Volatility Spillover effects among Oil price fluctuation, Foreign
Exchange Rates and the Nigerian Stock Exchange:
A VAR-MEGARCH Analysis
Osabuohien-Irabor Osarumwense
Mathematics/Statistics Department
Ambrose Alli University, Ekpoma, Nigeria
[email protected], [email protected]
Abstract:
This study investigates the returns and volatility transmission among currency market (EXR),
stock market (STX), and the price of crude oil (OPR) using daily data for period of April 2nd,
2012 to September 26th, 2014. Empirical analysis and results reveals that OPR has a
significant asymmetry impact which increases negative innovations about twice more than
positive innovation. Besides, the existence of a unidirectional return spillover, there is
evidence of persistence of volatility in EXR and STX and a unidirectional volatility spillover
effect of shock from STX to EXR. Our finding also reveals that volatility in domestic markets
are more important than volatility from cross-market spillover shocks. This paper contributes
to literature by applying the “Koutmos (1996)” multivariate VAR-EGARCH model to the
variables under consideration.
Key words: Volatility, exchange rate, stock market, spillover, oil price.
JEL Classification: C22; C52; C58; G32; Q43
1.0 Introduction
Following the presentation of seminal papers of Krugman (1983) and Golub (1983),
confirming oil shocks as a major factor that contributed to the recession in the United States,
studies on the connections between oil prices and macro-economy variable have become of
great concern to researchers, financial practitioner and market participants. Much attention
has been attracted to the relationship between oil price shocks and the stock market, and
recently, the relationship between oil price shocks and the exchange rate. However, bulk of
these literatures has focused on developed countries and developed markets. This present
study therefore is an attempt to add to the few literatures on developing countries and
emerging markets in the Nigeria’s perspective. To this end, investigation on the relationship
among oil prices (West Texas Intermediate (WTI)), Naira/USD exchange rate, and the
Nigerian stock markets (NSE-30). Since Nigeria is a mono-product economy with net export,
the main stay of the country’s economy is crude oil, changes in the price of crude oil have
economy implication. In particular, the currency market and the stock market movements. Oil
price volatility has been found to have effects on the Nigeria Stock market but more on the
exchange rate of Naira. This is because, crude oil export earning accounts for about 92
percents of Nigeria’s foreign exchange. While the value of Nigeria’s total expenditure in
2010 stood at US$70,579 millions, the income from petroleum exports of the total export
revenue was US$61,804 millions representing about 87.6 percent. This absolute dependence
of oil export revenue has accentuated to the level of Nigeria economy vulnerability to sudden
oil price movement. The stock market which help in long term financing of government
development projects, serves as a source of fund for private sector long term investment, and
as catalyst for bank consideration particularly during the 2004/2005 Nigerian banking system
consideration. An increase or decrease in the value of stock tends to have a corresponding
effect on the economy, mostly through the money market. An increase in stock prices
stimulates investment and increases the demand for credit which eventually leads to higher
interest rates in the overall economy. Since negative or positive shocks affecting one market
may be transmitted quickly to another through contagious effects, a thorough understanding
of the structures, drivers, linkages and transmission of the inter-relationship among
financial/macroeconomics market, stock market and the price of crude oil arising among
these markets would give invaluable insight to policy makers for the evaluation of policy
proposal aimed at maximizing economic gains and cushioning the adverse shocks to the
Economy.
The rest of the paper is organized as follows; Section 2 presents a review of relevant
empirical papers and Section 3 describes the materials and methods, including the data
preliminary analyses. While Section 4 discusses the model Specification analysis which
includes the VAR model for return and volatility model, Section 5 discusses the application
of data to the model as well as the empirical results of the application. Section 6 however
draws conclusion as well as the policy implication of the results.
2.0 Literature Review
For the past few decades there have been many mixed empirical studies and conclusions
concerning the relationship among financial/macroeconomics variable, the stock index
markets, and oil price which has been given much attention in the academic literatures. These
various studies have attempted to determine how one of these financial markets and the oil
price can predict the other and vice versa. But unfortunately, researches on developing
markets particularly African markets have been very few. In addition to the limited African
researches, most of the previous empirical analysis used the VAR, VECM, and SVAR (Zia
and Rahman, 2011, Ulku and Demirci, 2011, Kenani et al. 2012, Rahman and Uddin, 2009,
Oriakhi and Iyoha 2013), which have invalid statistical inference in the presence of spillover
and asymmetric effects. Therefore, this study adopts the VAR-EGARCH model formulated
by Koutmos, which allows for spillovers as well as asymmetry effects in financial markets.
Other prominent researchers that have added to the study of volatility transmission includes;
Piesse and Hearn (2005) found that the equity markets of South africa and Nigeria transmit
volatility to the equity markets of Botswana, Ghana, Kenya, Malawi, Mauritius, Namibia,
Zambia and Zimbabwe. They applied an EGARCH model on a data set spanning the period
of January 1993 to January 2000. VAR modeling and co-integration approach was also used
by Oraivwote and Eriemo (2012) in investigating the relationship between the real oil price
and the real exchange rate. Their findings suggested that a long run equilibrium relationship
between the real oil prices and the real exchange rate. Their results also suggest that there is
persistence of volatility between real oil prices and the real effective exchange rate.
Salisu and mobolaji (2013) model returns and volatility transmission between oil price and
USD/Naira exchange rate using the VAR GARCH model developed by Ling and McAleer
(2003) to captured the spillover effects in returns and volatility of oil price and exchange rate.
They examined the optimal weights of holding oil and foreign exchange asset, and compute
the hedging ratio in the presence of oil risk. Evidence of bidirectional returns and volatility
transmission between oil price and exchange rate market was also found. Ogundipe and
Ogundipe (2013) examined the relationship between oil price and exchange rate volatility in
Nigeria. They used the GARCH model by Tim Bolerslev (1986) and EGARCH model by
Daniel Nelson (1991). Their evidence revealed that there is a proportionate change in oil
price leads to a more than proportionate change in exchange rate volatility in Nigeria. Mordi
and Adebiyi used a structural VAR model in which the asymmetric impact of oil shocks on
output and price is analyzed in a unifying model. Their empirical results shows that the
impact of oil price shocks on output and price is asymmetric in nature with the impact of oil
price decrease significantly greater than oil price increase.
Boqiang, Wesseh Jr, and Appriah (2014) examined the dynamic volatility and volatility
transmission between oil price on one hand and the Ghanaian stock market plus the Nigerian
stock market on the other hand. Using the multivariate VAR-GARCH model developed by
Ling and McLeer (2003), empirical results shows that there appear to be a significant
interdependence and spillover in the conditional volatility between oil price and both stock
markets (Ghana and Nigeria) returns were detected . While there appears to be stronger
spillover effects for Nigeria, the transmission of volatility is much more apparent from oil
price to stock than from stock to oil price in the case of Ghana. Model results were also used
to compute and analyzed the optimal freight and hedge ratios for oil-stock portfolio holding.
Significance of the multivariate asymmetric effects makes it clear that the assumptions of
symmrtric effects and dynamic conditional correlation were not supported empirically. They
also found that optimal portfolios in Ghana and Nigeria’s stock out weight oil price assets.
Salisu and Fasanya (2012), compares the performance of volatility for oil price using the
daily return of WTI for three sub-periods of before, during and after global financial crisis. A
comparative performance of both the symmetric and asymmetric volatility model for oil price
was analyzed. It was revealed that oil was most volatile during the global financial crisis
compare to other sub samples. It was also discovered that the asymmetric GARCH model
appears superior to the symmetric ones dealing with volatility. This finding indicates
evidence of leverage effects in the oil market. The models compared are; the AR(1)GARCH(1,1), AR(1)-TGARCH(1,1), and AR(1)-EGARCH(1,1).
Wei and Chen (2014), examined whether the volatility of the West Texas intermediate oil
spot returns is affected by the Texas light sweet oil futures returns, USD/Euro exchange rate
returns and the S&P 500 energy returns, and if any of these index have change over time.
Using the multivariate BEKK-GARCH(1,1) model, results indicates that the West Texas
intermediate returns is significantly affected by its own past volatility and the volatility of
other index. Manish Kumar (2014), examined simultaneously, the impact of oil price shock
on indian stock market and foreign exchange market using daily data. The spillover index
methodology of Diebold and Yilmaz (2009) was used to analyzed the variables from these
markets. Results indicate that oil price affects the stock market and exchange rate, and the
stock market affecting the exchange rate. Volatility transmission was captured using the
BEKK-GARCH(1,1) model, and the optimal weight and hedge ratio for portfolio computed.
Kanas (2000) was one of the first studies which analysed volatility spillovers from stock
returns to exchange rate changes in the USA, the UK, Japan, Germany, France and Canada.
He found evidence of spillovers from stock returns to exchange rate changes for all countries
except Germany, suggesting that the asset approach to exchange rate determination is valid
when formulated in terms of the second moments of the exchange rate distribution for the
countries included in his analysis. Volatility spillovers from exchange rate changes to stock
returns were insignificant for all countries.
In terms of volatility transmission, this study will also stand in the gap as a reference paper
that simultaneously modeled the Exchange rate, the stock exchange and the price of crude oil
from Nigeria’s perspective.
3.0 Materials and methods
3.1 Data description and preliminary Analysis
The data sets used in this study contains the daily foreign exchange rates of Naira/USD, the
closing prices of the Nigerian Stock Exchange for thirty leading companies (NSE-30)
represented in this study as (STX) and the daily prices of the West Texas intermediate crude
oil (WTI) in U.S. dollar spot price per barrel also represented as OPR. Data used in this study
were obtained from the Bloomberg information network. The full sample period under study
is from April 2nd, 2012 to September 26th, 2014, yielding a total of 650 observations each.
But an initial observation was lost due to differencing thereby making each data used for
analysis to be 649. This study made use of daily data because, more information will be
captured than using the weekly or monthly data. However, missing data arising from holiday
and special events were filled using Neaime (2012) recommendations. Figure 1, shows the
historical time series for the three variables under investigation. The price of the STX was
low during the first quarter of 2012, but kept rising till May, 2013. While it is evident that
there was a clear upward trend for STX series for the period under investigation, the EXR
and OPR were very unpredictable. This fluctuation is more visible in the EXR during June
2013 - May 2014 where the dollars had a high exchange rate of about =N=166 as against the
calm period of August 2012 - May 2012.
3.2 Statistical Properties of data
The continuously compounded daily returns expressed in logarithmic difference of the series
considered in percentage were used. Where 𝑡 and 𝑡 − 1 represent the current day’s close total
index and the previous day’s close total return index respectively. The daily return series used
in this paper are;
𝑅𝑒𝑡𝑢𝑟𝑛𝐸𝑋𝑅 = 100 𝑋 𝑙𝑜𝑔(𝐸𝑥𝑐ℎ𝑎𝑛𝑔𝑒 𝑅𝑎𝑡𝑒 𝑖𝑛 𝑑𝑎𝑦 𝑡/ 𝐸𝑥𝑐ℎ𝑎𝑛𝑔𝑒 𝑅𝑎𝑡𝑒 𝑖𝑛 𝑑𝑎𝑦 𝑡 − 1)
𝑅𝑒𝑡𝑢𝑟𝑛𝑂𝑃𝑅 = 100 𝑋 𝑙𝑜𝑔( 𝑂𝑖𝑙 𝑝𝑟𝑖𝑐𝑒 𝑖𝑛 𝑑𝑎𝑦 𝑡/ 𝑂𝑖𝑙 𝑝𝑟𝑖𝑐𝑒 𝑖𝑛 𝑑𝑎𝑦 𝑡 − 1)
{
𝑅𝑒𝑡𝑢𝑟𝑛𝑆𝑇𝑋 = 100 𝑋 𝑙𝑜𝑔(𝑆𝑡𝑜𝑐𝑘 𝑀𝑎𝑟𝑘𝑒𝑡 𝑖𝑛 𝑑𝑎𝑦 𝑡/ 𝑆𝑡𝑜𝑐𝑘 𝑀𝑎𝑟𝑘𝑒𝑡 𝑖𝑛 𝑑𝑎𝑦 𝑡 − 1)
(1)
166.0
162.0
EXR
158.0
154.0
A
M
J
J
A
S
O
N
D
J
F
M
A
M
J
2012
J
A
S
O
N
D
J
F
M
A
M
J
2013
J
A
S
J
A
S
J
A
S
2014
115.0
105.0
95.0
OPR
85.0
75.0
A
M
J
J
A
S
O
N
D
J
F
M
A
M
J
2012
J
A
S
O
N
D
J
F
M
A
M
J
2013
2014
2000.0
1750.0
1500.0
STX
1250.0
1000.0
750.0
A
M
J
J
A
S
O
N
D
J
F
M
A
M
J
2012
J
A
S
O
N
D
J
F
M
A
M
J
2013
2014
Figure 1: Daily Index Series (2nd April 2012 - 26th September 2014)
The compounded daily returns appear to fluctuate around a constant level, showing various
quite and volatile periods of the series. Large changes in the returns tend to cluster together,
and small changes tend to cluster together as shown in figure 2 and more evidently in figure
3. This implies volatility clustering and evident of conditional heteroscedasticity. Table 1
contains the summary of statistical properties of return for EXR, OPR and STX. In panel A of
Table 1, the means of all the variables analyzed have zero unconditional means and appears
to be serially uncorrelated as shown in different modified L-B (lines 7, 8 and 9). This is in
conformity with Engle and Bollerslev (1986), Baillie and Bollerslev (1989), Diebold and
Nerlove (1989), West and Cho (1995), Dahiru Bala and Joseph Asemota (2013) etc
submissions. A normal distribution is not skewed and is defined to have a coefficient of
kurtosis of 3. While the coefficient of excess kurtosis, is equal to the coefficient of kurtosis
minus 3. Thus, a normal distribution has a coefficient of excess kurtosis equal to zero (Brook,
2008). The Bera and Jarque (1981), formalizes the ideas of testing normality based on the
skewness and kurtosis combined or jointly zero. Table 1 contains a comprehensive analysis
of returns and squares return of EXR, OPR and STX. And as shown, there is evidence of
skewness (negatively) for the EXRand STX (line 3).
2.0
1.0
0.0
EXR
-1.0
-2.0
A
M
J
J
A
S
O
N
D
J
F
M
A
M
J
2012
J
A
S
O
N
D
J
F
M
A
M
J
2013
J
A
S
J
A
S
J
A
S
2014
10.0
5.0
OPR
0.0
-5.0
A
M
J
J
A
S
O
N
D
J
F
M
A
M
J
2012
J
A
S
O
N
D
J
F
M
A
M
J
2013
2014
3.0
1.0
-1.0
STX
-3.0
-5.0
A
M
J
J
2012
A
S
O
N
D
J
F
M
A
M
J
J
A
S
O
N
D
J
F
M
2013
Figure 2: Daily Return for log Index (2nd April 2012 - 26th September 2014)
A
M
J
2014
Table 1: Summary Statistics
EXR
OPR
Panel A: Returns
1. Mean
0.006
-0.016
[0.010]
[0.049]
2. Standard
0.323
1.268
deviation
[0.019]
[0.078]
3. Skewness
-0.284
0.323
[0.326]
[0.526]
4. Excess
4.108
4.826
kurtosis
[1.037]
[2.705]
5. Jarque
474.215
653.280
Bera
(0.000)
(0.000)
6. Modified
18.175
7.102
L-B(5)
(0.003)
(0.213)
7. Modified
25.526
9.506
L-B(10)
(0.004)
(0.485)
8. Modified
61.566
54.787
L-B(50)
(0.126)
(0.298)
9. Modified 107.036
93.154
L-B(90)
(0.106)
(0.389)
10. Minimum -1.549
-4.761
STX
0.106
[0.036]
0.791
[0.042]
-0.211
[0.307]
2.910
[1.031]
238.807
(0.000)
14.732
(0.012)
22.677
(0.012)
56.185
(0.254)
97.584
(0.274)
-4.197
11. Q1
-0.123
-0.724
-0.305
12. Median
13. Q3
14. Maximum
0.000
0.143
1.811
0.040
0.705
9.001
0.031
0.524
2.915
1.608
[0.198]
4.196
[1.218]
0.637
[0.065]
1.381
[0.228]
Panel B: Returns squared
15. Mean
0.105
[0.013]
16. Standard
0.258
deviation [0.038]
17. L-B(10)
61.696
(0.000)
66.660
(0.000)
92.575
(0.000)
18. L-B(50)
102.243
(0.000)
153.375
(0.000)
158.242
(0.000)
19. L-B(90)
159.236
(0.000)
201.326
(0.000)
184.035
(0.000)
20. ARCH LM
8.262
(0.000)
3.663
(0.000)
8.937
(0.000)
Source: Author’s Calculation. 5% level of significance. Bracket values are level of significance
Author’s computation from www.investing.com & www.quandl.com.
The excess kurtosis exhibits large values and the Jarque-Bera test Statistics that the Null
hypothesis is zero, is highly significant for the three variables Considered (lines 4 and 5).
This indicates that the distributions of all returns is not normal and are in the form of
Leptokurtosis. With the exception of the EXR, the change in standard deviation of other
variables is about 1% per day, but with OPR showing much volatility than the EXR and STX.
The maximum and minimum changes in the entire sample size are generally three or more
standard deviation away from the mean while the interquarter range is much less than two.
Panel B of table 1 contains the summary statistics on squared returns of EXR, OPR and STX.
Since the Ljung-Box Q test for lack of serial correlation has asymptotics based on
assumptions, this become impossible to hold if we have higher order dependence one of the
most general implication of the Brownian motion). Thus, to check for serial dependence of
higher order, the McLeod-Li test given by McLeod and Li (1983) and as used by West and
Cho (1995), which specifically test for null of lack of higher order dependence was applied in
this study.
3.5
90.0
90.0
80.0
80.0
70.0
70.0
60.0
60.0
50.0
50.0
40.0
40.0
30.0
30.0
20.0
20.0
10.0
10.0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
2012
2013
2014
0.0
2012
EXR
2013
2014
2012
OPR
2013
2014
STX
Figure 3: Squared Returns for Naira/USD Rate, Crude Oil price, and NSE-30 respectively
This part of table 1(panel B) particularly lines 17 - 19, suggests in stark contrast to the levels
described in panel A, the squares residual of EXR, OPR and the STX are highly serially
correlated. However, lines 15 and 16 (Mean and standard deviation) are redundant because
the actual point estimates for the statistics can be deduced from panel A. The Arch Lagrange
multiplier test for twelve (12) lags as shown in Table 1, line 20, was conducted to detect if
there are any “ARCH-effects” left in the residual of EXR, OPR and STX. Arch-LM tests
whether the coefficients in the regression;
u    u
2
t
0
1
are zero, where
hypothesis is;
2

      p u t  p   t
2
t 1
t
(2)
is the observed series which we want to test for ARCH effects. So the null
 0   1         p  0 . If hypothesis is accepted then we can say that series
have no ARCH effects. But if it is rejected then one or more coefficients are non zero and we
say that there are ARCH effects. It has the chi-squared variant which uses 𝑇𝑅 2 with degree
of freedom equal to the number of tested lags, and the
F  test variant which test the lagged
square using the standard F  test . As revealed in Table 2, the test statistics for twelve lags,
the chi-squared variant and the F-test variant are statistically not different from zero and thus
resoundingly rejecting the “No ARCH effect” hypothesis for the EXR, OPR and STX.
Table 2: ARCH LM Test
EXR
Lags Statistic Sign. L.
1
85.105 0.00000
2
43.349 0.00000
3
28.811 0.00000
4
24.939 0.00000
5
19.888 0.00000
6
16.614 0.00000
7
14.178 0.00001
8
12.410 0.00001
9
11.132 0.00002
10
9.973 0.00003
11
9.049 0.00007
12
8.262 0.00011
F (12,623)  8.2648
OPR
Statistic Sign. L
15.353 0.00010
11.790 0.00001
8.297 0.00002
6.828 0.00002
6.338 0.00001
6.800 0.00000
5.936 0.00000
5.234 0.00000
4.868 0.00000
4.359 0.00001
3 .990 0.00001
3.663 0.00002
F (12,624)  3.6619
STX
Statistic Sign. L
39.541 0.00000
19.777 0.00000
15.159 0.00000
11.326 0.00000
9.307 0.00000
10.659 0.00000
9.107 0.00000
11.956 0.00000
11.741 0.00000
10.545 0.00000
9.737 0.00000
8.937 0.00000
F (12,624)  8.9374
Signif  0.0000
Signif  0.0000
Signif  0.0000
 2 (12)  87.3215
Signif  0.0000
 2 (12)  41.9076
Signif  0.0000
 2 (12)  93.4263
Signif  0.0000
Note: The autoregressive process of order 2,1,1 were used for the mean models for 12 lags
Author’s computation from www.investing.com & www.quandl.com.
Therefore with this statistical characteristics described in table 1 particularly Panel B, and
table 2, the assumption of constant variance (Homoskedasticity) is inappropriate. This
implies that these results clearly favor models that incorporate ARCH/GARCH features.
3.3 Unit Roots Test
In other not to have a spurious regression and persistence infinite shock, it is necessary as a
pre-requite to examine the stationarity of the original time series of EXR, OPR and STX
variables. This study applied the two basic and commonly used test of stationarity–the
Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) test of unit root. The ADF test
equation is given as;
 yt    t   y t 1   d  y t 1  
k
i 1
i
(3)
t
Where 𝜀𝑡 is a white noise error term and ∆𝑦𝑡−1 = 𝑦𝑡−1 − 𝑦𝑡−2 , ∆𝑦𝑡−2 = 𝑦𝑡−2 − 𝑦𝑡−3 , etc.
equation (3) tests the null hypothesis of a unit root against a trend stationary alternative. The
Philips-Perron (PP) test is similar to the ADF test, but incorporate an automatic correction to
the ADF test to allow for autocorrelated residuals. The test often gives the same conclusion as
the ADF test. The test regression for the PP test is,
y     yt 1    yt 1          yt  p  
t
t
Where

1
t
may be 0,
φ or φ + βt.
p
t
(4)
Table 3 Panel A and B, examines the stationarity or non-stationarity of the daily index and
returns of EXR, OPR and STX using the critical values and T-statistics. In panel A, the null
hypothesis of presence of a unit root in the original data is tested against the alternative
hypothesis of stationarity (no unit root). And as shown in Table 3 panel A, the non rejection
of the null hypothesis led to using equation (1) and subsequent rejection of the null
hypothesis at 1%, 5% and 10% in panel B, confirmed that the three return series of EXR,
OPR and STX has no unit root.
Table 3: Unit Root Test
ADF Test
Variables
T-Stat Crit Value
Panel A: Series
-3.44280
EXR
-2.68811 -2.86633
-2.56932*
OPR
-3.44280
-2.46408 -2.86633
-2.56932
PP Test
T-Stat Crit Value
-3.44280
-2.17799 -2.86633
-2.56932
-3.44280
-2.48657 -2.86633
-2.56932
-3.44280
STX
-2.01054 -2.86633
-2.56932
Panel B: Return
-3.4428***
EXR
-30.2466 -2.8663**
-2.5693*
-3.44280
-1.90527 -2.86633
-2.56932
-3.4428***
-30.9619 -2.8663**
-2.5693*
OPR
-3.4428***
-27.0341 -2.8663**
-2.5693*
-3.4428***
-27.0371 -2.8663**
-2.5693*
STX
-3.4428***
-22.2426 -2.8663**
-2.5693*
-3.4428***
-22.4140 -2.8663**
-2.5693*
Source: Author’s calculation, * Denote statistical significance at the 1% level,
** Denote statistical significance at the 5% level, *** Denote statistical significance at the 10% level
Author’s computation from www.investing.com & www.quandl.com.
4.0 Model Specification
4.1 Lags Selection
The Bayesian/Schwarz information criterion (SBC/SIC) and the Hannan-Quinn information
criterion (HQ) performed well with high probability of selecting the true DGP in the presence
of GARCH effect for lower dimension. While AIC specification is found consistent in the
selection of higher order GARCH process, Farrukh and Panagiotis (2013). These two
information criteria were used as a guide to choose the most appropriate lag order for the
VAR model. In Table 4 and as indicated, HQ criterion of lag 1 being the smallest values of
the two criteria considered was selected as the most appropriate lag order for the model.
Table 4: VAR Lag Selections
Lags
SBC/BIC
HQ
0
4032.68606* 4024.50004
1
4051.29798
4018.55387*
2
4099.59446
4042.29227
3
4123.50232
4041.64206
4
4164.83026
4058.41192
5
4215.05047
4084.07406
6
4267.02818
4111.49368
7
4308.91423
4128.82166
8
4356.01822
4151.36756
9
4395.04687
4165.83814
10
4439.94355
4186.17674
Source: Author’s Calculation
4.2 Test of Asymmetry
In order to investigate the existence of leverage effect, the sign and size bias tests for
asymmetry in volatility proposed by Engle and Ng (1993) were conducted in this study. This
test is used to determine whether an asymmetric model is required for a given series, or
whether the symmetric GARCH model can deemed adequate. The Engle-Ng test is usually
applied to the residuals of a Garch fit to return data. The test for sign bias is based on the
significance of
𝑢̂𝑡2 =
Where

t

  S
0
S
1

1
in;

t 1
t
(5)
is an indicator dummy that takes the value one if
t 1
is an IID term error. If positive and negative shocks to
conditional variance, then
u
u
t 1
t 1
 0 and zero otherwise.
impact different on the
 as shown in equation (5) will be statistically significant. To
1
determine whether the magnitude or size of the shock affect response of volatility to shock is
symmetric or not, the negative size bias test is conducted. The negative size bias is said to be
present if
𝑢̂𝑡2 =
 in equation (6) is statistically significant in the regression,
1
  S u

0
1

t 1
t 1
t
(6)

Where the S t 1  1  S t 1 is the observation with positive innovations. And the joint test for
sign and size based regression is given as;
+
−
−
𝑢̂𝑡2 = ∅0 + ∅1 𝑆𝑡−1
+ ∅2 𝑆𝑡−1
𝑢𝑡−1 + ∅3 𝑆𝑡−1
𝑢𝑡−1 + 𝜀𝑡
In equation (7) the significant of

1
(7)
indicates the presence of sign bias, where positive and
negative shocks have different impacts on future volatility. And the significant of

2
and

3
suggests the presence of size bias, where not only the sign but magnitude of the shock is
important. A joint test statistics is the calculating of 𝑇𝑅 2 from equation (7) which follows a
𝜒 2 distribution with 3 degree of freedom under the null hypothesis of no asymmetric effects.
As revealed in Table 5, the conditional volatility of returns are sensitive to both the sign and
size of shock to volatility in EXR and OPR. Though the results of this test are not
resoundingly, but are enough motivation that suggests the use of a model that can capture the
asymmetric effects. Hence, the EGARCH (1,1) model.
Table 5: Test for sign and size Bias
Statistics
Size bias
Positive size Negative size Joint Test
Bias
Bias
F(3,644)
EXR
0.1380
(0.4025)
0.0847
(0.5443)
-0.0610
(0.6361)
2.3860
(0.4962)
OPR
0.2254
(0.1103)
-0.2230
(0.0744)**
-0.1953
(0.0803)**
4.4652
(0.2154)
STX
0.1189
(0.4248)
-0.0090
(0.9384)
-0.1399
(0.2819)
1.5283
(0.6757)
Source: Author’s Calculation, *Denote Significance at 5%, **Denote Significance at 10%
Author’s computation from www.investing.com & www.quandl.com.
4.3 VAR Returns and Volatility Models
The Autoregressive Conditional Heteroskedasticity model of Engle (1982) and the
Generalized ARCH (GARCH) model of Bollerslev (1986) are popular in modeling second
moment dynamics in returns because they are parsimonious, and they capture stylized facts
such as thick tail, volatility clustering and persistence in returns. They are also very flexible
in terms of allowing different parameterization, but they cannot detect sign/size-bias
asymmetry and the non-negativity constraints could limit the application of the model.
Nelson (1991), introduced the Exponential GARCH (EGARCH) model which imposes no
parameter and sign restriction, allows for the oscillatory behavior of the variance and captures
the asymmetry behavior generated by the innovations within and across markets. In this
study, the vector autoregressive (VAR) model was combined with the exponential
generalized autoregressive conditional heteroskedasticity (EGARCH) models proposed and
used by Koutmos (1996). This models are specified by the following equation;
r i ,t  
 
3
i ,0
j 1
i, j
r
i , t 1
  i ,t for i  1,2,3
 i,t  exp  i,0   i, j f j ( z j ,t 1)   ln( i,t 1)
3
2
2
i
j 1
f j ( z j ,t 1)  ( z j ,t 1  E z j ,t 1  

i , j ,t

i, j
 
i ,t
j ,t
j
z
(8)
for i  1,2,3
), for j  1,2,3
j , t 1
(9)
(10)
(11)
From the mean equation shown in equation (8), the dynamic relationship in returns are
captured by using a vector Autoregressive (VAR) model, where r i,t is the own past return
for market i ,  captures the lead-lag relationship between returns in different markets, the
ij
information set Ω𝑡−1 contains all the information up to
time
t
. The

i ,0
t 1 , and  t
is the innovation at
represents the long-term drift coefficient. The first-order VAR is adopted
to show how the different variables (EXR, OPR and STX) respond to one another in the
market. Thus, to model the return of EXR, equation (8) can be written as;
r

OPR,t
OPR, 0

t ,OPR
r
OPR,t 1

i , EXR
r
EXR,t 1

i , STX
r
STX ,t 1
  OPR,t
(12)
The conditional variance equation in (9) expresses the conditional variances in each market
as an exponential function of the past standardized innovation, ( Z j ,t 1 
its own market and other markets. The estimated value of



j ,t 1
) that is from
j ,t 1
measures the persistence of
i
volatility which has to be less than one for the unconditional variance to be finite. If

i
 1,
then the unconditional variance does not exist and the conditional variance follows an
integrated process of order one. The spillovers are captured by the coefficient of  ij while
asymmetric implies negative  j . The asymmetric influence of innovation on the conditional
n
 f (Z
variance is captured by the (
j 1
In

2
OPR,t
i , EXR
ij
i
)) .We modeled the conditional variance of OPR;
j ,t 1
  OPR,t   i ,OPR Z OPR,t 1   OPR Z OPR,t 1  
In
i
In(
2
OPR,t 1
) 
2
i , EXR
In
2
EXR

(13)
EXR
Equation (13) indicate EGARCH (1,1) model for price of crude oil, represented by OPR. In
similar fashion, using equation (12) and (13), other variables (EXT and STX) can be
constructed. With the assumption of normality, the loglikelihood function for the multivariate
EGARCH model in equation (14) can be expressed as;
T
L( )  (0.5)( NT ) In(2 ) (0.5)  ( In S t
t 1

S )
'
1
t
t
(14)
t
Where N is the number of equation, T is the number of observation,  is the 33 x 1
parameter vector to be estimated, S t is the 3 x 3 time varying conditional variancecovariance matrix, and 𝜀𝑡′ = [𝜀𝑡𝐸𝑋𝑅 𝜀𝑡𝑂𝑃𝑅 𝜀𝑡𝑆𝑇𝑋 ] is the vector of innovations from the
three variables market at time t
5.0 Empirical Results and discussion
To assess the impact of returns and volatility spillover effects among the financial market, the
stock index markets, and oil price, the simultaneous parameterization of equation (8) - (10)
was done. The Broyden, Fletcher, Goldfarb and Shanno (BFGS) algorithm was used to
maximize the L( ) via Quasi-Maximum Likelihood Estimation (QMLE) which is robust to
the distribution of the disturbance term. The results of the estimation converged in one
iteration with a functional value (log-likelihood) as -1831.9628. Own and cross, first and
second moment dynamic among the three markets were examined. Table 6, contains the
return, the variance models with asymmetry volatility spillover parameter and the diagnostics
analysis, shows the estimation results of VAR (1)-MEGARCH model.
Regarding the interdependence of return in equation (8), shown in Table 6 panel A, there is a
unidirectional negative return spillover effect from the price of crude oil (OPR) to the foreign
exchange market (EXR). And about 29 percent of unidirectional past return innovations
spillover from STX is transmitted to EXR. However, we find no significance effects of
returns spillover between OPR and STX. The diagonal parameters of return particularly the
EXR and STX are statistically different from zero indicating that their return does not depend
on their first lags.
Furthermore, the impact of own innovation in current volatility in EXR, OPR and STX, are
all positive and significantly not different from zero. This indicates the absence of own
volatility spillover in these markets. Interestingly, the returns and volatility transmission
results do not support linkages between OPR and STX. As shown in Table 6, the diagonal
elements of

i
and

i
are statistically highly significant, indicating a strong GARCH
effects. In other words, own past shock and volatility affects the conditional variance of all
the variables considered in this paper.
Moreover, Table 6 panel B, shows the volatility transmission mechanism to be asymmetry in
OPR. The implication of this is that bad news increases volatility more than good news does.
However, asymmetric coefficient is not significant in EXR and STX, indicating that there is
no difference between positive and negative innovations for both markets. If the relative
importance of the asymmetry or, leverage effect, is measured by the ratio  1   j (1   j )
(Koutmos 1996), therefore, the degree of asymmetry impact for OPR, on the basis of the
estimated  coefficients, is 1.860. This means that negative innovations increases volatility
approximately 1.860 times more than positive innovations. Analysis also reveals that OPR
has weak volatility persistence as shown in Table 6, panel B, when compare to that of EXR
and STX which is close to unity. But results show that the degrees of persistence for the three
markets are statistically significant. The residual based diagnostic test shown in Table 6 panel
C is completely robust. This shows that the VAR-MEGARCH model satisfactorily explains
the interactions among the variables used in this study.
Table 6: Estimation of the VAR-EGARCH model
EXR
Panel A: VAR(1) for Returns
 0.0090
1, 0
OPR

(0.3569)

1,1
0.0036
1, 2
-0.1829

1, 3
0.0082
-0.0212
2,1

2, 2
0.2670

0.0316
2,3
(0.0000)*
1,1
0.5690
1, 2
0.0527

1, 3
0.2859

2, 2
1
0.8847
(0.0000)*
0.4882
1

3, 2

2,3
-0.0975

3, 3
2
0.4224
0.0243
(0.2419)

-0.0189
3, 0
(0.0000)*

0.0290
3,1
(0.2306)

-0.0146
3, 2
(0.3672)

3, 3
(0.1499)

-0.0572
(0.5233)
(0.0000)*
0.1793
(0.0000)*

3
(0.0000)*
-0.0010
 2 -0.3007
(0.9803)
(0.0000)*
Panel C: Diagnostics test checking
EXR
OPR
-0.0031
0.0071
Mean
0.9941
1.0028
Variance
L  BQ (12)
19.0837
14.4132
2
13.4735
30.4739ª
L  BQ (12)
42.8674*
ARCH  LM (12) 10.5864

0.1096
(0.0000)*
(0.0332)*
(0.0000)*

-0.1422
2 ,1
(0.2723)

3,1
(0.0000)*
(0.0000)*


(0.4659)
Panel B: EGARCH(1,1) for Volatility
 1,0 -0.2391
 2,0 0.1965
0.1664
(0.000)
(0.0124)*
(0.1516)

3, 0
(0.6060)
(0.0000)*


(0.7914)
(0.5985)

-0.0120
2,0
STX
0.9466
(0.0000)*

3
-0.1185
(0.1383)
STX
-0.0262
0.9967
40.0873
8.5825
1.2490
Note: EXR, OPR and STX are the returns for Naira/Dollar exchange rate, crude oil price and NSE-30 respectively.
* Indicate significance at 5%, **Denote Significance at 10%, ª Indicate serial dependence
Author’s computation from www.investing.com & www.quandl.com.
The estimated mean and the variance of the cross product of the standardized residual are
approximately equal to zero and one respectively. The Ljung-Box statistics up to 12 lags
show no evidence of serial correlation or autocorrelation in the standardized and squared
standardized residuals for EXR at 5% level of significance. But some dependence still
persists in the standardized residuals of OPR. This may be due to the restriction imposed i.e.,
zero mean and variance interactions. However, this shows no serious evidence against the
benchmark specification. In addition to the proof of adequacy of the model, the ARCH-LM
test also shows no ARCH effects exist.
6.0 Conclusions and Policy Implication
This paper examined the return dynamics and volatility transmission among financial
variable, the stock market and the price of crude oil, by using the multivariate VARMEGARCH model. The empirical analysis and results in section 5 of this paper provides
several interesting conclusions;
The price of crude oil (OPR) exhibits asymmetry volatility transmission, implying that
negative shocks from the price of crude oil has a greater impact on volatility than positive
shocks of equal magnitude. This is consistent with the notion of leverage effects for growing
list of researchers with empirical evidence of asymmetry effects in the price of crude oil,
particularly the West Texas Intermediate crude oil (WTI). Thus negative innovations of crude
oil price have impact on its conditional volatility about 2 times larger than positive
innovations. Our finding also show that price of crude oil (OPR) has no influence on the
Nigeria stock market (STX) and vice versa. This findings corroborates Maghyereh (2004),
Hammoudeh,Choi (2006) etc assertion on the relationship between price of crude oil(OPR)
and the Nigerian stock market (STX). Since oil price change is usually very sensitive to
events around the world, changes caused by oil prices have implications on the values of
USD/Naira exchange rate. This can cause a misalignment of the Naira. Empirical evidence
shows that there is a negative return of oil price (OPR) to USD/Naira exchange rate (EXR).
This means a fall in oil price will affect negatively the USD/naira exchange rate. This may
drives down the price on non-traded goods in the economy and also leads to large shifts in
current account balance and portfolio reallocation, Killian et al (2007). The volatility
spillover from the stock market to the foreign market is statistically significant and
unidirectional. This result is also consistent to the finding of Kanas (2000), that volatility
changes in the stock returns have impacts on the movement of foreign currency. This
provides supportive evidence for the “asset approach” to exchange rate determination
(Branson 1983, and Frankel 1983). Again, apart from the price of crude oil which has poor
volatility persistence of about 40% when compare to EXR and STX which is close to unity,
results shows that the degree of persistence for the three markets are statistically significant.
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