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I
nt. J. Global Energy Issues, Vol. 36, Nos. 5/6, 2013 277
Copyright © 2013 Inderscience Enterprises Ltd.
Non-linear cointegration between crude oil and stock
markets: evidence from Asia-Pacific countries
Sufang Li*
School of Statistics and Mathematics,
Zhongnan University of Economics and Law,
Wuhan 430073, China
E-mail: bbs8.8@163.com
*Corresponding author
Huiming Zhu
College of Business Administration,
Hunan University,
Changsha 410082, China
E-mail: zhuhuiming1999@aliyun.com
Rong Li
Department of Economics,
Huaihua College,
Huaihua 418000, China
E-mail: lirong917@aliyun.com
Abstract: This paper aims to investigate the possible linkage and non-linear
interaction between crude oil and stock markets using cointegration and
causality tests. Daily data on world crude oil prices and the stock indices of
Japan, Taiwan, South Korea, Australia, Indonesia, India, Singapore and
Malaysia are selected for this study. Unlike the previous literature, by
conducting a non-linear cointegration analysis, we obtain evidence of
non-linear long-term equilibrium between crude oil and stock markets for the
eight Asia-Pacific markets. The Granger causality tests demonstrate the
existence of bidirectional short-run Granger causality between crude oil and
these Asia-Pacific stock markets. The long-run Granger causality between them
is unidirectional. However, the speed of adjustment toward long-run
equilibrium is relatively slow.
Keywords: cointegration; non-linear analysis; crude oil; stock market; granger
causality.
Reference to this paper should be made as follows: Li, S., Zhu, H. and Li, R.
(2013) ‘Non-linear cointegration between crude oil and stock markets:
evidence from Asia-Pacific countries’, Int. J. Global Energy Issues, Vol. 36,
Nos. 5/6, pp.277–292.
Biographical notes: Sufang Li is a Lecturer at School of Statistics and
Mathematics, Zhongnan University of Economics and Law, China.
278 S. Li et al.
Huiming Zhu is a Professor at College of Business Administration, Hunan
University, Changsha, China.
Rong Li is a Lecturer at the Department of Economics, Huaihua College,
Huaihua, China.
1 Introduction
Oil price shocks are often considered to be a significant macroeconomic indicator that
has implications for economic activities, for example, the stock market, aggregate
demand and real economic growth (Cunado and Gracia, 2005). There is a substantial
body of applied research on the relationships between oil prices and a country’s macro
economy (for example, Huang et al., 2005; Bachmeier, 2008). Most of the existing
research on the relationships between energy prices and the macro economy has mainly
focused on the relationship between crude oil prices and economic growth (Hamilton,
2003; Bachmeier, 2008). However, it is surprising that only a relatively small amount of
the literature has studied the relationship between oil price shocks and stock markets
(Jones and Kaul, 1996; Huang et al., 1996; Huang and Guo, 2008). These studies fail to
reach a unanimous conclusion. Therefore, further research may be necessary on the
relationship between oil and stock markets.
While a wealth of studies thus far have been undertaken to focus on the relationship
between crude oil prices and the stock markets in American and European countries, few
studies have been conducted on the interaction between crude oil prices and stock
markets in other countries. Unlike most existing research, this paper intends to investigate
the linkage between crude oil prices and the main stock markets in Asia-Pacific countries.
The Asia-Pacific countries have been important oil consumers in the world and they have
played an increasingly significant role in the world oil markets. In recent years, crude oil
consumption has significantly increased in many Asia-Pacific countries, and an inclining
trend in oil consumption in Asia-Pacific countries was reported in the British Petroleum
Energy Review in 2011. Specifically, oil consumption in the Asia-Pacific region now
accounts for almost 31.5% of the total world oil consumption, with growth considerably
higher than the world growth in oil demand. Moreover, there are several important
emerging economies in the Asia-Pacific regions and these economies have expanded at a
rapid pace, which may inevitably attract global investors to these Asia-Pacific stock
markets. Thus, studies on the relationship between crude oil and stock markets in the
main Asia-Pacific countries are of global concern from both the theoretical and practical
points of view.
The main aim of this paper is to investigate the possible non-linear cointegration
between crude oil and stock markets in the major Asia-Pacific countries. The
implications of non-linearity in the relationship between crude oil and stock markets are
potentially very meaningful. In financial markets, for example, because there exist many
noise traders who tend to chase the market, bullish and bearish markets behave
differently and non-linearity may be considered. Likewise, stock markets may respond in
different ways to the rise and fall of the crude oil price, which can be attributed to
frictions in the financial markets, the differences in the availability of future contracts,
Non-linear cointegration between crude oil and stock markets 279
and institutional and regulatory constraints in financial markets among other factors.
Ignoring the non-linearity in the relationship between crude oil and stock markets may
lead to biased inferences and misleading outcomes. While linear cointegration techniques
have already been conducted to analyse the linkage between crude oil and stock markets,
studying the non-linear relationship between them could also provide more information
on the oil-stock nexus.
The major contributions of this study are as follows. First, this paper examines the
potential non-linear relationship between crude oil and stock markets for Asia-Pacific
countries. More specifically, it tests for non-linear cointegration in the oil-stock nexus
using the procedure proposed by Granger and Hallman (1991) and Granger (1991), which
transforms the crude oil price and stock variables through the non-parametric alternating
conditional expectations algorithm. In addition, one of the major advantages of this
procedure is that the general non-linear long-term equilibrium relationship between crude
oil and stock markets could be confirmed. Second, this paper uses the Bayesian Markov
Chain Monte Carlo method to conduct the non-linear cointegration study. Because the
Bayesian estimation method combines prior information, it may obtain more precise
cointegration test and make the cointegration inference more reliable.
The remainder of the paper is as follows. Section 2 presents a brief literature review.
Section 3 describes the econometric methodology. Section 4 presents the data sources
and its basis statistics. Section 5 provides the empirical results and discussion. Section 6
presents this paper’s conclusions.
2 Literature review
In the previous literature, studies on the relationship between crude oil price and stock
markets have mainly concentrated on American or European markets (Sadorsky, 1999;
Papapetrou, 2001; Hammoudeh et al., 2004; Park and Ratti, 2008; Chang et al., 2013).
For example, Sadorsky (1999) reveals the relationship between oil prices and the US
stock market. Papapetrou (2001) employs an error-correction representation of a vector
autoregressive macroeconomic model to investigate the interactions between oil price
shocks and the stock market in Greece, finding that positive oil price shocks have
negatively affected Greek stock returns. Hammoudeh et al. (2004) explore long-run
relationships or comovements between two US markets of oil prices and S&P oil sector
stock indices using unit root tests, cointegration tests, error correction models and
generalised autoregressive conditional heteroskedasticity models with spillover effects.
Ghouri (2006) shows that there is a very strong negative link between the WTI crude oil
price and US monthly stock positions. Park and Ratti (2008) examine the linkage
between oil price shocks and stock markets in the US and 13 European countries. They
indicate that the stock market’s response to oil price shocks partly depends on whether
the country is oil-importing or oil-exporting. Furthermore, Kilian and Park (2009)
identify the interaction between aggregate US real stock returns and the innovation of the
real oil price. Using four multivariate generalised autoregressive conditional
heteroskedasticity models, Chang et al. (2013) examine conditional correlations and
volatility spillovers between crude oil returns and stock index returns. The empirical
findings from the vector autoregressive moving average-generalised autoregressive
conditional heteroskedasticity and vector autoregressive moving average-asymmetric
280 S. Li et al.
generalised autoregressive conditional heteroskedasticity models indicate that there is
little evidence of dependence between the crude oil and financial markets.
More recently, some research suggests that the linear relationship between oil prices
and stock markets is not very evident in practice. The conventional approaches to
investigate the oil-stock nexus should be rethought. Therefore, the potential non-linear
relationship between crude oil and stock markets are further addressed. According to
Huang et al. (2005), the asymmetric and non-linear relationships between oil price shocks
and economic activities are confirmed by the multivariate threshold model. Bachmeier
(2008) considers the role that monetary policy plays in the transmission of oil shocks to
the US economy. Using the linear benchmark model and threshold model, he investigates
the response of stock prices to linear and non-linear measures of oil shocks. Chiou and
Lee (2009) confirm the non-linear effects of oil price shocks on the US stock market
using the autoregressive conditional jump intensity. Jammazi and Aloui (2010) employ
both a wavelet analysis and a Markov-switching vector autoregressive model for the
analysis of crude oil prices and the stock markets in the UK, France and Japan. Zhu et al.
(2011) studies the non-linear asymmetric cointegrating relationship between crude oil
price and stock markets using panel threshold cointegration techniques. Utilising quantile
regression with non-linear asymmetric effects, Lee and Zeng (2011) re-examine the links
between oil price and stock markets in G7 countries. Chen (2010) also shows the effects
of oil price shocks on the US stock market using time-varying, transition-probability
Markov-switching models. In addition, Zhang and Wei (2011) find that American, British
and Japanese market risks exert asymmetric shocks on the international crude oil market
in their up and down conditions.
However, as shown above, relatively few researchers have explored the oil-stock
nexus for Asia-Pacific countries in multi-country studies, which include several
important emerging economies. Thus, further exploration of the connection between
crude oil and stock markets in Asia-Pacific markets should be conducted. This paper
seeks to investigate the case of the main Asia-Pacific economies in a multi-country
context, which aims to fill this gap and enrich the literature on the relationship between
crude oil and stock markets. In addition, to the best of our knowledge, no work has
studied the possible non-linear linkages between crude oil prices and Asia-Pacific stock
markets. This paper combines linear cointegration and non-linear cointegration to analyse
the relationship between crude oil and stock markets for the main Asia-Pacific countries.
3 Methodology
It is of great importance to analyse the oil-stock nexus for Asia-Pacific economies in a
multi-country framework. Thus, to evaluate the dynamic relationship between crude oil
and stock markets in Asia-Pacific countries, both linear cointegration and non-linear
cointegration techniques are employed here to fully examine the mechanism.
3.1 Linear cointegration test
At first, we introduce the two-step procedure suggested by Engle and Granger (1987) to
test the linear cointegration between crude oil prices and stock markets in Asia-Pacific
countries. Considering the following cointegration regression
Non-linear cointegration between crude oil and stock markets 281
01 ,
ttt
sp oil ε=+ +
β
β
(1)
where
β
0 is the intercept,
β
1 is the regressive coefficient, εt is disturbance term, and spt
denotes the natural logarithm of the stock price index, oilt represents the natural
logarithm of the crude oil price. Specifically, spt and oilt are integrated processes of the
same order.
The linear cointegration test proposed by Engle and Granger (1987) is based on the
estimated residuals from an ordinary least squares estimation of the cointegration
regression. From equation (1), the estimated residuals are obtained by
01
ˆˆ
ˆ.
tt t
εsp oil=−−
ββ
(2)
To test for the null hypothesis of no cointegration, the unit root test is conducted on the
long-run residuals ˆt
ε of equation (2). If these residuals are found to be stationary, then
crude oil and the stock market are linearly cointegrated.
3.2 Non-linear cointegration test with alternating conditional expectations
transformation
To examine the possible non-linear cointegration between crude oil and stock markets,
the above-mentioned linear cointegration could be extended to a non-linear framework.
That is, spt and oilt are said to be non-linearly cointegrated if there are non-linear
measurable functions f(·) and g(·) such that f(spt) and g(oilt) are integrated of the same
order, and a linear combination of f(spt) and g(oilt) is stationary. Utilising the
non-parametric alternating conditional expectations algorithm suggested by Granger and
Hallman (1991), Granger (1991) and Meese and Rose (1991), this paper then tests for
non-linear cointegration between crude oil and stock markets based on the Engle-Granger
cointegration procedure.
As a non-parametric algorithm, alternating conditional expectations relies on
extremely weak distributional assumptions, and can handle various non-linear
transformations of the series. Suppose 11 2 2
( ( ), ( ), ( ), , ( ))
nn
f
ygx g x g x" are the
transformations for a set of variables 12
(, , , , )
n
yx x x" through the alternating conditional
expectations algorithm, which maximise the correlation between f(y) and
1
().
n
ii
i
g
x
=
∑
Equivalently, this algorithm is used to find the optimal non-linear transformations that
minimise
()
()
2
1
212
()
,, ,, [()]
n
ii
i
n
Efy gx
efgg g Var f y
=
⎡⎤
−
⎢⎥
⎢⎥
⎣⎦
=∑
" (3)
where 212
(, , , , )
n
efgg g" is the expected mean squared error of the non-linear additive
model
()
1
() .
n
ii
i
f
ygxe
=
=+
∑ (4)
282 S. Li et al.
To perform the alternating conditional expectations algorithm, the following steps can be
replicated:
Step 1 Initialise; set (0)
1/2
()
() () ,
[()]
yEy
fy f y Var y
−
== and (0)
() () 0,
ii i
i
gx g x
=
=
1, , .in="
Step 2 Iterate until 212
(, , , , )
n
efgg g" fails to decrease:
a for j = 1 to n, set (1)
() () () ()
j
jj iij
j
ij
g
xgxEfy gxx
≠
⎧
⎫
⎡⎤
== −
⎨
⎬
⎢⎥
⎣⎦
⎩⎭
∑
b calculate
(1)
11
() () ()| ,
nn
ii ii
ii
f
yE gxy gxy
==
⎡⎤
=⎢⎥
⎢⎥
⎣⎦
∑∑
and set f(y) = f(1)(y).
The above steps mean that (), 1, ,
ii
g
xi n
=
" are estimated conditional on a given choice
of f(y) and f(y) is then estimated conditionally for the estimate of (), 1, ,.
ii
g
xi n
=
" Note
that the transformations of all the variables except one are treated as fixed. Specifically,
using a non-parametric kernel smoothing technique, the optimal transformation of the
variable in question can be derived; the algorithm is then conducted to the next variable.
Step 1 through Step 2 can be iterated until (3) is minimised.
Similar to a linear cointegration, the detailed procedure for a non-linear cointegration
can then be proceeded by two steps. The estimation based on the following model should
be conducted first
01
() ( ) ,
ttt
fsp ααgoil u=+ + (5)
where f(spt) and g(oilt) are alternating conditional expectations transformed series of spt
and oilt,
α
0 is the intercept,
α
1 is the regressive coefficient, and the estimated residual
series ˆt
u could be obtained. The presence of cointegration between f(spt) and g(oilt) is
then examined via consideration of the integrated nature of the residual ˆt
u from (5):
1
1
ˆˆ ˆ
ΔΔ,
p
ttktkt
k
uc
ρ
uφuξ
−−
=
=+ + +
∑ (6)
where 2
~(0,), 2,,.
t
ξNσtp T=+" Test for the null hypothesis of no cointegration is to
test for the null hypothesis H0: ρ = 0 versus the alternative H1: ρ = 0. If H0: ρ = 0 can be
rejected, then there exists cointegration between f(spt) and g(oilt). It indicates the
presence of a non-linear cointegration relationship between the raw variables spt and oilt.
3.3 Bayesian Markov chain Monte Carlo methods
In this paper, we conduct a non-linear cointegration test from a Bayesian perspective,
which adopts Markov Chain Monte Carlo methods. Specifically, we will build a Bayesian
equivalent of the two-step approach of Engle and Granger. Bayesian methodologies have
an elegant approach to draw inference sabout the parameters by combing with prior
information. Furthermore, while most classical methods rely on asymptotic arguments to
make inferences, Bayesian Markov Chain Monte Carlo methods use the exact posterior
Non-linear cointegration between crude oil and stock markets 283
distribution of the parameters. Therefore, Bayesian Markov Chain Monte Carlo
techniques will yield more accurate results.
Bayesian approaches to statistical modelling and its applications in economics are
now established and indispensable. The basic idea of the Bayesian estimation is the
Bayes’ theorem, which states that the posterior joint distribution function of the
parameters in the model is proportional to the product of their prior distribution and the
likelihood of the data. More specifically, in the non-linear cointegration setting, let t
y
and t
x
be alternating conditional expectations transformed data and, 1
Ω(, , , , )
p
c
ρ
φφ
=
"
be the parameter vector to be estimated. Then we can write the likelihood function as
2
1
2
21
1ˆˆ ˆ
(Ω|,) exp ΔΔ.
2
p
T
tt ktk
tp k
Lyx ucρuφu
σ−−
=+ =
⎧⎫
⎛⎞
⎪⎪
∝− −−−
⎜⎟
⎨⎬
⎜⎟
⎪⎪
⎝⎠
⎩⎭
∑∑
(7)
Thus, after choosing prior distributions π(Ω) for the parameter vector Ω, we can obtain
the posterior density function of Ω as given by
(Ω|,) (Ω|,)(Ω),πyx L yxπ∝
(8)
and posterior inference is based on the above posterior density.
The Markov Chain Monte Carlo sampling algorithms provide a virtually useful
computational tool for generating random samples from the posterior distribution. In the
past 20 years or more, Markov Chain Monte Carlo techniques have particularly
influenced and popularised Bayesian methods. These techniques are based on the
construction of a Markov chain in the parameter space. Under some mild regularity
conditions (see Tierney, 1994), the chain asymptotically converges to the posterior
distribution. Therefore, the realised value of the chain could be employed to evaluate the
posterior distribution and make posterior inferences. This study employs two Markov
Chain Monte Carlo sampling algorithms to generate Markov chains: the Gibbs sampling
and the Metropolis-Hastings algorithm. For a detailed description of both techniques, the
reader is referred to Geweke (2005) and Franses et al. (1997).
To perform hypothesis testing, one way to use frequentist statistics is to use a 95%
confidence interval as the acceptance region for the corresponding hypothesis-testing
problem. A similar rule is available in the Bayesian framework, that is, to present a
central interval of 95% probability of which the upper and lower bounds correspond to
the 97.5% and 2.5% of the posterior distribution. An attractive advantage of this method
is that it is relatively easier to implement compared with the approach based on Bayesian
factors. To conduct a Bayesian non-linear cointegration, that is, to test for the hypothesis
H0: ρ = 0, we simply use the rule of rejecting the null hypothesis if 0 is not included in
the 95% posterior interval of ρ.
4 Data and descriptive statistics
The data selected in this study are composed of daily crude oil price and stock indices for
eight Asia-Pacific countries, including the Nikkei 225 Stock Index (NK225) of Japan, the
Taiwan Stock Exchange Weighted Stock Index (TWSI) of Taiwan, the Korea Composite
Price Index (KCPI) of South Korea, the All Ordinaries Stock Index (AOSI) of Australia,
284 S. Li et al.
the Jakarta Composite Index (JKSE) of Indonesia, the BSE 30 Index (BSESN) of India,
the Straits Times Industrial Index (STII) of Singapore and the Kuala Lumpur Composite
Price Index (KLCPI) of Malaysia. As for the oil price, we choose the US crude oil price
as a representative of world real oil price, the stock index data cover the period
1997/07/01 through 2013/07/01, and the US crude oil price series during the
corresponding period is collected. The international crude oil price used here is from the
West Texas Intermediate (WTI) index, the most widely used oil price index in the world,
published by the United States Energy Information Administration. And all the stock
indices data were downloaded from the website http://finance.yahoo.com. Considering
the fact that different countries have different holidays, we omit some observations. After
matching these daily observations, we use them to analyse the relationships between
crude oil and stock markets.
Figure 1 Time series plots of daily crude oil and stock price indices (1997/07/01–2013/07/01)
(see online version for colours)
Figure 1 provides a time series plot of the raw crude oil price and stock price series. It
shows that the stock price indices and the crude oil price are closely connected and the
co-movement of crude oil and stock markets in Asia-Pacific countries seems to be more
apparent in recent years. Specifically, it also reveals that the crude oil price reached its
peak in 2008, and stock price indices posted a substantial decrease at the same time.
Furthermore, all the data are in a natural logarithmic form, and Table 1 reports the
summary statistics for the stock price indices of these eight countries and the crude oil
price index. The skewness and kurtosis provide some insights on the characteristics of the
distributions of the price indices. The sample kurtosis shows that our time series do not
display fat tails, and sample skewness indicates that the data are fairly symmetrically
distributioned around the sample mean. The hypothesis of normality is strongly rejected
by the Jarque-Bera test for all price indices.
Non-linear cointegration between crude oil and stock markets 285
Table 1 Summary statistics for all series
Series Crude oil NK225 AOSI KCPI BSESN KLCPI JKSE TWSI STII
Mean 3.8147 9.4146 8.2626 6.9549 8.9575 6.8508 7.0391 9.6464 7.6852
Median 3.9273 9.3643 8.2702 6.9303 8.8724 6.8096 6.9986 9.6440 7.6808
Maximum 4.9789 9.9443 8.8289 7.7093 9.9525 7.4891 8.5593 10.3621 8.2625
Minimum 2.3814 8.8615 7.7403 5.6348 7.8633 5.5710 5.5484 8.8039 6.6909
SD 0.6356 0.2582 0.2478 0.5064 0.6952 0.3565 0.8572 0.3156 0.3132
Skewness –0.2945 0.1241 0.1864 –0.3431 0.0199 –0.1493 0.1838 –0.1779 –0.3492
Kurtosis 1.9104 1.7680 2.0194 2.1032 1.3545 2.4766 1.5780 2.0595 2.3925
Jarque-Bera 256.7328* 258.4301* 184.4205* 209.6824* 446.3210* 59.7358* 349.5871* 168.0529* 143.9609*
p-value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Observations 4,016 3,927 4,022 3,947 3,954 3,948 3,889 3,989 4,033
Note: *Denotes significance at the 1% level.
286 S. Li et al.
5 Empirical results
5.1 Unit root tests
Because there may be a long-run relationship between crude oil price and stock markets
in most of these Asia-Pacific countries, unit root tests should be conducted first to
confirm if the crude oil and stock price indices are stationary. When the financial time
series exhibits non-linearities, the conventional tests for stationarity such as ADF, PP and
KPSS unit root tests may be unable to detect the mean reverting characteristic of financial
time series variables. Recently, Ng and Perron (2001) have proposed a modified version
of the ADF and PP tests that try to solve the main problems presented in conventional
unit root tests. Here, as in the study by Esteve and Prats (2008), the unit root tests
proposed by Ng and Perron (2001) are adopted for crude oil and stock series at first.
Table 2 presents the unit root test results of crude oil price and stock price indices. The
results imply that crude oil and stock price indices of the selected Asia-Pacific countries
are all unit root processes.
Table 2 Ng and Perron modified unit root tests for crude oil and stock price indices
Region Series
GLS
MZ
α
statistic GLS
t
MZ statistic GLS
MSB statistic
Level Difference Level Difference Level Difference
World Crude oil –9.5749 –341.7661* –2.1537 –13.0695* 0.2249 0.0382*
Japan NK225 –1.8288 –290.6073* –0.6391 –12.0363* 0.3495 0.0414*
Australia AOSI –3.8258 –135.9510* –1.3726 –8.2378* 0.3588 0.0606*
Korea KCPI –3.7289 –638.0811* –1.3640 –17.8545* 0.3658 0.0280*
India BSESN –2.7127 –696.7362* –1.1466 –18.6504* 0.4227 0.0268*
Malaysia KLCPI –0.3762 –583.6731* –0.3104 –17.0761* 0.8252 0.0293*
Indonesia JKSE –1.3815 –621.3430* –0.7733 –17.6099* 0.5597 0.0283*
Taiwan TWSI –2.5365 –637.2501* –1.0247 –17.8410* 0.4040 0.0280*
Singapore STII –1.6782 –336.5940* –0.7191 –12.9726* 0.4285 0.0385*
Notes: The null hypothesis of Ng-Perron modified unit root tests is that a series has a unit
root. For the MSBGLS test, the null hypothesis is rejected in favour of stationarity
when the estimated value is smaller than the critical value.
*, ** and ***denote significance at the 1%, 5% and 10% levels, respectively.
Table 3 Linear cointegration test
Region Test statistic
Japan –0.4036
Australia –0.5309
Korea –2.9383
India –2.4378
Malaysia –3.3453**
Indonesia –1.8980
Taiwan –1.3027
Singapore –1.5460
Notes: The null hypothesis of the cointegration test is there is no cointegration between
the variables. The 1%, 5%, and 10% critical values are –3.900, –3.338 and –3.046,
respectively. **Denotes significance at the 5% level.
Non-linear cointegration between crude oil and stock markets 287
Table 4 Non-linear cointegration test
Region Parameters Mean SD MC error 2.5 percentile Median 97.5 percentile Sample
Japan ρ
J
0.2468 0.0994 0.0011 0.0530 0.2465 0.4442 10,000
Australia ρ
A
0.3638 0.1209 0.0013 0.1282 0.3628 0.6051 10,000
Korea ρ
K
–0.0110 0.0010 9.31E-5 –0.0131 –0.0109 –0.0093 10,000
India ρ
India
–0.0115 0.0015 1.407E-4 –0.0148 –0.0112 –0.0095 10,000
Malaysia ρ
M
–0.0138 9.045E-4 7.455E-5 –0.0156 –0.0137 –0.0122 10,000
Indonesia ρ
Indone
–0.0108 8.778E-4 7.473E-5 –0.0128 –0.0106 –0.0094 10,000
Taiwan ρ
T
0.2653 0.1031 0.0011 0.0639 0.2650 0.4704 10,000
Singapore ρ
s
–0.0163 8.064E-4 1.299E-5 –0.0179 –0.0163 –0.0148 10,000
288 S. Li et al.
5.2 Linear and non-linear cointegration
On the basis of the unit root results, we proceed to conduct linear cointegration and
non-linear cointegration tests. Firstly, we test the linear cointegration relationship
between the variables by the Engle-Granger method, of which the results are listed in
Table 3. It is clear from Table 3 that there is no cointegration relationship between crude
oil and the eight Asia-Pacific stock markets at the 1% significant level. Specifically, the
cointegration relationship between crude oil and the Malaysia stock market is statistically
significant at the 5% level, but the statistics are only slightly beyond the 5% significant
critical value. Therefore, it could be concluded that there is no linear cointegration
between crude oil and the eight Asia-Pacific stock markets. One possible explanation for
our findings is that the traditional linear specifications are too restrictive and could not
reproduce the non-linearities between crude oil prices and stock markets. Therefore, we
should explore whether our results are unconditioned to the assumption of a linear
relationship.
To investigate the possible non-linear interaction between crude oil prices and stock
markets in these Asia-Pacific economies, we then conduct our study with the non-linear
framework. Thus, a non-linear cointegration test based on the alternating conditional
expectations transformed crude oil price and stock price indices is also conducted.
According to equations (7) and (8), the parameters to conduct the non-linear
cointegration are estimated by the Markov Chain Monte Carlo algorithm. To obtain
Markov chains from stationary distribution and a more precise estimation, the Markov
Chain Monte Carlo algorithm is run for 15,000 iterations, and the initial 5,000 samples
are discarded. Table 4 lists the parameter estimates of non-linear cointegration tests and
their respective posterior confidence intervals.
The kernel density plots of parameters based on a Bayesian non-linear cointegration
test with alternating conditional expectations transformed series show that the posterior
mean of ρJ, ρA and ρT is far from zero. More specifically, from the parameter estimates in
Table 4, we can find that none of the posterior confidence intervals of ρ cover the value
ρ = 0, indicating that ˆt
u is stationary for all of these Asia-Pacific markets. The results
indicate that there exists non-linear cointegration between crude oil and stock markets for
these selected Asia-Pacific economies. That is, there exists non-linear long-run
equilibrium relationship between crude oil price and the eight Asia-Pacific stock price
indices.
5.3 Causality analysis
The cointegration methods test the existence or absence of a long-run relationship
between crude oil and Asia-Pacific stock markets. Given that crude oil and stock price
indices are cointegrated, the direction of Granger causality between the variables can then
be explored using a Granger causality test. Following the Engle and Granger (1987)
two-step procedure, we extend the error correction model to the general non-linear
framework. For the countries with evidence of general non-linear cointegration between
stock and oil markets, we proceed to determine the nature of the causal relation by
estimating the vector error correction models of alternating conditional expectations
transformed variables. The long-run parameters in equation (5) are first estimated using
the OLS procedure to obtain the estimated residuals ˆ.
t
u Defining the residuals from
Non-linear cointegration between crude oil and stock markets 289
equation (5) as the error correction term, the lagged residuals are then incorporated into
the vector error correction models. The general forms of the alternating conditional
expectations transformed vector error correction models can be written in the following
form:
111 12 111
11
ˆ
ΔΔΔ
mm
titiititt
ii
sp ηδsp δoil γuζ
−−−
==
=+ + + +
∑∑
(9.a)
221 22 212
11
ˆ
ΔΔΔ
mm
titiititt
ii
oil ηδsp δoil γuζ
−−−
==
=+ + + +
∑∑
(9.b)
where t
oil
and t
s
p
are non-parametric alternating conditional expectations transformed
crude oil and stock variables, respectively, and δ are the short-run adjustment
coefficients, ζ are disturbance terms, the optimal lag length m can be determined by the
Akaike Information Criterion or Schwarz Bayesian Information Criterion. The short-run
Granger causality can be tested on the basis of H0: δ = 0 and the significance exploration
of the adjustment speed γ indicates the possible existence of long-run Granger causality.
Table 5 reports the short-run and long-run Granger causality results for the Asia-Pacific
economies where general non-linear cointegration exists between crude oil and stock
markets, such as in Japan, Australia and the other markets.
Table 5 Granger causality test
Region Dependent
variable
Short run Long run
Δ
s
p
Δoil
ECT
Japan Δ
s
p
- 0.00327*(0.0000) 1.05E-6***(0.0552)
Δoil
0.1645*(0.0000) - –4.14E-5*(0.0000)
Australia Δ
s
p
- 0.0024*(0.0000) 3.70E-6**(0.0274)
Δoil
0.2125*(0.0000) - –7.05E-5*(0.0000)
Korea Δ
s
p
- 0.00204*(0.0000) –9.11E-8(0.8917)
Δoil
0.2571*(0.0000) –0.00014*(0.0000)
India Δ
s
p
- 0.00251*(0.0000) 3.56E-8(0.9473)
Δoil
0.2064*(0.0000) - –0.0001*(0.0000)
Malaysia Δ
s
p
- 0.00227*(0.0000) 5.57E-7(0.3046)
Δoil
0.2640*(0.0000) - –0.00015*(0.0000)
Indonesia Δ
s
p
- 0.00199*(0.0000) 2.62E-7(0.7102)
Δoil
0.26397*(0.0000) - –0.000209*(0.0000)
Taiwan Δ
s
p
- 0.00305*(0.0000) 1.31E-7(0.8106)
Δoil
0.18815*(0.0000) - –7.91E-5*(0.0000)
Singapore Δ
s
p
- 0.00238*(0.0000) 2.84E-7(0.6043)
Δoil
0.23705*(0.0000) - –0.00011*(0.0000)
Notes: ECT represents the coefficients of the error correction term. The values in
parentheses are probability values.
*, **, and ***indicate significance at the 1%, 5% and 10% levels, respectively.
290 S. Li et al.
With the crude oil price as the dependent variable, the error correction terms are
statistically significant at the 1% level and the error correction coefficients are negative.
This result indicates the existence of long-run Granger causality running from these Asia-
Pacific stock markets to the crude oil price. However, the speed of adjustment to
equilibrium is relatively slow when the change of the deviation is negative. On the other
hand, for the Japanese and Australian stock price indices, the error correction terms are
statistically significant at the 10% and 5% levels, respectively, but the signs of the error
correction terms are positive. These results may suggest absence of long-run Granger
causality from the crude oil price to the Japanese and Australian stock markets.
Meanwhile, for the other Asia-Pacific stock markets, there is also no long-run Granger
causality from the crude oil market to the stock markets. In contrast, the short-run results
for Granger causality running from the crude oil market to the stock markets and from the
stock markets to the crude oil market are both highly significant for all eight Asia-Pacific
economies. This implies that the Granger causality between the crude oil price and these
Asia-Pacific stock markets is bidirectional in the short run. The surprising result is the
unidirectional long-run Granger causality from the eight Asia-Pacific stock markets to the
crude oil price. One possible explanation can be attributed to the portfolio diversification
strategies of global investors or speculators. Investment in the Asia-Pacific stock markets
has increased dramatically in recent years, which results in a higher integration between
the international oil market and the Asia-Pacific stock markets.
6 Concluding remarks
In this article we contribute to the debate on the non-linear relationships between the
crude oil market and the selected eight main stock markets of Asia-Pacific economies.
The general non-linear equilibrium is confirmed by non-linear cointegration with the
non-parametric alternating conditional expectations algorithm. The finding suggests that
there is general non-linear long-run equilibrium between crude oil and these eight Asia-
Pacific stock markets. Moreover, the non-linear Granger causality based on alternating
conditional expectations transformed variables is further implemented to explore the
existence of Granger causal links between crude oil price and Asia-Pacific stock markets.
For all eight Asia-Pacific economies, there is unidirectional, long-run Granger causality
running from the stock markets to crude oil prices. However, the speed of adjustment to
equilibrium is very slow. The surprising impact of the stock market on crude oil prices is
consistent with what Narayan and Narayan (2010) found for the Vietnamese stock
market.
In the short run, the Granger causality between crude oil prices and Asia-Pacific stock
markets is bidirectional. In this case, by utilising the information from the crude oil
market, an investor can speculate in the Asia-Pacific stock markets in the short term, and
vice versa. This finding also suggests that Asia-Pacific stocks may be attractive
destinations for hedging hikes in crude oil price. Overall, there exists a non-linear
interaction between crude oil and stock markets in Asia-Pacific economies, which has
important implications for empirical research in financial consequences of crude oil price
shocks. The relationship between other substitute energy sources, such as coal,
electricity, nuclear energy and stock markets can be considered in future studies.
Non-linear cointegration between crude oil and stock markets 291
Acknowledgements
The authors wish to thank the editor and the two reviewers for their very constructive
comments. Financial supports from the National Natural Science Foundation of China
under grant Nos. 71301166, 71171075, Ministry of Education in China Project of
Humanities and Social Sciences under grant No. 13YJC910007 and China Postdoctoral
Science Foundation under grant No. 2013M540623 is also acknowledged.
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