A New Test of Multivariate Nonlinear Causality
Zhidong Bai
KLASMOE and School of Mathematics and Statistics,
Northeast Normal University, China
Yongchang Hui***Corresponding author: Yongchang Hui, School of Mathematics and
Statistics, Xi’an Jiaotong University. No.28, Xianning West Road, Xi’an,
Shaanxi, P.R. China. Tel: (86)-029-82663170, Email<EMAIL_ADDRESS>
School of Mathematics and Statistics,
Xi’an Jiaotong University, China
Zhihui Lv
KLASMOE and School of Mathematics and Statistics,
Northeast Normal University, China
Wing-Keung Wong
Department of Finance, Asia University, Taiwan
Department of Economics, Lingnan University, Hong Kong
Shurong Zheng
KLASMOE and School of Mathematics and Statistics,
Northeast Normal University, China
Zhen-Zhen Zhu
KLASMOE and School of Mathematics and Statistics,
Northeast Normal University, China

A New Test of Multivariate Nonlinear Causality

Abstract  The multivariate nonlinear Granger causality developed by Bai et al.
(2010) plays an important role in detecting the dynamic interrelationships
between two groups of variables. Following the idea of Hiemstra-Jones (HJ)
test proposed by Hiemstra and Jones (1994), they attempt to establish a
central limit theorem (CLT) of their test statistic by applying the
asymptotical property of multivariate $U$-statistic. However, Bai et al.
(2016) revisit the HJ test and find that the test statistic given by HJ is NOT
a function of $U$-statistics which implies that the CLT neither proposed by
Hiemstra and Jones (1994) nor the one extended by Bai et al. (2010) is valid
for statistical inference. In this paper, we re-estimate the probabilities and
reestablish the CLT of the new test statistic. Numerical simulation shows that
our new estimates are consistent and our new test performs decent size and
power.

Keywords:  nonlinear Granger causality, Hiemstra-Jones test, multivariate

##  1 Introduction

After the pioneering work of Granger (1969), Granger causality tests have been
developed into a set of useful methods to detect causal relations between time
series in economics and finance. Linear Granger causality tests within the
linear autoregressive model class have been developed in many directions,
e.g., Hurlin et al. (2001) proposed a procedure for causality tests with panel
data, Ghysels et al. (2016) test for Granger causality with mixed frequency
data based on the multiple-horizon framework established by Dufour and Renault
(1998) and Dufour et al. (2006). Though linear tests of Granger causality have
been investigated very deeply, they are limited in their capability to detect
nonlinear causality.

The real world is “almost certainly nonlinear” as Granger (1989) notes, so it
is more important to test the nonlinear causality. Baek and Brock (1992)
develop a nonlinear Granger causality test which is modified by Hiemstra and
Jones (1994) later on to study the bivariate nonlinear causal relationship
between two series. Among the various tests of nonlinear Granger causality,
the Hiemstra-Jones test (hereafter, the HJ test) proposed by Hiemstra and
Jones (1994) is the most cited by scholars and the most frequently applied by
practitioners in economics and finance. There were over 1100 Google Scholar
hits by September 2016, which illustrates its significance in the economics
and finance literatures.

Bai et al. (2010) extend the HJ test from bivariate setting to multivariate
setting catering to the practical needs that economic and financial factors
usually move together and influence others in groups. This extension
encourages a large amount of applications. For example, Lam et al. (2012)
suggest to use such technics to make better investment decisions. Zheng and
Chen (2013) proposed a complete double selection method in identifying
external influential factors for a particular stock market. Choudhry et al.
(2015) investigates the nonlinear dynamic co-movements between gold returns,
stock market returns and stock market volatility during the recent global
financial crisis. Choudhry et al. (2016) investigate the relationship between
stock market volatility and the business cycle in four major economies US,
Canada, Japan and the UK.

However, several works note that counterintuitive results are obtained from
the HJ test, Diks and Panchenko (2005, 2006) find that the HJ test is
seriously over-rejecting in simulation studies. In accordance with the
evidence presented by Diks and Panchenko (2005, 2006), Bai et al. (2016)
reinvestigate the HJ test and reveal some of the underlying reasons for the
questionable performance of HJ test. They find that the estimators of the
probabilities in the definition are not $U$-statistics as Hiemstra and Jones
(1994) claimed and the central limit theorem of the test statistics is not
valid. Bai et al. (2016) propose a set of consistent estimators of the
probabilities in the definition of Hiemstra and Jones (1994) and provide a new
test statistic with its asymptotic distribution.

Considering the significant importance of the multivariate nonlinear Granger
causality test, there is an urgent need to reinvestigate Bai et al. (2010) and
extend Bai et al. (2016) to multivariate setting. The remainder of this paper
is organized as follows. In Section 2, we simply review the procedure of the
multivariate nonlinear Granger causality test (here after BWZ test) extended
by Bai et al. (2010). In Section 3, we re-estimate the probabilities in the
definition of Bai et al. (2010) and establish the asymptotic distribution of
the new test statistics. Simulation results are presented in Section 4.
Finally, we provide some concluding remarks in Section 5.

##  2 The Multivariate Nonlinear Causality Test Extended from HJ Test

Bai et al. (2010) consider two strictly stationary and weakly dependent vector
time series processes
${X}_{t}=({X}_{1,t},{X}_{2,t},\cdots,{X}_{n_{1},t})^{\prime}$,
${Y}_{t}=({Y}_{1,t},{Y}_{2,t},\cdots,{Y}_{n_{2},t})^{\prime}$. The
$m_{x_{i}}$-length lead vector of $X_{i,t}$ is defined as
${X}^{m_{x_{i}}}_{i,t}\equiv(X_{i,t},X_{i,t+1},\cdots,X_{i,t+m_{x_{i}}-1}),m_{x_{i}}=1,2,\cdots,t=1,2,\cdots,\\\
$ similarly $L_{x_{i}}$-length lag vector of $X_{i,t}$ , and
$L_{y_{i}}$-length lag vector of $Y_{i,t}$ are defined as
${X}^{L_{x_{i}}}_{i,t-L_{x_{i}}}\equiv(X_{i,t-L_{x_{i}}},X_{i,t-L_{x_{i}}+1},\cdots,X_{i,t-1}),L_{x_{i}}=1,2,\cdots,t=L_{x_{i}}+1,L_{x_{i}}+2,\cdots,\\\
$
${Y}^{L_{y_{i}}}_{i,t-L_{y_{i}}}\equiv(Y_{i,t-L_{y_{i}}},Y_{i,t-L_{y_{i}}+1},\cdots,Y_{i,t-1}),L_{y_{i}}=1,2,\cdots,t=L_{y_{i}}+1,L_{y_{i}}+2,\cdots$
.
Denote $M_{x}=(m_{x_{1}},\cdots,m_{x_{n_{1}}})$,
$m_{x}=\max(m_{x_{1}},\cdots,m_{x_{n_{1}}})$,
$L_{x}=(L_{x_{1}},\cdots,L_{x_{n_{1}}})$,
$l_{x}=\max(L_{x_{1}},\cdots,L_{x_{n_{1}}})$,
$L_{y}=(L_{y_{1}},\cdots,L_{y_{n_{2}}})$,
$l_{y}=\max(L_{y_{1}},\cdots,L_{y_{n_{2}}})$. For given $M_{x},L_{x},L_{y},e$,
Bai et al. (2010) define that

$\displaystyle\\{\|{X}^{M_{x}}_{t}-{X}^{M_{x}}_{s}\|<e\\}\equiv$
$\displaystyle\\{\|{X}^{m_{x_{i}}}_{i,t}-{X}^{m_{x_{i}}}_{i,s}\|<e,\quad
for\quad all\quad i=1,2,\cdots,n_{1}\\}$
$\displaystyle\\{\|{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\|<e\\}\equiv$
$\displaystyle\\{\|{X}^{L_{x_{i}}}_{i,t-L_{x_{i}}}-{X}^{L_{x_{i}}}_{i,s-L_{x_{i}}}\|<e,\quad
for\quad all\quad i=1,2,\cdots,n_{1}\\}$
$\displaystyle\\{\|{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\|<e\\}\equiv$
$\displaystyle\\{\|{Y}^{L_{y_{i}}}_{i,t-L_{y_{i}}}-{Y}^{L_{y_{i}}}_{i,s-L_{y_{i}}}\|<e,\quad
for\quad all\quad i=1,2,\cdots,n_{2}\\}\,,$

where $\|\cdot\|$ denotes the maximum norm defined as
$\|X-Y\|=\max\big{(}|x_{1}-y_{1}|,|x_{2}-y_{2}|,\cdots,|x_{n}-y_{n}|\big{)}$
for any two vectors $X=\big{(}x_{1},\cdots,x_{n}\big{)}$ and
$Y=\big{(}y_{1},\cdots,y_{n}\big{)}$.

######  Definition 2.1.

The vector time series $\\{Y_{t}\\}$ does not strictly Granger cause another
vector time series $\\{X_{t}\\}$ if

$\displaystyle
P\left(\parallel{X}^{M_{x}}_{t}-{X}^{M_{x}}_{s}\parallel<e\big{|}\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e,\parallel{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\parallel<e\right)$
$\displaystyle=P\left(\parallel{X}^{M_{x}}_{t}-{X}^{M_{x}}_{s}\parallel<e\big{|}\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e\right)\,,$
(1)

where $P(\cdot|\cdot)$ denotes conditional probability.

Using the notation

$\displaystyle{C}_{1}(M_{x}+L_{x},L_{y},e)$ $\displaystyle\equiv
P\left(\parallel{X}^{M_{x}+L_{x}}_{t-L_{x}}-{X}^{M_{x}+L_{x}}_{s-L_{x}}\parallel<e,\parallel{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\parallel<e\right)$
$\displaystyle{C}_{2}(L_{x},L_{y},e)$ $\displaystyle\equiv
P\left(\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e,\parallel{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\parallel<e\right)$
$\displaystyle{C}_{3}(M_{x}+L_{x},e)$ $\displaystyle\equiv
P\left(\parallel{X}^{M_{x}+L_{x}}_{t-L_{x}}-{X}^{M_{x}+L_{x}}_{s-L_{x}}\parallel<e\right)$
$\displaystyle{C}_{4}(L_{x},e)$ $\displaystyle\equiv
P\left(\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e\right)\,,$

Bai, et al. (2010) re-express Equation (1) as

$\frac{{C}_{1}(M_{x}+L_{x},L_{y},e)}{{C}_{2}(L_{x},L_{y},e)}=\frac{{C}_{3}(M_{x}+L_{x},e)}{{C}_{4}(L_{x},e)}\,.$
(2)

For two sets of simultaneous samples
$\\{x_{i,t},i=1,\cdots,n_{1},t=1,\cdots,T\\}$ and
$\\{y_{i,t},i=1,\cdots,n_{2},t=1,\cdots,T\\}$, they propose the following test
statistic

$\sqrt{n}\left(\frac{C_{1}\big{(}M_{x}+{L_{x}},L_{y},e,n\big{)}}{C_{2}\big{(}{L_{x}},L_{y},e,n\big{)}}-\frac{C_{3}\big{(}M_{x}+{L_{x}},e,n\big{)}}{C_{4}\big{(}{L_{x}},e,n\big{)}}\right)\,,$
(3)

where

$\displaystyle C_{1}\big{(}M_{x}+{L_{x}},L_{y},e,n\big{)}$
$\displaystyle\equiv\frac{2}{n(n-1)}\operatorname*{\sum\sum}_{t<s}\prod^{n_{1}}_{i=1}I\left({x}^{m_{x_{i}}+L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{m_{x_{i}}+L_{x_{i}}}_{i,s-L_{x_{i}}},e\right)\cdot\prod^{n_{2}}_{i=1}I\left({y}^{L_{y_{i}}}_{i,t-L_{y_{i}}},{y}^{L_{y_{i}}}_{i,s-L_{y_{i}}},e\right),$
$\displaystyle
C_{2}\big{(}{L_{x}},L_{y},e,n\big{)}\equiv\frac{2}{n(n-1)}\operatorname*{\sum\sum}_{t<s}\prod^{n_{1}}_{i=1}I\left({x}^{L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{L_{x_{i}}}_{i,s-L_{x_{i}}},e\right)\cdot\prod^{n_{2}}_{i=1}I\left({y}^{L_{y_{i}}}_{i,t-L_{y_{i}}},{y}^{L_{y_{i}}}_{i,s-L_{y_{i}}},e\right),$
$\displaystyle
C_{3}\big{(}M_{x}+{L_{x}},e,n\big{)}\equiv\frac{2}{n(n-1)}\operatorname*{\sum\sum}_{t<s}\prod^{n_{1}}_{i=1}I\left({x}^{m_{x_{i}}+L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{m_{x_{i}}+L_{x_{i}}}_{i,s-L_{x_{i}}},e\right),$
$\displaystyle
C_{4}\big{(}{L_{x}},e,n\big{)}\equiv\frac{2}{n(n-1)}\operatorname*{\sum\sum}_{t<s}\prod^{n_{1}}_{i=1}I\left({x}^{L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{L_{x_{i}}}_{i,s-L_{x_{i}}},e\right),$
and $\displaystyle I(x,y,e)=\begin{cases}0,&\text{if $\|x-y\|>e$}\\\
1,&\text{if $\|x-y\|\leq e$}\end{cases}\,.$

Remark: Following the instruction of Hiemstra and Jones (1994), Bai et al.
(2010) take $C_{j}(*,n)$s as multivariate $U$-statistic estimators of their
counterparts $C_{j}(*)$s and apply the asymptotic property of $U$-statistic to
show the limiting results for the test statistics (3). However the
$C_{j}(*,n)$s are not $U$-statistics, because the expectations of the general
terms are not the same. Moreover, the $C_{j}(*)$s are related to the indices
$t$ and $s$ (in fact, related to $|t-s|$ for strongly stationary processes),
while the $C_{j}(*,n)$s were independent of $t$ and $s$ for summing up over
them. Therefore, the $C_{j}(*,n)$ estimators are neither consistent nor
asymptotic normal estimators of their counterparts $C_{j}(*)$.

##  3  A New Multivariate Nonlinear Causality Test

We first remind the reader that the pair $(s,t)$ (in fact, $|t-s|$ for
strongly stationary processes) in Equation (1) of Definition 2.1 is a key
parameter of the probabilities $C_{j}(*)$. In fact, both Hiemstra and Jones
(1994) and Bai et al. (2010) note this, and there is no problem in Equation
(1) of Definition 2.1. However, it seems that they overlooked this fact in
their proposed estimation of $C_{j}(*)$. The improper estimators $C_{j}(*,n)$
thus lead to an invalid asymptotic distribution of the test statistic.

We now begin to state the procedure for our new test. For any given pair
$(s,t)$, we denote

$\displaystyle{C}_{1}(M_{x}+L_{x},L_{y},e;t,s)\equiv
P\left(\parallel{X}^{M_{x}+L_{x}}_{t-L_{x}}-{X}^{M_{x}+L_{x}}_{s-L_{x}}\parallel<e,\parallel{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\parallel<e\right)$
$\displaystyle{C}_{2}(L_{x},L_{y},e;t,s)\equiv
P\left(\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e,\parallel{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\parallel<e\right)$
$\displaystyle{C}_{3}(M_{x}+L_{x},e;t,s)\equiv
P\left(\parallel{X}^{M_{x}+L_{x}}_{t-L_{x}}-{X}^{M_{x}+L_{x}}_{s-L_{x}}\parallel<e\right)$
$\displaystyle{C}_{4}(L_{x},e;t,s)\equiv
P\left(\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e\right)$

Under the assumption of the stationary, for the given pair $(s,t)$, if
$s-t=l$, we denote
${C}_{1}(M_{x}+L_{x},L_{y},e;t,s)\equiv{C}_{1}(M_{x}+L_{x},L_{y},e;t,l)$,
which does not depend on t, so we can write ${C}_{1}(M_{x}+L_{x},L_{y},e;l)$
instead of ${C}_{1}(M_{x}+L_{x},L_{y},e,t;l)$, the same to the others. So
under the assumption of strictly stationary, for each $l>0$, we examine
whether there is nonlinear Granger causality from $\\{Y_{t}\\}$ to
$\\{X_{t}\\}$ by testing the following hypothesis

$H_{0}\ :\ \
\frac{{C}_{1}(M_{x}+L_{x},L_{y},e;l)}{{C}_{2}(L_{x},L_{y},e;l)}=\frac{{C}_{3}(M_{x}+L_{x},e;l)}{{C}_{4}(L_{x},e;l)}\,.$
(4)

If we consider two sets of simultaneous samples111To implement the test, each
series is standardized so that all series share a common standard deviation,
and thereby share a common scale parameter.
$\\{x_{i,t},i=1,\cdots,n_{1},t=1,\cdots,T\\}$ and
$\\{y_{j,t},j=1,\cdots,n_{2},t=1,\cdots,T\\}$, we first provide the consistent
estimators of ${C}_{1}(M_{x}+L_{x},L_{y},e;l)$, ${C}_{2}(L_{x},L_{y},e;l)$,
${C}_{3}(M_{x}+L_{x},e;l)$ and ${C}_{4}(L_{x},e;l)$ are

$\displaystyle{\hat{C}}_{1}(M_{x}+L_{x},L_{y},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{m_{x_{i}}+L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{m_{x_{i}}+L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)\cdot\prod^{n_{2}}_{j=1}I\left({y}^{L_{y_{j}}}_{j,t-L_{y_{j}}},{y}^{L_{y_{j}}}_{j,t+l-L_{y_{j}}},e\right)$
$\displaystyle{\hat{C}}_{2}(L_{x},L_{y},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)\cdot\prod^{n_{2}}_{j=1}I\left({y}^{L_{y_{j}}}_{j,t-L_{y_{j}}},{y}^{L_{y_{j}}}_{j,t+l-L_{y_{j}}},e\right)$
$\displaystyle{\hat{C}}_{3}(M_{x}+L_{y},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{m_{x_{i}}+L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{m_{x_{i}}+L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)$
$\displaystyle{\hat{C}}_{4}(L_{x},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)$

where $L_{xy}=\max(L_{x},L_{y})$, $n=T-L_{xy}-l-m_{x}+1$ and

$I(x,y,e)\equiv\begin{cases}0,&\text{if $\|x-y\|>e$ }\\\ 1,&\text{if
$\|x-y\|\leq e$ }\end{cases}.$

The consistency of our proposed estimators can be shown straightforwardly and
the detail of the proof is omitted. We use a simple numerical study to show
that our estimators are consistent whereas those of Bai et al. (2010) are not.
Let

$\left(\begin{array}[]{c}X_{1,t}\\\ X_{2,t}\\\
\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)\left(\begin{array}[]{c}a_{1,t-1}\\\ a_{2,t-1}\\\
\end{array}\right)+\left(\begin{array}[]{c}a_{1,t}\\\ a_{2,t}\\\
\end{array}\right),$

where $\\{a_{1,t}\\},\\{a_{2,t}\\}$ are i.i.d. and mutually independent random
variables generated from $N(0,1)$, while {$Y_{t}$} could be any stationary
sequence. Let $l=1$, $L_{x}=L_{y}=M_{x}=1$. We can calculate the exact values
of ${C_{4}\big{(}{L_{x}},e;l\big{)}}$, which are $0.2709$ and $0.5057$,
respectively, when $e=1$ and $e=1.5$. For simplicity, we denote the true value
of ${C_{4}\big{(}{L_{x}},e;l\big{)}}$, the estimate proposed by Bai et
al.(2010) and our new estimate as $C_{4}$, $\hat{C}_{4}^{BWZ}$ and
$\hat{C}_{4}$, respectively, in Table 1. Additionally, Table 1 provides the
estimated values with their corresponding relative estimation errors in
brackets when $T=1000,2000\,\textrm{and}\,4000$. It is obvious that
$\hat{C}_{4}^{BWZ}$ is not consistent.

Table 1: ${C_{4}\big{(}{L_{x}},e;l\big{)}}$ and estimated values.

|  | $e=1$ |  | $e=1.5$ |
---|---|---|---|---|---
$T=$ |  | $C_{4}$ | $\hat{C}_{4}$ | $\hat{C}_{4}^{BWZ}$ |  | $C_{4}$ | $\hat{C}_{4}$ | $\hat{C}_{4}^{BWZ}$ |
1000 |  | 0.2709 | 0.2497(7.83%) | 0.0154(94.32%) |  | 0.5057 | 0.4774(5.60%) | 0.0732(85.53%) |
2000 |  | 0.2709 | 0.2639(2.58%) | 0.0176(93.50%) |  | 0.5057 | 0.4847(4.15%) | 0.0795(84.28%) |
4000 |  | 0.2709 | 0.2692(0.63%) | 0.0193(92.88%) |  | 0.5057 | 0.4909(2.93%) | 0.0820 (83.78%) |

Note: The true value of ${C_{4}\big{(}{L_{x}},e;l\big{)}}$ is denoted $C_{4}$,
the BWZ estimate and our new estimate are denoted $\hat{C}_{4}^{BWZ}$ and
$\hat{C}_{4}$, respectively. The relative estimation errors are in the
accompanying brackets.

Now, we propose

$T_{n}=\sqrt{n}\left(\frac{\hat{C}_{1}\big{(}{M_{x}}+{L_{x}},L_{y},e,l\big{)}}{\hat{C}_{2}\big{(}{L_{x}},L_{y},e,l\big{)}}-\frac{\hat{C}_{3}\big{(}{M_{x}}+{L_{x}},e,l\big{)}}{\hat{C}_{4}\big{(}{L_{x}},e,l\big{)}}\right)$
(5)

as the test statistic, and we derive the following asymptotic distribution of
$T_{n}$ for the Granger causality test.

######  Theorem 3.1.

Stationary sequences $\\{x_{i,t},i=1,\cdots,n_{1},t=1,\cdots,T\\}$ and
$\\{y_{j,t},j=1,\cdots,n_{2},t=1,\cdots,T\\}$ are strong mixing, with mixing
coefficients satisfying the conditions of Lemma 1 presented in Appendix, for
given values of $l,L_{x},L_{y},M_{x}$ and $e>0$, under the null hypothesis
that $\\{Y_{t}\\}$ does not strictly Granger cause $\\{X_{t}\\}$, then the
test statistic is defined in (5)

$\sqrt{n}\left(\frac{\hat{C}_{1}\big{(}{M_{x}}+{L_{x}},L_{y},e,l\big{)}}{\hat{C}_{2}\big{(}{L_{x}},L_{y},e,l\big{)}}-\frac{\hat{C}_{3}\big{(}{M_{x}}+{L_{x}},e,l\big{)}}{\hat{C}_{4}\big{(}{L_{x}},e,l\big{)}}\right)\overset{d}{\longrightarrow}N\big{(}0,\sigma^{2}(M_{x},{L_{x}},L_{y},e,l)\big{)}\,.$

The asymptotic variance $\sigma^{2}(M_{x},{L_{x}},L_{y},e,l)$ with its
consistent estimator $\hat{\sigma}^{2}(M_{x},{L_{x}},L_{y},e,l)$ and the proof
of theorem 3.1 are given in the Appendix. The hypothesis $H_{0}$ defined in
(4) is rejected at $\alpha$ if

$\displaystyle\big{|}T_{n}\big{|}/{\hat{\sigma}(M_{x},{L_{x}},L_{y},e,l)}>z_{{\alpha/2}},$

where $z_{{\alpha/2}}$ is the up ${\alpha/2}$ quantile of the standard normal
distribution. In this situation, we will conclude that there exists nonlinear
Granger causality from {$Y_{t}$} to {$X_{t}$}.

There are several possible methods to estimate the asymptotic covariance
$\sigma^{2}(M_{x},{L_{x}},L_{y},e,l)$. A model-based approach uses known laws
of {$X_{t}$} and {$Y_{t}$} to calculate the expectations in the formula given
in the Appendix and simply substitutes $C_{j}(*),j=1,2,3,4$ with their
corresponding estimates. However, in practice, we can hardly avoid model
misspecification and may obtain improper laws of {$X_{t}$} and {$Y_{t}$}. We
suggest the use of bootstrap methods as in the simulation studies we use to
test hypothesis $H_{0}$.

##  4 Simulation

In this section, we perform numerical studies using simulations to illustrate
the applicability and superiority of the new multivariate nonlinear Granger
causality test developed in Section 3. Let $R$ be the times of rejecting the
null hypothesis that $Y_{t}$ does not strictly Granger cause $X_{t}$
nonlinearly in 10,000 replications at the $\alpha$ level, and thus, the
empirical power is ${R}/{10,000}$. In our simulation, the level $\alpha=0.05$,
we standardized the series and chose the same lag length and lead length:
$L_{x}=L_{y}=M_{x}=1$. We set three situations of $l$ and two situations of
$e$: $l=1$, $l=2$, $l=3$ and $e=1$, $e=1.5$.

Consider the following model:

$\displaystyle X_{t}=\beta Y_{1,t-1}Y_{2,t-1}+\varepsilon_{t}\,,$ (6)

where $\\{(Y_{1,t},Y_{2,t})^{\prime}\\}$ are i.i.d. and mutually independent
random variables generated from standard normal distribution $N(0,1)$,
$\\{\varepsilon_{t}\\}$ is Gaussian white noise generated from $N(0,0.1)$ and
independent of $\\{Y_{1,t}\\},\\{Y_{2,t}\\}$. There is no nonlinear Granger
causality from $Y_{t}$ to $X_{t}$ when $\beta=0$, and causality strengthens
when $\beta$ increases.

From the results displayed in Table 2, we conclude first that our test
possesses decent size, as we can see when $\beta=0$ the empirical size are all
closed to the test level 0.05 for different settings of parameters and sample
size. Second, our test possesses very appropriate power, as we see that
empirical power increases as $\beta$ increases, especially when sample size is
500 the empirical power sharply increase to 1. Further, we find that different
settings of $e$ may influence the test results. Though the influence is little
in our simulation, we still suggest that practitioners choose a couple of
different values of $e$.

Table 2: Test multivariate nonlinear Granger causality form $Y_{t}$ to $X_{t}$

$T=200$ |  | $e=1$ |  | $e=1.5$ |
---|---|---|---|---|---
|  | $l=1$ | $l=2$ | $l=3$ |  | $l=1$ | $l=2$ | $l=3$ |
$\beta=0\ \ $ |  | 0.0419 | 0.0441 | 0.0432 |  | 0.0444 | 0.0506 | 0.0438 |
$\beta=0.1$ |  | 0.1509 | 0.2341 | 0.2184 |  | 0.3647 | 0.5209 | 0.5121 |
$\beta=0.2$ |  | 0.5629 | 0.7425 | 0.7284 |  | 0.8352 | 0.9416 | 0.9416 |
$\beta=0.3$ |  | 0.8178 | 0.9323 | 0.9267 |  | 0.929 | 0.9810 | 0.9825 |
$\beta=0.4$ |  | 0.8994 | 0.9712 | 0.9719 |  | 0.9512 | 0.9878 | 0.9889 |
$\beta=0.5$ |  | 0.9366 | 0.9812 | 0.9808 |  | 0.9586 | 0.9914 | 0.9915 |
$\beta=0.6$ |  | 0.9475 | 0.9870 | 0.9862 |  | 0.9640 | 0.9918 | 0.9940 |
$\beta=0.7$ |  | 0.9574 | 0.9875 | 0.9874 |  | 0.9664 | 0.9921 | 0.9942 |
$\beta=0.8$ |  | 0.9615 | 0.9888 | 0.9882 |  | 0.9688 | 0.9926 | 0.9953 |
$\beta=0.9$ |  | 0.9633 | 0.9896 | 0.9901 |  | 0.9711 | 0.9923 | 0.994 |
$T=500$ |  | $e=1$ |  | $e=1.5$ |
|  | $l=1$ | $l=2$ | $l=3$ |  | $l=1$ | $l=2$ | $l=3$ |
$\beta=0\ \ $ |  | 0.0560 | 0.0560 | 0.0478 |  | 0.0501 | 0.0529 | 0.0423 |
$\beta=0.1$ |  | 0.3969 | 0.5081 | 0.5344 |  | 0.7835 | 0.9043 | 0.9017 |
$\beta=0.2$ |  | 0.9622 | 0.9908 | 0.9900 |  | 0.9994 | 1.0000 | 1.0000 |
$\beta=0.3$ |  | 0.9989 | 1.0000 | 0.9997 |  | 0.9999 | 1.0000 | 1.0000 |
$\beta=0.4$ |  | 0.9998 | 1.0000 | 1.0000 |  | 1.0000 | 1.0000 | 1.0000 |
$\beta=0.5$ |  | 1.0000 | 1.0000 | 1.0000 |  | 1.0000 | 1.0000 | 1.0000 |
$\beta=0.6$ |  | 1.0000 | 1.0000 | 1.0000 |  | 1.0000 | 1.0000 | 1.0000 |
$\beta=0.7$ |  | 1.0000 | 1.0000 | 1.0000 |  | 1.0000 | 1.0000 | 1.0000 |
$\beta=0.8$ |  | 1.0000 | 1.0000 | 1.0000 |  | 1.0000 | 1.0000 | 1.0000 |
$\beta=0.9$ |  | 1.0000 | 1.0000 | 1.0000 |  | 1.0000 | 1.0000 | 1.0000 |

Note: $Y_{t}$ and $X_{t}$ are from the model present in equation (6).
$L_{x}=L_{y}=M_{x}=1$ in our test. Simulation is conducted with the test level
$\alpha=5\%$, and 10,000 replications.

##  5 Conclusion and Remarks

In this paper, we reinvestigate the multivariate nonlinear Granger causality
test extended by Bai et al. (2010) which attempt to uncover significant
nonlinearities in the dynamic interrelationships between two groups of
variables. We find that Bai et al. (2010) as well as Hiemstra and Jones (1994)
take the estimators of the probabilities in their definition as $U$-statistics
and establish a CLT of the test statistic by applying the asymptotic property
of $U$-statistics. After revealing that the estimators proposed by Bai et al.
(2010) is not $U$\- statistics, we show that their estimators are also not
consistent.

The procedure of our new test begins with presenting consistent estimators of
probabilities in the definition. Numerical study supports that our estimators
are consistent, further our new test possesses admirable properties both in
size and power.

There are still amounts of appealing aspects in nonlinear Granger causality
test. It is worth noting that Diks and Wolski (2015) extend the test in Diks
and Panchenko (2006) which highlight a need for substitutions for the
relationship tested in the HJ test.

## Appendix

A1: Central Limit Theorems for strong mixing stationary sequence

$\\{\left(Z_{t},\mathcal{F}_{t}\right),-\infty<t<\infty\\}$ is a stochastic
process defined on the probability space $(\Omega,\mathcal{F},P)$. The history
and the future of $Z_{t}$ are $\sigma$-algebras
$\mathfrak{M}_{t}^{\infty}=\\{\mathcal{F}_{s},s>t\\}$ and $\sigma$-algebras
$\mathfrak{M}_{-\infty}^{t}=\\{\mathcal{F}_{s},s<t\\}$ respectively.

Let {$\left(Z_{i},\mathcal{F}_{i}\right)$} be a stationary sequence with
$E(Z_{i})=0$, $E({Z_{i}}^{2})<0$,and set
$S_{n}^{m}=\sum\limits_{i=m+1}^{n+m}Z_{i}$,
${\sigma_{n}}^{2}=Var(S_{n}^{m})$.We shall say that the sequence satisfies the
central limit theorem if

$\displaystyle\lim_{n\to\infty}P\\{\frac{S_{n}^{m}}{\sigma_{n}}<z\\}=(2\pi)^{-\frac{1}{2}}\int_{-\infty}^{z}{e^{-\frac{1}{2}u^{2}}du}=\Phi(z)\
.$

Definition A1: A stationary process {$Z_{t}$} is said to be strongly mixing
(completely regular) if
$\alpha(\tau)=\sup\limits_{A\in{\mathfrak{m}_{-\infty}^{0}},{B\in\mathfrak{m}_{\tau}^{\infty}}}|P(AB)-P(A)P(B)|\to
0$ as $\tau\to\infty$ through positive values.

Lemma A1: Let the stationary sequence {$Z_{i}$} satisfy the strong mixing
condition with mixing coefficient $\alpha(n)$, and let
$E|Z_{i}|^{2+\delta}<\infty$ for some $\delta>0$. If
$\sum\limits_{n=1}^{\infty}{\alpha(n)}^{\delta/(2+\delta)}<\infty$, then
$\sigma^{2}=E({Z_{0}}^{2})+2\sum\limits_{j=1}^{\infty}E(Z_{0}Z_{j})<\infty$,
and if $\sigma\neq 0$, then
$\lim\limits_{n\to\infty}P\\{\sigma^{-1}n^{-\frac{1}{2}}\sum\limits_{i=1}^{n}Z_{i}<z\\}=\Phi(z)$.

Readers can be referred to Ibragimov (1971) for a proof and detailed
discussion.

A2: Proof of Theorem 3.1

Assume $\\{x_{i,1},x_{i,2},\cdots,x_{i,T}\\}$ and
$\\{y_{j,1},y_{j,2},\cdots,y_{j,T}\\},i\in\\{1,2,\cdots,n_{1}\\},j\in\\{1,2,\cdots,n_{2}\\}$
are both strong mixing stationary sequences whose mixing coefficient
satisfying the conditions in Lemma 1. Then the following four sequences

$\displaystyle\\{Z_{1t}=n^{-1}\Big{(}\prod^{n_{1}}_{i=1}I\big{(}x_{i,t-L_{x_{i}}}^{m_{x_{i}}+L_{x_{i}}},x_{i,t+l-L_{x_{i}}}^{m_{x_{i}}+L_{x_{i}}},e\big{)}\cdot\prod^{n_{1}}_{j=1}I\big{(}y_{j,t-L_{y_{j}}}^{L_{y_{j}}},y_{j,t+l-L_{y_{j}}}^{L_{y_{j}}},e\big{)}-{C}_{1}\big{(}M_{x}+{L_{x}},L_{y},e;l\big{)}\Big{)}\\}\,,$
$\displaystyle\\{Z_{2t}=n^{-1}\Big{(}\prod^{n_{1}}_{i=1}I\big{(}x_{i,t-L_{x_{i}}}^{L_{x_{i}}},x_{i,t+l-L_{x_{i}}}^{L_{x_{i}}},e\big{)}\cdot\prod^{n_{1}}_{j=1}I\big{(}y_{j,t-L_{y_{j}}}^{L_{y_{j}}},y_{j,t+l-L_{y_{j}}}^{L_{y_{j}}},e\big{)}-{C}_{2}\big{(}{L_{x}},L_{y},e;l\big{)}\Big{)}\\}\,,$
$\displaystyle\\{Z_{3t}=n^{-1}\Big{(}\prod^{n_{1}}_{i=1}I\big{(}x_{i,t-L_{x_{i}}}^{L_{x_{i}}},x_{i,t+l-L_{x_{i}}}^{L_{x_{i}}},e\big{)}\cdot\prod^{n_{1}}_{j=1}I\big{(}y_{j,t-L_{y_{j}}}^{L_{y_{j}}},y_{j,t+l-L_{y_{j}}}^{L_{y_{j}}},e\big{)}-{C}_{2}\big{(}{L_{x}},L_{y},e;l\big{)}\Big{)}\\}\,,$
$\displaystyle\\{Z_{4t}=n^{-1}\Big{(}\prod^{n_{1}}_{i=1}I\big{(}x_{i,t-L_{x_{i}}}^{L_{x_{i}}},x_{i,t+l-L_{x_{i}}}^{L_{x_{i}}},e\big{)}-{C}_{4}\big{(}{L_{x}},e;l\big{)}\Big{)}\\}\,,$
$\displaystyle t=L_{xy}+1,\cdots,T-l-L_{xy}-m+1,$

satisfy the conditions of Lemma 1, where $n=T-L_{xy}-l-m_{x}+1$ and

$\displaystyle{C}_{1}(M_{x}+L_{x},L_{y},e;t,s)\equiv
P\left(\parallel{X}^{M_{x}+L_{x}}_{t-L_{x}}-{X}^{M_{x}+L_{x}}_{s-L_{x}}\parallel<e,\parallel{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\parallel<e\right)\,,$
$\displaystyle{C}_{2}(L_{x},L_{y},e;t,s)\equiv
P\left(\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e,\parallel{Y}^{L_{y}}_{t-L_{y}}-{Y}^{L_{y}}_{s-L_{y}}\parallel<e\right)$
$\displaystyle{C}_{3}(M_{x}+L_{x},e;t,s)\equiv
P\left(\parallel{X}^{M_{x}+L_{x}}_{t-L_{x}}-{X}^{M_{x}+L_{x}}_{s-L_{x}}\parallel<e\right)$
$\displaystyle{C}_{4}(L_{x},e;t,s)\equiv
P\left(\parallel{X}^{L_{x}}_{t-L_{x}}-{X}^{L_{x}}_{s-L_{x}}\parallel<e\right)\,.$

So {$Z_{1t}$}, {$Z_{2t}$}, {$Z_{3t}$} and {$Z_{4t}$} satisfy the central limit
theorem.

Further, for any real number $a_{1},a_{2},a_{3}$ and $a_{4}$, the sequence
{$Z_{t}=a_{1}Z_{1t}+a_{2}Z_{2t}+a_{3}Z_{3t}+a_{4}Z_{4t},t=L_{xy},\cdots,T-l-
L_{xy}-m+1$} also satisfies the conditions of Lemma 1 which implying that

$\displaystyle\sqrt{n}\left[\begin{array}[]{c}\hat{C}_{1}\big{(}m+{L_{x}},L_{y},e;l\big{)}-{C}_{1}\big{(}m+{L_{x}},L_{y},e;l\big{)}\\\
\hat{C}_{2}\big{(}{L_{x}},L_{y},e;l\big{)}-{C}_{2}\big{(}{L_{x}},L_{y},e;l\big{)}\\\
\hat{C}_{3}\big{(}m+{L_{x}},e;l\big{)}-{C}_{3}\big{(}m+{L_{x}},e;l\big{)}\\\
\hat{C}_{4}\big{(}L_{x},e;l\big{)}-{C}_{4}\big{(}{L_{x}},e;l\big{)}\end{array}\right]\overset{d}{\longrightarrow}N(0,\mathbf{\Sigma}),$

where

$\displaystyle{\hat{C}}_{1}(M_{x}+L_{x},L_{y},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{m_{x_{i}}+L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{m_{x_{i}}+L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)\cdot\prod^{n_{2}}_{j=1}I\left({y}^{L_{y_{j}}}_{j,t-L_{y_{j}}},{y}^{L_{y_{j}}}_{j,t+l-L_{y_{j}}},e\right)$
$\displaystyle{\hat{C}}_{2}(L_{x},L_{y},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)\cdot\prod^{n_{2}}_{j=1}I\left({y}^{L_{y_{j}}}_{j,t-L_{y_{j}}},{y}^{L_{y_{j}}}_{j,t+l-L_{y_{j}}},e\right)$
$\displaystyle{\hat{C}}_{3}(M_{x}+L_{y},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{m_{x_{i}}+L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{m_{x_{i}}+L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)$
$\displaystyle{\hat{C}}_{4}(L_{x},e;l)\equiv\frac{1}{n}\sum^{T-l-
m_{x}+1}_{t=L_{xy}+1}\prod^{n_{1}}_{i=1}I\left({x}^{L_{x_{i}}}_{i,t-L_{x_{i}}},{x}^{L_{x_{i}}}_{i,t+l-L_{x_{i}}},e\right)$

and $\mathbf{\Sigma}$ is a $4\times 4$ symmetric matrix. Denote

$h_{1}(L_{x},L_{y},M_{x},l,k)=I(x^{L_{x}+M_{x}}_{L_{xy}+1+k-L_{x}},x^{L_{x}+M_{x}}_{L_{xy}+1+k+l-L_{x}},e)\,,$
$h_{2}(L_{x},L_{y},l,k)=I(y^{L_{y}}_{L_{xy}+1+k-L_{y}},y^{L_{y}}_{L_{xy}+1+k+l-L_{y}},e)\
,$

We have

$\displaystyle\mathbf{\Sigma}_{11}=E\left[\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}^{2}\right]$
$\displaystyle+\sum\limits_{k=1}^{n-1}2(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}\Big{]}\
,$
$\displaystyle\mathbf{\Sigma}_{12}=E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}\Big{]}\
,$
$\displaystyle\mathbf{\Sigma}_{13}=E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}({M_{x}+L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)-{C}_{3}({M_{x}+L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}({M_{x}+L_{x}},e;l)\big{)}\Big{]}\
,$
$\displaystyle\mathbf{\Sigma}_{14}=E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,k)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{1}(M_{x}+{L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}\ ,$
$\displaystyle\mathbf{\Sigma}_{22}=E\left[\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}^{2}\right]$
$\displaystyle+\sum\limits_{k=1}^{n-1}2(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}\Big{]}\
,$
$\displaystyle\mathbf{\Sigma}_{23}=E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}\Big{]}\
,$
$\displaystyle\mathbf{\Sigma}_{24}=E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,0)h_{2}(L_{x},L_{y},l,0)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,k)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,k)h_{2}(L_{x},L_{y},l,k)-{C}_{2}({L_{x}},L_{y},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}\ ,$
$\displaystyle\mathbf{\Sigma}_{33}=E\left[\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}^{2}\right]$
$\displaystyle+\sum\limits_{k=1}^{n-1}2(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}\Big{]}\
,$
$\displaystyle\mathbf{\Sigma}_{34}=E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,0)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,k)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}$
$\displaystyle+\sum\limits_{k=1}^{n-1}(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},M_{x},l,k)-{C}_{3}(M_{x}+{L_{x}},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}\ ,$
$\displaystyle\mathbf{\Sigma}_{44}=E\left[\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}^{2}\right]$
$\displaystyle+\sum\limits_{k=1}^{n-1}2(1-\frac{k}{n})E\Big{[}\big{(}h_{1}(L_{x},L_{y},0,l,0)-{C}_{4}({L_{x}},e;l)\big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big{(}h_{1}(L_{x},L_{y},0,l,k)-{C}_{4}({L_{x}},e;l)\big{)}\Big{]}\ .$

Under the null hypothesis, applying the delta method (Serfling, 1980), we have

$\displaystyle\sqrt{n}\left(\frac{\hat{C}_{1}\big{(}M_{x}+{L_{x}},L_{y},e,l\big{)}}{\hat{C}_{2}\big{(}{L_{x}},L_{y},e,l\big{)}}-\frac{\hat{C}_{3}\big{(}M_{x}+{L_{x}},e,l\big{)}}{\hat{C}_{4}\big{(}{L_{x}},e,l\big{)}}\right)\overset{d}{\longrightarrow}N\big{(}0,\sigma^{2}(M_{x},{L_{x}},L_{y},e,l)\big{)}\,,$

where
$\sigma^{2}(M_{x},{L_{x}},L_{y},e,l)=\nabla^{\prime}{\mathbf{\Sigma}}\nabla$,
in which

$\displaystyle\nabla$
$\displaystyle=\left(\frac{1}{C_{2}\big{(}{L_{x}},{L_{y}},e,l\big{)}}\,,\,-\frac{C_{1}\big{(}M_{x}+{L_{x}},{L_{y}},e,l\big{)}}{C_{2}^{2}\big{(}{L_{x}},{L_{y}},e,l\big{)}}\,,\,-\frac{1}{C_{4}\big{(}{L_{x}},e,l\big{)}}\,,\,\frac{C_{3}\big{(}{M_{x}}+{L_{x}},e,l\big{)}}{C_{4}^{2}\big{(}{L_{x}},e,l\big{)}}\right)^{\prime}\
.$

An consistent estimator $\hat{\sigma}^{2}(M_{x},{L_{x}},L_{y},e,l)$ of the
asymptotic variance can be got by replacing all the parts in the sandwich
$\nabla^{\prime}{\mathbf{\Sigma}}\nabla$ by their empirical estimates.

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