z “cause” X in (9) while our findings in (a) and (c) show that our proposed

multivariate test performs better than its bivariate counterpart in both size and power.

We turn to examine both size and power when we increase the sample size to 100.

The results are displayed in Table 2. As the sample size is larger, we report results with

longer lag length scale including Lx = Ly = Lz = 1, 3, and 5.4 Comparing with sample

size = 50, the simulation results show that our observations of (a), (b), and (c) for sample

size = 50 still hold for sample size of 100, but, as expected, both size and power for both

bivariate and multivariate tests have improved and our findings are consistent to show

that our proposed multivariate test performs better than its bivariate counterpart in both

size and power when sample size = 100.

Table 1: Size and power comparison between bivariate and multivariate nonlinear Grangercausality tests when sample size=50

Beta -0.5 -0.4 -0.3 -0.2 -0.1 -0.05 0 0.05 0.1 0.2 0.3 0.4 0.5Lags=1Bivariate 0.917 0.875 0.833 0.775 0.413 0.114 0.016 0.129 0.384 0.748 0.851 0.894 0.902

Multivariate 0.996 1.000 0.997 0.987 0.716 0.204 0.029 0.220 0.686 0.981 1.000 0.999 0.999Lags=2Bivariate 0.645 0.655 0.624 0.516 0.240 0.100 0.034 0.126 0.255 0.509 0.598 0.647 0.644


Multivariate 0.864 0.850 0.796 0.676 0.293 0.115 0.045 0.106 0.310 0.655 0.789 0.842 0.853

Bivariate 0.676 0.661 0.634 0.541 0.277 0.095 0.033 0.109 0.267 0.537 0.624 0.667 0.660Multivariate 0.948 0.943 0.916 0.844 0.479 0.154 0.035 0.164 0.488 0.839 0.915 0.940 0.945

Note: The last two rows display the average values for the case of “Bivariate” and “Multivariate”,respectively, for the corresponding value of β.

4We have examined other lag lengths and the results are consistent with our findings. Thus, we skipreporting other lag lengths to save space.


Table 2: Size and power comparison between bivariate and multivariate nonlinear Grangercausality tests when sample size=100

Beta -0.5 -0.4 -0.3 -0.2 -0.1 -0.05 0 0.05 0.1 0.2 0.3 0.4 0.5Lags=1Bivariate 1.000 1.000 0.996 0.989 0.781 0.266 0.029 0.271 0.807 0.994 0.999 0.997 0.999



Multivariate 0.865 0.823 0.776 0.676 0.256 0.105 0.049 0.104 0.264 0.645 0.788 0.842 0.870

Bivariate 0.811 0.799 0.776 0.727 0.455 0.154 0.041 0.150 0.474 0.702 0.779 0.803 0.792Multivariate 0.955 0.941 0.923 0.883 0.621 0.233 0.045 0.242 0.615 0.869 0.926 0.946 0.957

Note: The last two rows display the average values for the case of “Bivariate” and “Multivariate”,respectively, for the corresponding value of β.

5 Illustration

Qiao, et al (2008) examine the bivariate linear and nonlinear Granger causality rela-

tionships between pairs of daily returns from five indices: (a) Shanghai A shares (SHA)

and Shanghai B shares (SHB) from Shanghai Stock Exchange (SHSE), (b) Shenzhen A

shares (SZA) and Shenzhen B shares (SZB) from Shenzhen Stock Exchange (SZSE), and

(c) Hong Kong H shares (H) before and after February 19, 2001, the date the Chinese

Government allowed domestic citizens to trade B shares from this date onwards5.

As they only apply the bivariate Granger causality test to study the issue, their results

may not be able to capture the multivariate causality relationships among these indices.

To bridge the gap, in this paper we first apply the traditional multivariate linear Granger

causality test and thereafter apply our proposed multivariate nonlinear causality test

to examine the existence of multivariate linear and nonlinear causality relationships in

any of the following three groups for the segmented Chinese stock markets before and

after February 19, 2001: (a) A-share (including SHA and SZA) and B-share (including

SHB and SZB), (b) Shanghai stock market (including SHA and SHB) and Shenzhen

stock market(including SZA and SZB), and (c) domestic stock market (A-share) and

5Readers may refer to Qiao, et al (2008) for detailed information on SHA, SZA, SHB, SZB, and H.


foreigner-invested stock market (including B-share and H-share). The studying period

is from October 6, 1992 to December 31, 2007 and all data are taken from DataStream

International. For easy comparison, we follow Qiao, et al (2008) to use February 19, 2001

as the cut-off point so that the first sub-period is from October 6, 1992 to February 16,

2001 and the second sub-period is from February 19, 2001 to December 31, 2007.

We first adopt the VAR model in (4) or the ECM-VAR model in (7) to examine whether

there is any multivariate linear Granger causality relationship among the indices in any

of the groups mentioned above. We find that, in the first subperiod, there is only strong

unidirectional linear causality from Shenzhen stock market to Shanghai stock market.

On the other hand, in the second sub-period, we find that (a) there is unidirectional

linear causality from B-share to A-share, and (b) there are strong feedback causality

relationships between Shanghai stock market and Shenzhen stock market, and between

domestic stock market and foreigner-invested stock market6.

After checking the multivariate linear causalities, we apply our new proposed multi-

variate nonlinear Granger causality test to the error terms from the estimated VAR or

ECM-VAR models to investigate whether there is any remaining undetected multivariate

nonlinear relationship among the indices. We set the common lead length m = 1 and

the common lag length to be 1 to 10. A common scale parameter of e = 1.5σ is used,

where σ = 1 denotes the standard deviation of standardized series. The values of the

standardized test statistic of (8) are reported in Table 3. Under the null hypothesis of no

multivariate nonlinear Granger causality, the test statistic is asymptotically distributed as

N(0, 1). Therefore, very large or small values of the test statistic lead to the rejection of

null hypothesis, or equivalently, indicate the existence of multivariate nonlinear causality.

The results show that, in the first subperiod, (a) there is weak nonlinear causality

from B-share to A-share, and (b) there is strong bi-directional nonlinear causality between

Shenzhen and Shanghai stock markets, since the test statistics of SZ→SH are significant

6The detailed results of the multivariate linear causality test are available upon request.


at the 0.05 level for all lags, and the test statistics of SH→SZ are significant at the 0.05

level for the first half of lags. In the second sub-period, our results infer that (c) there

exists strong feedback nonlinear causality between Shenzhen and Shanghai stock markets

similar to the results of the first subperiod, (d) there is strong unidirectional nonlinear

causality from A-share to B-share, since the test statistics of A→B are significant at

the 0.05 level for some lags, and (e) there exists weak nonlinear causality from domestic

stock market to foreigner-invested stock market, as there are two test statistics to be

significant. Our findings infer that there are more nonlinear causality among the indices

after the Chinese Government introduced the policy on February 19, 2001.

6 Conclusion Remarks

In this paper, we first discuss linear causality tests in multivariate settings and thereafter

develop a nonlinear causality test in multivariate settings.

However, there is a disadvantage of the Hiemstra-Jones test. For example, Diks and

Panchenko (2005) point out that Hiemstra-Jones test might have an over-rejection bias

on the null hypothesis of Granger non-causality. Their simulation results show that rejec-

tion probability will goes to one as the sample size increases. Diks and Panchenko (2006)

address this problem by replacing the global test by an average of local conditional depen-

dence measures. Their new test shows weaker evidence for volume causing returns than

Hiemstra-Jones test does. Besides Hiemstra-Jones test, other forms of nonlinear causality

test has also been developed. For example, Marinazzo, Pellicoro, and Stramaglia (2008)

adopt theory of reproducing kernel Hilber spaces to develop nonlinear Granger causal-

ity test. And Diks and DeGoede (2001) develop an information theoretic test statistics

for Granger causality. They use bootstrap methods instead of asymptotic distribution

to calculate the significance of the test statistics. Thus, further extension of this paper

could include to develop multivariate settings for the more powerful linear and nonlinear

causality tests.


Table 3: Multiple Nonlinear Testing Results for China’s Stock Markets

First Sub-Period Second Sub-Periodlags B→A SZ→SH B,H→A B→A SZ→SH B,H→A1 1.620* 4.499*** 0.617 0.358 1.434* 1.2142 1.413* 4.882*** 0.344 0.947 3.361*** 1.0193 0.512 4.610*** -0.591 0.376 4.085*** 0.6094 0.546 4.179*** -0.418 -0.541 3.638*** -0.1785 -0.187 3.971*** -0.800 -1.217 3.359*** -1.3056 -0.704 3.523*** -1.024 -1.018 3.031*** -1.1587 -1.663 2.953*** -1.465 -0.961 2.846*** -0.6258 -1.858 2.863*** -1.249 -1.191 1.990** -0.8799 -1.859 2.375*** -1.058 -1.331 2.084** -1.42610 -2.186 1.887** -1.054 -1.283 1.733** -0.778

lags A→B SH→SZ A→B,H A→B SH→SZ A→B,H1 -2.130 3.071*** -1.638 2.098** 2.611*** 2.309**2 -2.423 2.983*** -1.171 2.311** 2.792*** 2.708***3 -2.052 1.896** -1.049 2.282** 2.599*** 1.2324 -1.901 2.381*** -1.263 1.673** 1.827** 0.2245 -1.104 1.754** -1.428 1.115 1.474* -0.6006 -1.201 1.505* -1.461 1.649** 1.308* -1.1287 -1.325 1.091 -0.943 1.569* 0.907 -0.7668 -0.961 0.793 -1.204 1.329* 0.933 -0.9439 -1.177 0.843 -1.165 1.122 0.726 -0.77410 -1.371 0.294 -0.636 1.222 0.913 -0.597

Note: ***, **, and * represent significance levels of 1%, 5%, and 10%, respectively. A includesSHA and SZA, B includes SHB and SZB, SH includes SHA and SHB, and SZ includes SZAand SZB. The first sub-period is from October 6, 1992 to February 16, 2001 while the secondsub-period is from February 19, 2001 to December 31, 2007.


At last, we note that linear and nonlinear causality tests could be used in medical

sciences, physical sciences, social sciences, and many other areas, especially in finance and

economics. The tests could be used to examine many important economics and financial

issues, see, for example, Fong et al (2005, 2008) and Broll et al (2006), and to explain

many financial anomalies, see, for example, Manzur et al (1999), Wan and Wong (2001),

and Lam et al (2010). The results drawn from the linear and nonlinear causality tests

could assist investors in their decision making of their investment. In order to obtain

better information in their decision making, investors may consider to incorporate the

findings from linear and nonlinear causality tests with other techniques7 to obtain better

investment decision making.

7For example, portfolio optimization (Leung and Wong, 2008, Bai et al., 2009a,b; Egozcue and Wong,2010), technical analysis (Wong et al; 2001, 2003; Kung and Wong, 2009), fundamental analysis (Thomp-son and Wong, 1991, 1006; Wong and Chan, 2004), and stochastic dominance theory, see, for example,Wong (2007), Wong and Chan (2008), Wong and Ma (2008), Sriboonchita et al (2009) and the referencestherein for more information.


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Multivariate causality tests with simulation and application

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