Journal of Finance and Accounting Research (JFAR)
Volume 3 Issue 1, Spring 2021
ISSN:(P) 2617-2232 ISSN:(E) 2663-838X
Journal DOI: https://doi.org/10.32350/jfar
Issue DOI: https://doi.org/10.32350/jfar/0301
Homepage: https://ojs.umt.edu.pk/index.php/jfar
Comparative Analysis of Naira/US Dollar Exchange Rate
Volatility using GARCH Variant Modeling
Article:
Journal QR
Atabani Adi Agya, Amadi W. Kingsley, David Vincent Hassan
Author(s):
Article QR
Affiliation:
Department of Economics, Faculty of Humanities, Management and Social
Sciences, Federal University, Wukari, Nigeria
Article DOI:
https://doi.org/10.32350/jfar.0301.02
Article History:
Received: January 7, 2021
Revised: February 1, 2021
Accepted: July 16, 2021
Citation:
Copyright
Information
Agya, A. A., Kingsley, A. W., & Hassan, D. V. (2021).
Comparative analysis of Naira/US dollar exchange rate
volatility using GARCH variant modeling. Journal of Finance
and Accounting Research, 3(1), 18–40. Crossref
This article is open access and is distributed under the terms of
Creative Commons Attribution 4.0 International License
A publication of the
Department of Finance, School of Business and Economics
University of Management and Technology, Lahore, Pakistan
Indexing
Comparative Analysis of Naira/US Dollar Exchange Rate Volatility Using
GARCH Variant Modeling
Atabani Adi Agya*, Amadi W. Kingsley and David Vincent Hassan
Department of Economics, Faculty of Humanities,
Management and Social Sciences, Federal University, Wukari, Nigeria
Abstract
This paper employed the GARCH variance models to examine the return
volatilities of official bank, interbank and Bureau de change. Using the monthly
exchange rate of Naira/USD from January 2004 to September 2020 (2004:12020:9), it was observed that the returns were not normally distributed and were
stationary at level. The power statistics of Ljung-Box Q and Ljung-Box Q2
transformed, using the powers 0.25, 0.5 and 0.75 for conditional
heteroscedasticity and
lags of 6, 12 and 20 to indicate conditional
heteroscedascity in all returns. The study also found exchange rate volatility in
official, interbank and Bureau de change, observing that exchange rate returns
were persistent. However, Bureau de change return was relatively more persistent
while official exchange rate return was the least persistent. Also, it can be said
that leverage effect exists in all the three exchange rate returns; while asymmetric
model was the best model to estimate the exchange rate, IGARCH was not a
suitable model to estimate the exchange rate return in Nigeria. There is also a
need to incorporate the impact of news when developing an exchange rate policy
by the monetary authority in Nigeria.
Keywords: exchange rate volatility, GARCH variant, leverage effects,
Naira/USD, persistency
JEL Code: G, G1, G12
Introduction
Nigeria is an open economy with trading partners worldwide. The stability of its
exchange rate or otherwise has far reaching implications on Nigeria’s current and
capital account, foreign direct investment and polio investment. Also, the stability
of the country’s currency plays an important role on the cross border currency
transactions especially when the investor usually weighs the risk associated with
the exchange rate when making international investments while also assessing the
political risks involved. While a country may think that depreciation of its
*
Corresponding Author: atabaniadi@yahoo.com
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Comparative Analysis of Naira/US Dollar Exchange…
currency is an opportunity to increase exports, it could adversely affect the
domestic output, especially when intermediate inputs are imported into the
country for production activities. Foreign investor weighs the exchange rate
volatility against the anticipated profit before investing in a particular economy.
Furthermore, export-import activities are significantly affected by the volatility of
the exchange rate, because following the depreciation/appreciation of exchange
rates the invoicing currency has an important implication on the importer’s cost
especially in terms of credit trade.
Foreign exchange market in Nigeria is divided into three markets with distinct
rates which are operational side by side. For instance, the official foreign
exchange market is operated by the Central bank of Nigeria (CBN) as the buyer
and seller of foreign exchange to banks through the weekly Wholesale Dutch
Auction system and Bureau de change operators. It’s also serving as the regulator
in the foreign exchange market. The interbank market is the market where foreign
exchange is bought and sold between banks in Nigeria, multinational oil company
(IOC), Nigeria national petroleum company (NNPC) and other companies dealing
in foreign trade. The last segment of foreign market is the Bureau de change;
which was established in 1989 to cater to the end user of foreign exchange in
Nigeria. It provides services such as personal travel allowance, school fee to
students studying abroad, medical bills and credit card payment among others.
The above arrangement was meant to ensure stability in foreign exchange
market in Nigeria by providing foreign exchange to those who need foreign
currencies. However, foreign exchange rate continues to be volatile with
unprecedented rates different from those in the markets. For instance, Emenike
(2016) compared volatility persistent in official, interbank and bureau de change
and found bureau de change market volatility was explosive while Oyinlola
(2018) examined the impact of past volatility on current volatility in interbank and
bureau de change and found past volatility played a significant role in the current
volatility in interbank and Bureau de exchange. The study examines three foreign
exchange markets in this present study. However, there is a need to account for
recent developments in the foreign exchange rate market, thus the impact of
structural breaks in these rates cannot be overemphasized. This is the gap, this
current study has identified to address in this research.
Following the introduction is the literature review, the next section deals with
the methodology, followed by analysis and discussion of the result while the last
section provides the concluding remarks and recommendations.
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Agya, Kingsley and Hassan
Literature review
Economic literature is replete with studies that examine the effect of exchange
rate volatility on economic growth and determine the exchange rate volatility
persistency between fixed and floating exchange rate system in Nigeria. For
instance, Ehikioya (2019) examined exchange rate volatility in Nigeria, using
monthly data for the period of January 1980 to December 2019. The study found
that the exchange rate volatility of Naira against US Dollar is persistent during the
period of analysis and has a negative impact on the economy of Nigeria. In the
same vein, Musyoki et al. (2012) used monthly data and employed GARCH and
generalized moment method to study the volatility of Kenya’s exchange rate, for
the period of January 1993 to December 2009. They found Kenya’s exchange rate
volatility was persistent throughout the period and thus had a negative impact on
its economic growth.
Kuhe et al. (2018) examined exchange rates returns of Naira vis-à-vis Euro,
UK Pound Sterling, CFA, US Dollar and West African Unit of Account (WAUA)
as well as Japanese Yen, using daily data for the period of 11th December 2001 to
13th April 2018. They employed symmetric and asymmetric GARCH methods
with non-Gaussian errors. The result from EGARCH (1.1) found CFA and US
Dollar has the highest and least volatility among the exchange rate returns
respectively. They also found the presence of volatility clustering and shocks
were persistent in all the six exchange rate returns. They also found evidence of
leverage effects in all return series. In a single country study, Oyinlola (2018)
examined exchange rate return volatility persistent and asymmetric of Naira
against US dollar exchange rate for interbank and Bureaux de exchange (BDC)
using monthly data from January 2004 to November 2017. The study employed
Threshold GARCH [T-GARCH (1.1)] and Exponential GARCH [E-GARCH
(1,1)] as well as Bai-Parron (2003) unit root with break to capture the impact of
structural break on the returns volatility. The study found two break dates in 2014
and 2015 and explosive volatility in BDC while the interbank was high but not
explosive. Also, it was found that symmetric model is best for interbank return
while asymmetric appears the best in BDC market respectively.
Emenike (2016) carried out a comparative analysis of the exchange rate
volatility in official and interbank markets as well as the Bureau de exchange
rate markets. The study employed GARCH (1, 1) and GJR-GARCH (1,1) for the
period of January 1995 to December 2014. The study found past volatility in
interbank and Bureaux de change rates to significantly influence their parent
volatility and also observed that volatility clustering was present in both markets.
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Comparative Analysis of Naira/US Dollar Exchange…
The study also, found volatility persistent and clustering was more common in the
Bureaux de change market than others markets. It also deduced that depreciation
of exchange rate aggravates volatility in immediate future in both interbank and
Bureau de change markets.
Ajayi et al. (2019) examined daily exchange rate returns of Naira against six
currencies, such as Chinese Yuan, Indian Rupees, Spain Euro, UK Pound and US
Dollar for the period of January 2012 to August 2019. The study employed
GARCH (1,1), EGARCH (1,1), TGARCH (1,1) and GJR-GARCH(1,1) models.
The study found high volatility and no leverage effect in all estimates without
break and GJR-GARCH was the best model for all the exchange rate returns.
Bala and Asemota (2013) examined exchange rate volatilities of Naira against
US Dollar and UK Pound for the period of January 1985 to July 2011 for
Naira/US Dollar, January 2004 to January 2011 for Naira/British Pounds and
Naira/Euro returns. The employed variant of GARCH models was examined with
and without break. They used exogenous to determine break for US Dollar. The
study found that volatility is persistent in all the three exchange rates and that all
asymmetry models without break reject leverage effect; while models with break
showed the presence of leverage effect in all the three currencies. The study
further advocates the inclusion of break on the estimate of volatility in exchange
rate returns as does the improved or reduced rate of volatility persistent. In a
related analysis, Musa et al. (2014) examined daily exchange rate of Naira against
US Dollar for the period of June 2000 to July 2011. They employed symmetry
and asymmetry GARCH models. The study found significant asymmetry effects
of exchange return and the loose function such as MAPE, MAE, RMAE while
Theil inequality coefficient found T-GARCH model is the best model for forecast
purposes. Also, Abdullah et al. (2017) examined daily exchange rate volatility for
Naira against US Dollar for the period of 1st January 2008 to 30th April 2015. The
study employed symmetry and asymmetry models. The study found in contrast to
normal distribution, student t-distribution improved the model forecast
performance and satisfied the diagnostics statistics. Afees (2011) examined the
extend of Naira exchange rate volatility against US Dollar, using daily return
series for the period of sustainable democracy based on sub-period of democratic
transition of 05/29/1999-05/28/2003; 05/29/2003-05/28/2007; and 05/29/200705/28/2011 and employed variant of GARCH models. The study found exchange
rate behavior change in short time, and that leverage and persistence vary over
time.
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Agya, Kingsley and Hassan
Methodology
The paper employed GARCH, EGARCH, APARCH, IGARCH, TARCH and
GARCH with structural break in volatility modeling; this is done to see whether
structural break will improve our result.
The GARCH model is an extension of the ARCH, thus the GARCH model
incorporates past conditional variances into current conditional variance equation.
The GARCH model is formulated as follows:
p
q
i 1
j 1
t2 i t2i j t2 j ...................................................................................1
Where p≥0, q˃0, >0, αi≥0, βj≥0, i=1,2…,p, j=1,2…,q.
Equation (1) is the GARCH (p,q) model where p and q denote the lags terms
of the squared error term and conditional variance respectively. This implies, the
current conditional variance is the function of past shocks (ARCH term) and past
variances (GARCH term). From equation (1) the trader predicts its current
volatility by taking the weighted average of the long term mean (the constant),
thus the information observed from previous period volatility (the ARCH term)
and forecasted variance from the previous period (The GARCH). Where is the
p
q
i 1
j 1
constant, i t2i is the ARCH term and GARCH effect j t2 j is the GARCH
term.
Equation (1) will be stationary if the sum of the ARCH and GARCH (
q
i j ) is less than 1.
p
i 1
j 1
Equation (1) can be extended by adding an exogenous variable or dummy
variable to account for structural break in the variance equation.
p
q
i 1
j 1
t2 i t2i j t2 j .....................................................................................2
Where dum1t, dumnt are dummy variables representing periods of key policy
changes in the foreign exchange market and exogenous shocks (0 for normal
periods and 1 for periods of high currency movements). We determined periods of
high currency movements by detecting sudden jumps or outliers resulting from
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Comparative Analysis of Naira/US Dollar Exchange…
exchange rate policy changes and other exogenous shocks. Consequently, a higher
order GARCH model, expressed as GARCH (p,q) is given by:
p
q
i 1
j 1
t2 i t2i j t2 j ....................................................................................3
Where p and q are lags order of ARCH term and GARCH term respectively
and k lag order of dummy variables.
In addition, the integrated GARCH (p,q) or IGARCH(p,q) model is expressed
as follows: Engle and Bollerslev (1986) extend a standard GARCH(1,1) model to
an IGARCH(1,1 ) model by imposing the restriction that α1 +β1 =1. An
IGARCH(p,q) is expressed thus;
p
q
i 1
j 1
t2 i t2i j t2 j ............................................................................................4
Such that
p
i 1
2
i t i
q
j t2 j 1
j 1
This model imposing restriction that α1 +β1 =1 and assuming the constant term
is equal to zero, for detailed exposition see (Nelson, 1990) when α1 +β1>1 and
constant is greater than zero ( > 0). Furthermore, Nelson's (1991) proposed an
EGARCH model to allow for asymmetric effects between positive and negative
shock to asset return. An EGARCH (p,q) model is expressed as;
p
log t2 i
i 1
q
r
t i
2
j log( t2 j ) k t k .......................................5
t i
t k
j 1
k 1
Where ω, αi, βj and γk are constant parameters. The EGARCH (p,q) model,
unlike the GARCH (p, q) model, indicates that the conditional variance is an
exponential function. The asymmetric effect of past shocks is captured by the γ
coefficient, which is usually negative, that is, positive shocks generate less
volatility than negative shocks (Longmore & Robinson, 2004). The leverage
effect can be tested if γ < 0. If γ ≠ 0 the news impact is asymmetric. Similarly,
TGARCH Model also known as GJR-GARCH is employed related to the
transformation to estimate the leverage effects on the conditional standard
deviation. This model takes the following form;
p
q
i 1
j 1
t2 ( i i Nt 1 ) t2i j t2 j .....................................................................6
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Agya, Kingsley and Hassan
Where N t 1 is an indicator of negative ɛt-i, that is;
1 for t i 0
N t i
0 for t i 0
Or
p
q
r
j 1
k 1
t i t2i j t2 j k t2k Itk ..............................................................7
i 1
t
Where I is a dummy variable, 1 if εt< 0 and 0 otherwise. In the GJRGARCH model, good news εt-i >0 and bad news, εt-i < 0, have differential effects
on the conditional variance; good news has an impact of αi while bad news has an
impact of αi + γ. If γi> 0, bad news increases volatility, and there is a leverage
effect for i-th order. If γ ≠0, the news impact is asymmetric. Also, TS-GARCH
model usually used to capture the information contained in the fat tails and is
characterized to return distribution of speculative prices. The model is thus
expressed as;
p
q
i 1
j 1
t i t i j t2 j ................................................................................8
The asymmetry power ARCH (APARCH) model developed by Ding et al.
(1993) also, allows for asymmetric effects of shocks on conditional volatility. The
APARCH (p, q) model is hereby expressed as follows:
p
q
i 1
j 1
t i ( t i i t i ) j t2 j ..................................................................9
Where δ>0, i 0 for i=,…,r, i >0 for all I>r, and r≤p if 0 shock impact
is not asymmetrical. The power parameter of the standard deviation can be
estimated rather than imposed, and γ parameters are added to capture asymmetry
of up to order r. The assumption of normality in modeling financial data, which
restricts d to either 1 or 2, is often a denial of reality due to significant skewness
and kurtosis (Longmore & Robinson, 2004).
Data Description and Source
The data for the study consists of monthly exchange rate of Naira/USD from
January 2004 to September 2020 (2004:1-2020:9) for official rate, interbank and
Bureau de change exchange rates observations. The exchange rates were obtained
from Central bank of Nigeria statistical bulletin. Here we employed continuously
compounding returns due to its advantages over the simple net returns as well as
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Comparative Analysis of Naira/US Dollar Exchange…
its attractive statistical properties. The returns are defined as rt=log(et/ et-1)
=log(et) -log(et-1 ), where rt is the exchange rate return, et is the spot rate of
Naira/USD at time t and et-1 is the spot rate of Naira/USD exchange rate at time t1.
Data Analysis and Result Discussion
Table 1
Descriptive Statistics and Autocorrelation of Naira exchange rate (Raw)
Statistics
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Official Rate
1.0003
1.0000
1.0293
0.9936
0.0033
6.1456
51.1674
13694.42
(0.000)
Observations 133
Ljung Box Q Statistics
Q(1)
0.399**
(0.000)
Q(5)
0.009**
1.0005
0.9999
1.0274
0.9929
0.0038
3.9111
25.2299
3077.602
(0.000)
133
Bureau de
change (BDC)
1.0007
1.0000
1.0303
0.9839
0.0051
1.9787
13.8430
738.3367
(0.000)
133
0.488**
(0.000)
0.371**
(0.000)
Interbank rate
-0.026**
0.024**
(0.001)
(0.000)
(0.000)
Q(10)
-0.019**
-0.063**
-0.061**
(0.001)
(0.000)
(0.000)
Note. figure in parentheses are p-value ** indicates significant at 5 percent level
Source: Authors’ computation
Table 1, shows the descriptive statistics of Naira/USD exchange rate, Bureau
de change has the highest mean while official rate has less mean value, the
official rate and Bureau de change has the highest median value of 1.000 while
interbank rate has the least median value (0.999). The maximum or the highest
value was for Bureau de change 1.03 while interbank rate has the least maximum
value (1.02). Also, Official rate has the highest minimum rate (0.993) while
interbank has the least minimum rate (0.992). Standard deviation which measures
the volatility of the rate showed that Bureau de change was the most volatile
26
Journal of Finance and Accounting Research
Volume 3 Issue 1, Spring 2021
Agya, Kingsley and Hassan
while official rate was the least volatile of the rates observed. The skewness of the
rates further showed that all the rates were positively skewed as against the
normal distribution (0 indicates skewness for the normal distribution of rates), an
indication of asymmetry distribution and Kurtosis were far greater than 3 for a
normal distribution of all the rates. Skewness also indicates a non-normal
distribution and the large kurtosis series are leptokurtic, providing evidence of fat
tails. The JB test further confirms the non-normality of distribution with a
probability of (0.000) for all rates. The Ljung Box Q statistics for lags of 1, 5 and
10 considered were significant at 5 percent, indicating autocorrelation (serial
correlation) in the rates for all exchange rate return. The Q-Q plot for official rate,
interbank rate and Bureau de change exchange rate returns and diagrams clearly
show a marked departure from the normality graphs.
Having found that our series are non-normal, the usual method of testing
conditional homoscedasticity by using autocorrelation in squared return series is
inappropriate. As opined by Mckenzie (1997) volatility clustering is not unique to
squared returns of assets price. Absolute changes in an assets price usually exhibit
volatility clustering, hence, inclusion of power term amplified relative period of
tranquility and volatility by identifying outliers in the returns.
Again, we perform conditional homoscedasticity by testing for autocorrelation
of power transformed for the exchange rate returns of the following: official,
interbank and Bureau de change using powers 0.25, 0.5 and 0.75. The Ljung-Box
Q20.25, Q20.5 and Q20.75 statistics for the three exchange rate returns at 5 percent
critical value are significant for all the lags and powers implying the presence of
conditional heteroscedasticity.
Figure 1
Volatility Clustering of Official Exchange Rate Return
DAS Q-Q Normality Plot
Quantiles of Normal
1.0050
1.0025
1.0000
0.9975
1.016
1.020
1.012
1.015
1.008
1.010
Quantiles of Normal
1.0075
Quantiles of Normal
BDC Q-Q Normality Plot
IBR Q-Q Normality Plot
1.0100
1.004
1.000
0.996
1.005
1.000
0.995
0.9950
0.992
0.990
0.9925
0.988
0.985
0.9900
0.99
0.984
0.99
1.00
1.01
1.02
Quantiles of DAS
Department of Finance
Volume 3 Issue 1, Spring 2021
1.03
1.00
1.01
1.02
1.03
Quantiles of IBR
1.04
1.05
0.980
0.97
0.98
0.99
1.00
1.01
1.02
1.03
1.04
Quantiles of BDC
27
Comparative Analysis of Naira/US Dollar Exchange…
Figure 2
Volatility Clustering of Interbank Exchange Rate Return
.020
.015
.010
.005
.020
.000
.015
-.005
.010
-.010
.005
.000
-.005
-.010
09M07
10M01
10M07
Residual
11M01
Actual
11M07
12M01
Fitted
Figure 3
Volatility Clustering of Bureau de Change Exchange Rate Return
.020
.06
.015
.04
.010
.005
.020
.000
.015
-.005
.010
-.010
.02
.06
.00
.04
-.02
.02
.005
.000
.00
-.005
-.010
-.02
09M07
10M01
10M07
Residual
11M01
Actual
11M07
12M01
Fitted
2009M07
2010M01
Residual
2010M07
Actual
2011M01
Fitted
Figure 1, 2 and 3 clearly show the presence of volatility clustering, where
periods of high volatility are followed by periods of high volatility while period of
low volatility are followed by period low volatility. The official return tends to be
more clustered with spike in 2009 while Bureau de change is relatively less
clustered of all the returns with spike in 2008.
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Journal of Finance and Accounting Research
Volume 3 Issue 1, Spring 2021
Agya, Kingsley and Hassan
Table 2
Autocorrelation of Power Transformed Return Series Using Powers of 0.25, 0.5
and 0.75
Ljung-Box Q0.25
statistics
Official rate
Interbank Rate
Bureau de
change
Box Q0.25(6)
0.09402*
(0.001)
-0.076*
(0.012)
-0.010*
(0.017)
-0.004*
(0.000)
-0.039*
(0.000)
0.081*
(0.000)
-0.117*
(0.000)
0.110*
(0.000)
-0.113*
(0.000)
0.020*
(0.001)
-0.075*
(0.012)
0.002*
(0.017)
-0.004*
(0.000)
0.0399*
(0.000)
0.081*
(0.000)
-0.116*
(0.000)
0.110*
(0.000)
-0.113*
(0.000)
Box Q0.25(12)
Box Q0.25(20)
Ljung-Box Q0.5
statistics
Box Q0.5(6)
Box Q0.5(12)
Box Q0.5(20)
Ljung-Box Q0.75
statistics
Box Q0.75(6)
-0.116*
-0.004*
(0.000)
(0.000)
Box Q0.75(12)
0.111*
-0.039*
(0.000)
(0.000)
Box Q0.75(20)
-0.013*
0.081*
(0.000)
(0.000)
Note. figure in parentheses are p-value * indicates significant
Source: Authors’ computation
-0.116*
(0.000)
0.111*
(0.000)
-0.013*
(0.000)
at 5 percent level
Table 3 displayed unit root test result which shows all returns were stationary
at level, this is discernable by comparing the ADF and PP test statistics with
critical value of 1%, 5% and 10% were greater than respective critical value at
level implying returns are integrated of order zero I (0).
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Comparative Analysis of Naira/US Dollar Exchange…
Table 3
Unit Root Test Result
Variables
Official
Rate
Statistics
ADF Critical Value
1%
5%
10%
-14.831* -4.949
-4.443 -4.193
(0.01)
Statistics
-7.343*
(0.000)
PP Critical Value
1%
5%
10%
-3.480
-2.883 -2.579
Interbank -11.839* -4.949
-7.800* -3.463
-4.443 -4.193
-2.876 -2.575
Rate
(0.01)
(0.000)
Bureau de -10.237* -4.949
-9.646* -3.466
-4.443 -4.193
-2.876 -2.575
change
(0.01)
(0.000)
Note. figure in parentheses are p-value * indicates significant at 5 percent level
Source: Authors’ computation
Table 4
Estimates of GARCH Models Official Rate Return, January 2004 –September
2020
GARCH GJREGARCH
GARCH
Mean equation
APARCH IGARCH TSGARCH
C
0.999
0.999
(2.280) (1.890)
Variance Equation
0.999
(2.790)
0.999
(2.550)
ϖ
1.950
(1.060)
0.811
(0.009)
0.069
(0.042)
0.226*
(0.110)
0.146
(0.100)
2.084
(0.139)
842.539
α
β
γ
2.750
(1.310)
0.701
(0.025)
0.148
(0.054)
2.130
(1.250)
0.812
(0.054)
0.024
(0.001)
5.941*
(3.102)
-3.271
(0.561)
0.762
(0.041)
0.201
(0.038)
-1.535*
(0.885)
2.406
(0.193)
827.679
2.208
(0.935)
835.758
2.012
(0.013)
9344.422
δ
V
LL
30
0.999
(1.905)
0.999
(2.490)
0.061
(0.026)
0.311
(0.026)
1.310
(1.610)
0.712
(0.036)
0.116
(0.073)
6.994
(5.340)
2.677
(0.130)
805.763
2.138
(0.177)
842.857
Journal of Finance and Accounting Research
Volume 3 Issue 1, Spring 2021
Agya, Kingsley and Hassan
GARCH GJREGARCH APARCH IGARCH TSGARCH
GARCH
Pers.
0.849
0.836
0.963
0.880
0.372
0.828
AIC
-12.371 -12.463 -11.697
-12.564
-12.072
-12.584
SC
-12.262 -12.457 -11.567
-12.412
-12.006
-12.453
HQC
-12.326 -12.521 -11.645
-12.502
-12.045
-12.531
N
133
133
133
133
133
133
Notes. Standard errors are in parentheses. * indicates significant at the 5% level.
LL, AIC, SC, HQC and N are the maximum log-likelihood, Akaike information
Criterion, Schwarz Criterion, Hannan-Quinn criterion and Number of
observations respectively. Source: Authors’ computation.
Table 5
Estimates of GARCH Models Interbank Rate Return, January 2004-September
2020
GARCH
GJRGARCH
EGARCH
APARCH IGARCH TSGARCH
0.999
(0.000)
1.000
(0.000)
Mean equation
C
1.000
1.000
(0.000)
(0.002)
Variance Equation
ϖ
4.120
3.206
(2.640)
(1.514)
α
β
-2.977
(0.0521)
1.690
(0.000)
0.156
(0.316)
0.656
(0.134)
0.057*
(0.010)
0.441
(0.049)
2.223*
(0.154)
1077.605
0.812
-10.759
0.209
(0.561)
0.511
(0.022)
0.890
(0.403)
0.021
(0.019)
2.613*
(1.215)
0.521
(0.206)
0.219
(0.046)
-4.160*
(2.434)
2.139*
(0.134)
1083.049
0.720
-10.834
2.145*
(0.407)
1009.647
0.911
-10.020
2.005*
(0.006)
1016.967
0.740
-10.161
γ
δ
V
LL
Pers.
AIC
Department of Finance
Volume 3 Issue 1, Spring 2021
1.000
(6.570)
1.000
(0.001)
4.620
(6.740)
-0.009
(0.002)
1.001
(0.012)
0.950
(0.140)
-0.160
(0.007)
1.312
(2.388)
2.261*
2.123*
(0.047)
(0.204)
977.8761002.049
0.992 0.790
-9.797 -10.016
31
Comparative Analysis of Naira/US Dollar Exchange…
GARCH
GJREGARCH APARCH IGARCH TSGARCH
GARCH
SC
-10.751
-9.919
-10.061
-10.644
-9.748 -9.917
HQC
-10.801
-9.985
-10.120
-10.712
-9.778 -9.976
N
199
199
199
199
199
199
Notes. Standard errors are in parentheses. * indicates significant at the 5% level.
LL, AIC, SC, HQC and N are the maximum log-likelihood, Akaike information
Criterion, Schwarz Criterion, Hannan-Quinn criterion and Number of
observations respectively. Source: Authors’ computation
Table 6
Estimates of GARCH Models for Bureau de Change Rate Return, January 2004September 2020
GARCH
GJR-GARCH
EGARCH APARCH IGARCH
TS-ARCH
Mean equation
C
0.999
1.000
0.999
0.999
0.999
0.999
(0.002)
(0.001)
(0.000)
(0.007)
(0.000)
(0.051)
0.000
4.130
-1.762
5.580
0.000
(0.013)
(6.690)
(0.317)
(0.004)
(0.014)
0.417
0.952
0.022
0.801
0.618
0.802
(0.272)
(0.267)
(0.107)
(0.410)
(0.024)
(0.340)
0.355
-0.107
0.859
0.026
0.361
-0.024
(0.081)
(0.059)
(0.025)
(0.007)
(0.024)
(0.003)
0.027*
0.579*
-0.667*
1.520*
(0.000)
(0.178)
(0.213)
(0.206)
Variance Equation
Π
Α
Β
Γ
Δ
0.470
(0.078)
V
32
2.001*
2.340*
2.349*
2.223*
3.397*
2.001*
Journal of Finance and Accounting Research
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Agya, Kingsley and Hassan
GARCH
GJR-GARCH
EGARCH APARCH IGARCH
TS-ARCH
(0.118)
(0.315)
(0.256)
(0.154)
(0.241)
LL
862.857
864.341
853.070
871.769
839.634871.570
Pers.
0.772
0.845
0.881
0.827
0.979
0.756
AIC
-8.622
-8.749
-8.513
-8.691
-8.408
-8.699
SC
-8.538
-8.580
-8.413
-8.575
-8.358
-8.599
HQC -8.588
-8.675
-8.473
-8.644
-8.388
-8.659
N
199
199
199
199
199
199
(0.105)
Notes. Standard errors are in parentheses. *indicates significant at the 5% level.
LL, AIC, SC, HQC and N are the maximum log-likelihood, Akaike information
Criterion, Schwarz Criterion, Hannan-Quinn criterion and Number of
observations respectively. Source: Authors’ computation
Table 4 shows the sum of α and β in the GARCH, GJR-GARCH, EGARCH,
APACRH model were less than 1, indicates the variance process are mean
reverting and that shocks to volatility will die down slowly, thus the variance
process revert slowly to their mean, except for IGARCH that has a rapid mean
reversion process to it mean. In table 5, the sum of α and β for GJR-GARCH and
IGARCH were close to 1 which is an indication of slow mean reverting process,
implying that shock to volatility will die down slowly while GARCH, EGARCH,
APARCH and TS-GARCH has fast mean reverting process and shock to their
variance means that it will revert quickly to their mean. In the same vein, table 6
shows the sum of α and β were all less than 1, indicating mean reverting process
and shock to volatility will die down relative slowly for GARCH, GJR-GARCH,
EGARCH, APARCH and TS-GARCH. However, IGARCH is close to 1 implying
a very sluggish mean reverting process and shock to volatility will die down
rather slowly. In a nutshell, bureau de-change volatility was most persistent,
followed by the official and interbank rate which were the least volatile of the
three returns examined within the period.
Table 4, 5 and 6 present γ coefficients, which measure symmetry and leverage
effects, in table 4, two were positive and statistically significant at 5% level in
GJR-GARCH and APARCH models and negative while significant in EGARCH
model. Leverage effect exists, if γ > 0 in the GJR-GARCH and APARCH models
and γ < 0 in EGARCH. In view of the above, we cannot reject null hypothesis of
leverage effect for GJR-GARCH, APARCH and EGARCH models, this implies
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33
Comparative Analysis of Naira/US Dollar Exchange…
that negative shock to volatility exerts more impact on volatility than positive
shock of equivalent magnitude. Table 5, also shows γ coefficients with positive
effect and significant in GJR-GARCH and APARCH and negative and significant
in EGARCH model. We cannot reject the null hypothesis of leverage effect in
GJR-GARCH, APARCH and EGARCH models; this implies that negative shock
exerts more impact on the interbank exchange return than positive shock of
equivalent magnitude. Furthermore, table 6, shows γ coefficients were positive
and significant in GJR-GARCH, EGARCH, APARCH, TS-GARCH and positive
and significant in EGARCH model. Since EGARCH is positive and we reject the
null hypothesis: because we need the negative significant for leverage effect to
exist, hence, we reject the leverage effect in EGARCH model but cannot reject
the null hypothesis of leverage effect in GJR-GARCH, APARCH and TSGARCH models. It implies that negative shock exerts more impact on Bureau de
change return than positive shock of equivalent magnitude. As expected bureau de
change return was the most volatile followed by official rate and the inter-bank
return being the least volatile. As seen in preliminary investigation in table 1, the
returns were not normally distributed, hence, we employed student t to estimate
our models and degree of freedom represented by V coefficients were statistically
significant at 5 percent level in all models as presented in tables 4, 5 and 6, thus
validating the use of student t instead of normality assumption.
Diagnostic Test
Table 7, 8 and 9 show the diagnostic tests for the returns of official, inter-bank
and bureau de change models. The Ljung-Box Q test statistics for autocorrelation
of standardized residuals at 5 percent significant level shows that autocorrelation
of standardized residuals are statistically insignificant for all lags. Hence, we
cannot reject the null hypothesis of no autocorrelation in standardized residuals.
The Ljung-Box Q2-statistics of squared standardized residuals in Tables 7, 8 and 9
are statistically insignificant at 5 percent significant level for all lags. Hence, we
cannot reject null hypothesis of no autocorrelation in squared standardized
residuals. The ARCH-LM test statistics presented in tables 7, 8 and 9 show that
the standardized residuals did not exhibit ARCH effect anymore or that the
ARCH effect has been adequately taken out. And Jarque-Bera statistics still
indicates standardized residuals were non-normally distributed.
34
Journal of Finance and Accounting Research
Volume 3 Issue 1, Spring 2021
Agya, Kingsley and Hassan
Table 7
Autocorrelation of Standardized Residuals, Autocorrelation of Squared
Standardized Residuals and ARCH LM and Normality test for Official return
Ljung-Box Q-Statistics
GARCH
GJR-ARCH
EGARCH
APARCH
IGARCH
TS-
Ljung-Box Q-Statistics
ARCH LM NML
Q(6)
Q(12)
Q(20)
Q2(6)
Q2(12)
Q2(20)
F
N*R2
JB
-0.009
-0.016
-0.009
-0.009
-0.009
-0.010
0.010
0.010
651
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.918)
(0.917)
(0.000)
0.010
0.015
0.009
-0.009
-0.010
-0.011
0.015
0.015
442
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.900)
(0.899)
(0.000)
0.002
-0.029
0.000
-0.011
-0.011
-0.011
0.010
0.011
426
(1.000)
(0.998)
(1.000)
(.000)
(1.000)
(1.000)
(0.917)
(0.916)
(0.000)
-0.010
-0.014
-0.008
-0.009
-0.010
-0.011
0.015
0.012
446
(1.00)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.901)
(0.900)
(0.000)
-0.009
-0.010
-0.014
-0.010
-0.010
-0.011
0.019
0.020
347
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.888)
(0.887)
(0.000)
0.010
0.014
0.009
-0.009
-0.010
-0.011
0.015
0.015
443
(1.000)
(1.000)
(1.000)
1.000
(1.000)
(1.000)
(0.900)
(0.999)
(0.000)
GARCH
Note. Figures in parentheses are p-values
Source: Authors’ computation
Table 8
Autocorrelation of Standardized Residuals, Autocorrelation of Squared
Standardized Residuals and ARCH LM and Normality test for interbank return
Ljung-Box Q-Statistics
GARCH
ARCH LM NML
Ljung-Box Q-Statistics
Q(6)
Q(12)
Q(20)
Q2(6)
Q2(12)
Q2(20)
F
N*R2
JB
-0.007
0.007
0.007
-0.006
-0.006
-0.006
0.007
0.007
208
(1.000)
(1.00)
(1.000)
(1.000)
(1.000)
(1.000)
(0.930)
(0.929)
(0.000)
0.003
-0.019
-0.044
-0.016
-0.017
-0.016
0.021
0.021
209
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.884)
(0.883)
(0.000)
-0.011
-0.020
0.011
-0.012
-0.012
0.026
0.027
516
GJRGARCH
EGARCH
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-0.012
35
Comparative Analysis of Naira/US Dollar Exchange…
Ljung-Box Q-Statistics
APARCH
IGARCH
TS-GARCH
ARCH LM NML
Ljung-Box Q-Statistics
Q(6)
Q(12)
Q(20)
Q2(6)
Q2(12)
Q2(20)
F
N*R2
JB
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.870)
(0.869)
(0.000)
-0.007
-0.007
-0.007
-0.006
-0.006
-0.006
0.008
0.008
196
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.928)
(0.9278)
(0.000)
-0.014
-0.028
0.044
-0.018
-0.018
-0.010
3.130
3.132
668
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.976)
(0.976)
(0.000)
0.003
0.019
-0.044
-0.016
-0.017
-0.016
0.021
0.210
120
(1.000)
(1.000)
(0.999)
(1.000)
(1.000)
(1.000)
(0.885)
(0.884)
(0.000)
Note. Figures in parentheses are p-values. Source: Authors’ computation
Table 9
Autocorrelation of Standardized Residuals, Autocorrelation of Squared
Standardized Residuals and ARCH LM and Normality test for Bureau de change
return
Ljung-Box Q-Statistics
GARCH
ARCH LM NML
Ljung-Box Q-Statistics
Q(6)
Q(12)
Q(20)
Q2(6)
Q2(12)
Q2(20)
F
N*R2
JB
-0.046
0.019
0.017
-0.005
-0.006
-0.007
0.007
0.007
195
(0.997)
(1.00)
(1.000)
(1.000)
(1.000)
(1.000)
(0.931)
(0.931)
(0.000)
0.028
0.060
0.014
-0.006
-0.000
-0.007
0.009
0.009
146
(0.991)
(0.996)
(0.995)
(1.000)
(1.000)
(1.000)
(0.923)
(0.923)
(0.000)
-0.050
-0.007
0.020
-0.004
-0.006
0.015
0.015
620
(0.862)
(0.982)
(0.999)
(1.000)
(1.000)
(1.000)
(0.902)
(0.901)
(0.000)
-0.043
-0.019
-0.018
-0.005
-0.006
-0.007
0.007
0.007
195
(0.996)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0931)
(0.930)
(0.000)
-0.023
-0.020
-0.013
-0.005
-0.006
-0.006
0.005
0.005
263
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(1.000)
(0.939)
(0.939)
(0.000)
0.028
0.060
0.014
0.005
-0.006
-0.006
0.005
0.005
GJRGARCH
EGARCH
APARCH
IGARCH
TS-GARCH
-0.010
Note. Figures in parentheses are p-values, Source: Authors’ computation
Table 10 presents the ranked model used in this study, based on Maximum
Log-likelihood ratio, Akaike information criteria, Schwartz information criteria
36
Journal of Finance and Accounting Research
Volume 3 Issue 1, Spring 2021
Agya, Kingsley and Hassan
and Hannan-Quinn criterion. Table 10 shows EGARCH was the best model,
followed by TS-GARCH, APARCH, GJR-GARCH, GARCH and I-GARCH
models respectively. It implied EGARCH model is the best model for forecasting
purposes in official exchange rate return market. In like manner, table 11 shows
GARCH is the best model followed by APARCH, EGARCH, GJR-GARCH, TSGARCH and I-GARCH models respectively, hence, GARCH is best model for
forecasting purpose in inter-banks exchange rate return market while Table12,
ranked shows GJR-GARCH is the best, followed by TS-GARCH, APARCH,
GARCH, EGARCH and I-GARCH models respectively. It implies that GJRGARCH model is the best for forecasting purpose in Bureau de change exchange
rate return market. In summary, asymmetric models are best suited for exchange
rate return estimate of volatilities in Nigeria foreign exchange market and
IGARCH is the worst of all models.
Table 10
Official return Models Ranking in Order of maximum log-likelihood, Akaike
information Criterion, Schwarz Criterion, Hannan-Quinn criterion.
LL
AIC
SC
HQC
Ranking
GARCH
827.679
-12.371
-12.262
-12.326
5th
GJR-GARCH
835.758
-12.463
-12.457
-12.521
4th
EGARCH
9344.422
-11.697
-11.567
-11.645
1st
APARCH
842.539
-12.564
-12.412
-12.502
3rd
IGARCH
805.763
-12.072
-12.006
-12.045
6th
TS-GARCH
842.857
-12.584
-12.453
-12.531
2nd
Source. Authors’ computation
Table 11
Interbank return Models Ranking in Order of maximum log-likelihood, Akaike
information Criterion, Schwarz Criterion, Hannan-Quinn criterion.
LL
AIC
SC
HQC
Ranking
GARCH
1083.049
10.834
10.751
10.801
1st
GJR-GARCH
1009.647
10.020
9.919
9.985
4t
EGARCH
1016.967
10.161
10.061
10.120
3rd
APARCH
1077.605
10.759
10.644
10.712
2nd
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Comparative Analysis of Naira/US Dollar Exchange…
LL
AIC
SC
HQC
Ranking
IGARCH
977.876
9.797
9.748
9.748
6th
TS-GARCH
1002.049
10.016
9.917
9.976
5th
Source. Authors’ computation
Table 12
Bureau de change return Models Ranking in Order of maximum log-likelihood,
Akaike information Criterion, Schwarz Criterion, Hannan-Quinn criterion.
LL
AIC
SC
HQC
GARCH
862.857
GJR-GARCH
Ranking
-8.622
-8.538
-8.588
4th
864.341
-8.749
-8.580
-8.675
1st
EGARCH
853.070
-8.513
-8.413
-8.473
5th
APARCH
871.769
-8.691
-8.575
-8.644
3rd
IGARCH
839.634
-8.408
-8.358
-8.388
6th
TS-GARCH
871.570
-8.699
-8.599
-8.659
2nd
Source. Authors’ computation
Conclusion
The paper examined the foreign exchange market volatility of Naira/US
Dollar for official rate, interbank rate and Bureau de change markets. Using
monthly exchange rate of Naira/USD from January 2004 to September 2020
(2004:1-2020:9), the returns were not normally distributed and stationary at level.
Ljung-Box Q statistic and Ljung-Box Q2 statistics of power transformed using
power 0.25, 0.5 and 0.75 for conditional heteroscedasticity for lags of 6, 12 and
20 indicates present of conditional heteroscedascity in all returns.
The sum of α and β in the GARCH, GJR-GARCH, EGARCH, APACRH
model were less than 1, indicating that the variance process are the mean reverting
and that shocks to volatility will die down slowly, thus the variance process
reverts slowly to their mean, except for IGARCH that has a rapid mean reversion
process. Also, the sum of α and β for GJR-GARCH and IGARCH were close to
1, an indication of slow mean reverting process, shock to volatility will die down
slowly while GARCH, EGARCH, APARCH and TS-GARCH has fast mean
reverting process and shock to their variance reverts quickly to their mean in
interbank return. In the same vein, the sum of α and β were all less than 1,
38
Journal of Finance and Accounting Research
Volume 3 Issue 1, Spring 2021
Agya, Kingsley and Hassan
indicating that mean reverting process and shock to volatility die down relatively
slowly for GARCH, GJR-GARCH, EGARCH, APARCH and TS-GARCH.
However, IGARCH is close to 1, and implied a very sluggish mean reverting
process and indicating that shock to volatility will die down rather slowly in
bureau de change. In sum, bureau de-change volatility was the most persistent,
followed by official and interbank rates thus this was the least volatile of the
three. Shocks to volatilities were asymmetric in the three exchange rate returns,
that is, negative shock of the same magnitude has more impact on volatilities than
positive shocks. Both Ljung-Box Q test statistics for autocorrelation of
standardized residuals and Ljung-Box Q2-statistics of squared standardized
residuals shows there were no autocorrelation in standardized and squared
standardized residuals and no ARCH effect in residuals.
The ranks of the model show that EGARCH model is best for forecasting
purposes in official exchange rate return market, whereas GARCH is best for
forecasting purposes in inter-banks exchange rate return market while GJRGARCH model is best for forecasting purpose in Bureau de change exchange
rate return market. In summary, asymmetric models were best suited for the
estimates of exchange rate return volatilities, IGARCH being the worst in Nigeria
foreign exchange return market.
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Journal of Finance and Accounting Research
Volume 3 Issue 1, Spring 2021