GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTICITY (GARCH) MODELS AND OPTIMAL FOR NIGERIAN STOCK EXCHANGEG. A Eriyeva ^{1}, C.N. Okoli ^{2}^{} ^{1, 2 }Department of Statistics, Chukwuemeka Odumegwu Ojukwu
University, Anambra State, Nigeria.





Received 5 November 2021 Accepted 16 December 2021 Published 31 December 2021 Corresponding Author G.
A Eriyeva, eriyevag@gmail.com DOI 10.29121/granthaalayah.v9.i12.2021.4426 Funding:
This
research received no specific grant from any funding agency in the public,
commercial, or notforprofit sectors. Copyright:
© 2021
The Author(s). This is an open access article distributed under the terms of
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original author and source are
credited. 
ABSTRACT 


This
paper focused on comparative performance of GARCH models, ascertaining the
best model fit, estimating the parameters and making prediction from optimal
model. The study used UBA daily stock exchange prices sourced from the
official websites of www.investing.com,on the daily basis of the Nigeria
stock exchange rate over a period of ten years from 06/06/2012 – 04/06/2021.
Five GARCH models (SGARCH, GJRGARCH or TGARCH, EGARCH, APGARCH and IGARCH)
were fitted to the secondary data set of the Nigerian Stock exchange market
for the period of June 2012 June 2021 and the results of the findings were
obtained. The AIC results were SGARCH (1,1) (6.1784), GJRGARCH (1,1)
(6.1778), EGARCH (1,1) (6.1714), APGARCH (1,1) (6.1245) and IGARCH (1,1)
with the value of AIC 6.1793. The EGARCH (1, 1) was found to be the optimal
model with AIC value of 6.1714. The
further findings indicated volatility clustering and leverage effect. The
result of the analysis equally showed parameter estimates of the EGARCH (1,1)
model and all the parameters were significant including mean and alpha.
Prediction using the optimal model was made with an initial out of sample of
200 and n ahead of 200 with predicted values within the 95% confidence
interval resulting there is no sign of volatility and clustering. Based on the findings of the study, other
time series packages should be compared with GARCH models, data should be
making available for easy access and investors should be encouraged to invest
in United Bank for Africa (UBA, Nigeria). 


Keywords: Variants,
Volatility, Heterogeneity, GARCH, Information Criteria. 1. INTRODUCTION Stock markets are
subject to irregular growth and decline. This is due to market crashes which
are difficult to comprehend, and these unexpected fluctuations affect the
dynamics of the data temporarily or permanently Foley (2014). An
increase or decrease in the value of stock tends to have a corresponding
effect on the economy, mostly through the money market. A fluctuation in
stock prices stimulates investment and increases the demand for credit, which
eventually leads to higher interest rates in the overall economy Spiro (1990). The stock
market is very sensitive to any central bank movement since it represents the
future exchange rate. The stock market is a gauge of the economy's wealth.
The global financial crisis had such a negative impact on the Nigerian stock
market that investors lost faith in it. The stock market impact on return
rate has both long and shortterm. Any sharp depreciation in the foreign
exchange rate increases liabilities in terms of the domestic currency,
increasing the chances of defaults and creating room for financial crisis,
and eventually influencing the value of the firm, due to its negative effects
on banks and enterprise balance sheets, which are usually denominated in
foreign currency. 


No work has compared performance of five GARCH type models for Nigerian stock exchange. Therefore, this current study will compare the performance of GARCH type models, ascertain the best fit model, estimate parameters and make forecast with the optimal model.
2. LITERATURE REVIEW
Vitor (2015) employed GARCH
family models to investigate the sensitivity of shock persistence and
asymmetric effects in the international stock market during the global
financial crises using daily data of twelve stock indexes over the period from
October 1999 to June 2011. The results showed that the Subprime crisis period
turned out to have bigger impact on stock market volatility with high shock
persistence and asymmetric effects.
Tabajara
et al. (2014) compared the
stock market behaviour of Brazil, Russia, India and China (BRIC) emerging
economies to those of the industrialized economy of USA, Japan, United Kingdom
and Germany in the light of 2008 global financial crisis using GARCH, EGARCH
and TARCH univariate models. The stock market behaviours of the BRIC’s emerging
markets and the industrialized economies in terms of shock persistence effects
on volatility, asymmetry and delayed reaction of volatility to stock market
changes were found to be similar in both markets. However, the BRIC’s stock
markets showed less persistence of shocks, less asymmetric effects and faster
volatility reactions to market changes.
Ding et al. (1993) extended the standard deviation GARCH model
proposed by Taylor (1986) and Schwert
(1989) named Power
GARCH (PGARCH). This model multiplies
the power of the conditional standard deviation d (positive exponent) with the function of the lagged conditional
standard deviation and lagged absolute innovation. When the positive exponent is set to 2, this expression becomes the standard GARCH model. The provision of switching power supplies increases the flexibility of the model.
Miron and Tudor (2010) studied several types of asymmetric GARCH
models (EGARCH, PGARCH, and TGARCH)
using the stock indexes of the United States and Romania from 2002 to 2010. They proved that the
estimation of volatility comes from the
application of the EGARCH model and
is
more reliable than the estimation
of
other models. EGARCH records the
asymmetry between returns, volatility
and aims to compensate for the three main shortcomings of the GARCH model. (i) Ensuring the parameter constraints of
positive conditional variance; (ii) Insensitive to the
asymmetric response of volatility to shocks; and (iii) It
is difficult to measure persistence in highly stable series. The logarithm of the
conditional variance in the EGARCH model means that the leverage is exponential rather than quadratic. Specifying
the
volatility in the form of logarithmic
transformation means that the parameters are unrestricted to ensure the positiveness of the variance (Majose, 2010),
which is the decisive advantage of the EGARCH model over the symmetric GARCH
model.
Chang (2010) analyzes the
effect of the economic and financial crisis on Chinese stock return volatility
using daily data from 2000 to 2007 as the precrisis period and 2007 to 2010 as
the during crisis period. The findings show that the EGARCH model fits the data
better than the GARCH model in modeling the volatility of Chinese stock
returns. The result also indicated that volatility is more persistent during
crisis period than in precrisis period.
Aliyu
(2011) assessed the
innovations of monetary policy in Nigerian stock market during the global
financial crisis period using monthly data for the period of January 2007 to
August 2011. He employed EGARCH model and regressed stock market returns
against money stock (M1 and M2) and monetary policy rate (MPR). The empirical
findings from the study revealed that, unlike the anticipated components of the
monetary innovation, the unanticipated component of the policy innovations on
M2 and MPR exerted destabilizing effect on Nigerian stock returns.
Franses and Van Dijk (1996) compared three
models of the GARCH family (GARCH, QGARCH, and GJRGARCH) to predict the weekly volatility of several
European stock indexes. At the end of this study, they found that the nonlinear model could not beat the standard
GARCH model.
Jayasuriya
(2002) examined the
effect of stock market liberalization on stock returns volatility in Nigeria
and fourteen other emerging market data, from December 1984 to March 2000 to
estimate symmetric GARCH model. The study found that positive (negative)
changes in prices have been followed by negative (positive) changes in
volatility. The Nigeria portion of the result indicates more of business cycle
behavior of stock return rather than volatility clustering. In studying
volatility behavior of stock returns for emerging markets, Ogum et al. (2005) apply the
Nigerian and Kenya stock data on EGARCH model. The finding differed from that
of Jayasuriya
(2002). Although
volatility persistence was found in both markets; volatility responds more to
negative shocks in the Nigeria market and the reverse is the case for Kenya
market.
3. MATERIAL AND METHODS
3.1. METHOD OF DATA COLLECTION
The study used secondary data and is all about UBA
daily stock exchange prices in Nigeria. The dataset used for this study was
collected from the official websites of www.investing.com,on the daily basis of
the Nigeria stock exchange rate over a period of ten years. (06/06/2012 –
04/06/2021).
3.2.
METHODS OF DATA ANALYSIS
3.2.1. AUGMENTED DICKEYFULLER TEST (ADF)
The Augmented DickeyFuller (ADF) test was employed in the study. This is an augmented version of the Dickey Fuller test for data which is considered large and complicated for time series models. The testing for ADF is similar to the DickeyFuller and can be applied to the regression model:
∆𝑦_{𝑡 }= + + 𝜃𝑦_{𝑡−1 }+_{ }
_{ }
Where ∆ is difference operator, 𝛽_{0} is the constant, is the coefficient on a time trend and 𝛾_{𝑖} is the lag of autoregressive process, and 𝑎_{𝑡} is error term.
The hypothesis formulated
H_{0}: =1 (There is unit root)
H_{1}:< 1(There is no unit root)
The ADF test statistic is given by
ADF =
Where denotes the least square estimate of 𝛾 is as well know Augmented Dickey fuller test.
The null hypothesis of the unit root is accepted if the pvalue is greater than critical value.
3.3. MODEL OF SELECTION CRITERION
3.3.1. AKAIKE INFORMATION CRITERION (AIC)
The Akaike Information Criterion (AIC) is a measure of the relative goodness of fit of a statistical model. Akaike (1974) suggested that the goodness of fit of a given model should be measured by weighing the fitting error and the number of parameters in the model. When a specific model is used to describe reality, it provides a level of information loss. Given a data, several competing models may be ranked according to their AIC, with the one having the lowest AIC being the best model.
Mathematically,
The AIC is defined as = 2(k−In(L) 3.3
When k is the number of parameters in the statistical model, and L is the maximized value of likelihood function for the estimated model.
3.3.2. BAYESIAN INFORMATION CRITERION
Bayesian Information criterion (BIC) or Schwarz Criterion is a criterion for model selection among
parametric model classes
with different parameters.
Choosing a model to optimize BIC is a form of regularization. BIC assumes that the data distribution is an asymptotic result of the exponential distribution. Mathematically expression
Let BIC =2𝑙𝑜𝑔_{𝑒}L +K𝑙𝑜𝑔_{𝑒}n 3.4
Where n = sample size, k = the number of free parameters to be estimated L = the maximized value of the likelihood function for the estimated model. In this study ADF will employed in this paper
3.4. TEST FOR NORMALITY
The test for normality is done using the JarqueBera
test statistic. According to Dikko et
al. (2015), JarqueBera could be
defined as points test of skewness and Kurtosis to examine whether data series
exhibit normal distribution or not. The test statistics was developed by Jarque and Bera (1980). and this is defined as
Test statistic:
J B = 3.5
Where n = sample size, S= skewness coefficient, and K =kurtosis. Under the null hypothesis of a normal distribution, the JarqueBera statistic is distributed as with 2 degrees of freedom. The probability that a JarqueBera statistic exceeds (in absolute value) the observed value under the null hypothesis—a small probability value leads to the rejection of the null hypothesis of a normal distribution.
3.5. TEST FOR RANDOMNESS
The adequacy of the GARCH model implies that any of a group of autocorrelations of a time series is different from zero. The test investigates the overall randomness based on a number of lags. Ljung–Box (Ljung and Box 1978) test is employed in this research to test for the adequacy of the optimal GARCH family model. The Ljung–Box statistic is defined as
Q
Where n is sample size, is the sample autocorrelation at lag k and h is the number of the lags being tested.
3.6. TEST HETEROSCEDASTICITY (ARCH)
In this study, we employ the use of ARCH effect test. ARCH LM was proposed by Engle (1982). ARCH LM test is a langrage multiplier test to assess the significance of ARCH effect. Hence, the conditional heteroscedasticity in a variance is equal to the autocorrelation in the squared innovation process, is given by the regression equation
3.7
Where is a white noise error process and lag 𝑚 is a prespecified positive integer? The null hypothesis is
𝐻_{0}: ∝_{0}=∝_{1}=, ⋯, ∝_{𝑚}= 0
The null hypothesis state that there is no ARCH effect (since the pvalue is greater than the chosen significance level at 5%) Do not rejected the null hypothesis, there we conclude that there is no ARCH effect in time series.
3.7. ESTIMATION OF GARCH MODEL WITH DISTRIBUTION ERROR
Several methods of error distributions we be comparing in the estimate of GARCH family model (symmetric and Asymmetric). We introduced normal, student t distribution (std), generalized error distribution to analyze the GARCH family models using the daily UBA log return stock exchange in Nigeria. In this paperwork, we employed student t distribution.
3.7.1. NORMAL DISTRIBUTION
Standard normal error distribution the random variable (Z) following loglikelihood function needs to be maximized
3.7.2. STUDENT T DISTRIBUTION ERROR
According (Shamiri and Isa, 2009), the probability density function of is given asstandardized Student t distribution (std)
f  3.9
Where ν is the number of degrees of freedom and Г denotes the Gamma function.
3.7.3. GENERALIZED ERROR DISTRIBUTION
Generalized Error Distributionas proposed by Nelson (1991) is more interesting in terms of satisfying stationarity compared to the studentt distribution. Just like in the case of the student’st error distribution the unconditional means and variances may not be finite in the EGARCH. The loglikelihood function for the standard generalized error distribution is defined as
)
3.10
3.11
The generalized error distribution (GED) incorporates
both normal error distribution when (V = 2), Laplace distribution when (v = 1),
and the unique distribution for v = .
GARCH models
In the study, we used the univariate GARCH family model approach and also compared the different GARCH family models in analyzing the daily UBA stock exchange in Nigeria, which are indicated as follows; EGARCH, SGARCH, TGARCH or GJRGARCH, APARCH and IGARCH models
3.8. ARCH/GARCH FAMILY MODELS
The autoregressive conditional heteroscedastic (ARCH) model of Engle (1982), the generalized ARCH (GARCH) model of Bollerslev (1986). ARCH models which report the conditional variance, which depend on past return is that the shock at of an asset return is serially uncorrelated, but dependent,
The ARCH(q) model can be expressed as
=
Where ~N (0,1) is a white noise process; 𝜇 is the constant mean of the returns. The conditional variance, is a function of past squared residuals of returns, which scales the process .
In the ARCH (q) process proposed by Engle (1982),
3.12
Where and implied that is strictly positive; is the error of return estimation at time t. With these nonnegative restrictions (particularly 𝜔> 0), if a major shock happened onelagged period, twolagged period or up to j periods ago, the impact would increase recent conditional variance. However, no matter the market movement is positive or negative, due to the squared return shock on the righthand side
3.8.1. GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY (GARCH) MODEL
The standard GARCH model is proposed by Boleslaw (1986) to solve the problematic caused by the long lag structure in the ARCH process. Now GARCH (p, q) process, the conditional variance depends not only on q lagged error square, but also on p lagged historical conditional variances. The GARCH Model can be defined as the follows
is a discrete – time stochastic defined to be =𝑍_{𝑡}𝜎_{𝑡 }give 𝑍_{𝑡} ~ iid. N (0,1) and 𝜎_{𝑡} is the conditional standard deviation of return at time t. All parameters 𝜔, 𝛼_{1, 𝛽1} are nonnegative. The stationary condition of 𝛼 + 𝛽 < 1 should hold to ensure weakly stationarity of GARCH process.
𝛼_{1} indicate the shortrun persistency of shock while 𝛽 implies the long run persistency
3.8.2. INTEGRATED GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEOSKEDASTICITY (IGARCH) MODEL
Integrated generalized autoregressive conditional heteroscedasticity (IGARCH) model. IGARCH model are unit root GARCH model. Similar to ARIMA models, a key feature of IGARCH model is that the impact of past squared shocks.for is persistent. The volatility processes in these markets are purely random walk. Therefore, IGARCH is introduced to use for such kind of nonstationary volatility process. The IGARCH (1, 1) model is specified in Tsay (2005) and Grek (2014) as
3.14
Where, Ali (2013) used to denote. The model is also exponential smoothing for the series. To rewrite this model as
By substitution,
Which is the wellknown exponential smoothing formulation with being the discounting factor (tsay,2005)
3.8.3. THRESHOLD GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY (TGARCH) MODEL
Threshold GARCH has been developed by Zakoian (1990) and Glosten et al. (1993) it is also known as GJR GARCH. This model defines the conditional variance as a piecewise function and captures the asymmetric effect Zhang (2016).
TGARCH (p, q) model specification for conditional variance is given by
where_{𝑡}_{−1} if _{𝑡}^{2}< 0 and 0 otherwise
In TGARCH model, good news implied that and bad news implies that and these two shocks of equal size have differential effect on the conditional variance. Good news has an impact of 𝛼_{𝑖,} and bad news has an impact of 𝛼_{𝑖}_{ }+ 𝛾_{𝑖}. Bad news increase volatility when 𝛾_{𝑖}_{ }> 0, which implied the existence of leverage effect in the ith other and when 𝛾_{𝑖}_{ }≠ 0 the news impact is asymmetric. However, the first order representation is of TGARCH (p, q) is
3.8.4. GLOSTENJAGANATHAN AND RUNKLE GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY (GJR GARCH) MODEL
The GJR GARCH model is another nonlinear extension of the standard GARCH model and was first proposed by Glosten et al. (1993) and have developed the GJR –GARCH model which estimates effect of good news and bad news in financial stock markets. Therefore, to take into account this kind of effect, a dummy variable is introduced into the symmetric GARCH model. This model covers asymmetric or leverage effect confidently a with long memory.
GJRGARCH model can be expressed as
where S, is a dummy variable that takes the value "1" when the error term and 1 is negative and
"O" and when the error term is positive Raza et al. (2015).
3.8.5. Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH)Model
Exponential GARCH (EGARCH) model of was proposed by Nelson (1991) There are various ways to express the EGARCH model. However, it allows for asymmetric effects between positive and negative asset returns, he considered the weighted innovation.
The EGARCH specification as specifies log volatility as
Where is iid with a generalized error distribution (GED) which nest the Gaussian and allow for slightly fatter tail. Due the specification of log volatility, no parameter restriction is necessary to keep volatility positive. However, the condition for weak and strong stationarity coincides. Here, that if 𝜃 = 0, then Cov (𝑦_{𝑖}^{2}, 𝑦𝑡_{−}_{𝑗}) ≠ 0 such that a leverage effect can be captured
3.8.6. ASYMMETRY POWER AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY (APARCH) MODEL
Asymmetry power ARCH(APARCH) model was developed byDing et al. (1993).
PGARCH (p, q) it can mathematical written as
3.23
Note, 𝛿> 0 𝑎𝑛𝑑ℝ^{+},_{ }< 1 establish the existence of leverage effect. If 𝛿 = 2, the PGARCH (p,q) replicate a GARCH(p,q) with a leverage effect. If 𝛿 = 1, standard deviate is modeled.
The first order PGARCH can be mathematical equation
The failure to accept the null hypothesis that 𝛾_{𝑖}_{ }≠ 0, shows the
presence of leverage effect. The impact of news on volatility in PGARCH is
similar to that of TGARCH when 𝛿 𝑖𝑠 1
4. DATA ANALYSIS AND RESULTS
The plot time showed the direction of movement of the daily stock exchange prices in Nigeria over the period of ten years

Figure 1 
Figure 1 present the original time plot of daily stock price in Nigeria (UBA). The plot shows the movement of daily stock prices over period of time between 20122021.The data exhibits the presence of persistence, trend and nonstationary in the observed data. The stock price rose steadily to peak level in the middle of 2013, maintaining its fluctuation till the middle of 2013 and 2014, it finally dropped at 2015/2016. Later, between 2017 there is fluctuation in stock price. It also maintained a peak level between 2017/2018, at beginning 2018 the stock price starting decline.

Figure 2 
Figure 2 Present the ACF plot of Daily Stock Exchange Price in Nigeria (UBA), it can be observed from the autocorrelation function plot, the data decaying slowly and nonstationary.

Figure 3 
Figure 3 present PACF plot of Daily Stock Exchange Price in Nigeria (UBA), it can be observed from the partial autocorrelation function plot, that the data is sinuous in nature and nonstationary.
4.1. RESULT OF ROOT TEST
The ADF test is (2.1001), at the chosen level of significance (0.05), the pvalue of the test (0.5359) is greater than 0.05. The null hypothesis is not rejected. This implies that, Nigeria’s daily stock exchange price is nonstationary.
Table 1 Descriptive statistics of daily price and log return in Nigeria 20122021 

Statistics 
DSE
prices 
DSE
log return prices 
Number of observations 
2225 
2224 
Minimum 
2.59 
0.042324 
Maximum 
13 
0.088493 
Mean 
6.638908 
0.000138 
Median 
6.75 
0 
Sum 
14771.57 
306531 
Standard deviation 
2.202832 
0.012671 
Skewness 
0.414509 
0.012671 
Kurtosis 
0.2299624 
4.267005 
Show the summary descriptive statistics for the
daily stock exchange price and log return series. The skewness is
0.285355 for the return series and 0.414509 for the
daily stock price. And this is an
indication of a positive skewness. This implies that most of the values of the
series are concentrated on the right side of the mean. Furthermore,
kurtosis value (4.267005) is greater than that of normal distribution which is
3. It shows that the distribution has a fat tail and also indicates the distribution
has a small outlier.
On the contrary, the kurtosis value (0.229624) is less than that of normal
distribution. It shows that the distribution has a flat tail which is one of
the (features)
of stock returns.

Figure 4 
Figure
4 present the plot of the
stationary of DSE log return. The
variance and mean of this series are now fluctuating around a common location
with no indication of structural breaks. The time plot we
observe that the returns vary along the zero line with the largest log return
of stock prices observed around 2013,2015, 2016 and 2019 having a value of
0.04, while 2017 having value 0.05. During the years
2016,
20192020, there is spike in volatility indicating
nonconstant conditional volatility
4.2. RESULT OF TEST OF RANDOMNESS (LJUNGBOX TEST)
statistic (Chisquared) = 19.301, df = 1, pvalue =1.116e05
Conclusion: The daily log return price auto correlated at 5% level of significance.
4.3. Test for Heteroskedasticity (ARCH LM) test
Test statistic (Chisquared) =247.26, Degree freedom=100, pvalue=1.792e14
Conclusion: Since the pvalue (pvalue=2.2e16) is less than 0.05, Therefore we reject the null hypothesis and conclude that is there is an ARCH effect presence
4.4. Fitting of the GARCH series (symmetric and asymmetric) family models
Table 2 Present ARMA (0,0) SGARCH (1,1,) with different Distribution error 

Normal 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIG 
MLE 
SGARCH (1,1) 

0.000033 
0.8854 
6.0688 
6.0577 
6148.632 

0.000024 
0 


0.269099 
0 


0.597631 
0 

Student’s t
distribution 

Mode 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
SGARCH (1,1) 

0.000098 
0.602714 
6.1784 
6.1645 
6260.621 

0.000019 
0.000046 


0.370307 
0 


0.611955 
0 

Generalize error
distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
SGARCH (1,1) 

0 
0.999959 
6.2178 
6.2039 
6300.524 

0.000019 
0.000239 


0.364403 
0 


0.617922 
0 
Table 3 Present ARMA (0,0) APARCH (1,1,) with different Distribution error 

Normal Distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
APARCH (1,1) 

0.000131 
0.579306 
6.0631 
6.0465 
6144.909 

0 
0.437921 


0.210776 
0 


0.553791 
0 


0.06902 
0.049151 


3.185452 
0 

Student’s t
distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
APARCH (1,1) 

0.000056 
0.77202 
6.1769 
6.1575 
6261.122 

0.000042 
0.5035 


0.361589 
0 


0.622841 
0 


0.038706 
0.43348 


1.830679 
0 

Generalize error
Distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
APARCH 

0 
1 
6.1871 
6.1677 
6271.466 

0 
0.80291 


0.056049 
0 


0.892102 
0 


0.064414 
0.301 


2.765914 
0 
Table 4 Present ARMA (0,0) GjrGARCH (1,1,) with different Distribution error 

Normal 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
GjrGARCH (1,1) 

0.000131 
0.58348 
6.0688 
6.0549 
6149.665 

0.000025 
0 


0.23531 
0 


0.597383 
0 


0.061727 
0.15275 

Student’s t
Distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
GjrGARCH (1,1) 

0.00006 
0.758414 
6.1778 
6.1611 
6260.998 

0.00002 
0.000039 


0.338261 
0.000001 


0.611086 
0 


0.060385 
0.389174 

Generalize Error
Distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
GjrGARCH (1,1) 

0 
1 
6.1705 
6.1705 
6253.601 

0 
0.8053 


0.08859 
0 


0.915618 
0 


0.023208 
0.11747 
Table 5 Present ARMA (0,0) IGARCH (1,1,) with different Distribution error 

Normal 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
IGARCH (1,1) 

0.000102 
0.64349 
6.0554 
6.0471 
6134.062 

0.000018 
0 


0.393722 
0 


0.606278 
Na 

Student’s t
distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
IGARCH (1,1) 

0.000098 
0.599815 
6.1793 
6.1682 
6260.551 

0.000019 
0.000064 


0.386911 
0 


0.613089 
Na 

Generalize Error
Distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
IGARCH (1,1) 

0 
0.999721 
6.2187 
6.2076 
6300.449 

0.000019 
0.000142 


0.384428 
0 


0.615571 
Na 
Table 6 Present ARMA (0,0) EGARCH (1,1,) with different Distribution error 

Normal 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
EGARCH (1,1) 

0.000284 
0.23213 
6.0564 
6.0425 
6137.059 

1.631956 
0 


0.02403 
0.25277 


0.81224 
0 


0.433991 
0 

Student’s t Distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
EGARCH (1,1) 

0.000074 
0.697497 
6.1714 
6.1548 
6254.573 

1.188998 
0.000017 


0.020313 
0.48702 


0.863507 
0 


0.521023 
0 

Generalize Error
Distribution 

Model 
Parameter 
Estimate 
Pr(>t) 
AIC 
BIC 
MLE 
EGARCH (1,1) 

0 
0.999997 
6.2129 
6.1963 
6296.562 

1.186107 
0.000115 


0.022655 
0.485002 


0.864843 
0 


0.512627 
0 
Table 2, 3, 4 , 5, 6 show the results obtained from the estimation of GARCH family model with their corresponding error distribution
4.5. MODEL SELECTION
Table 7 shows the information criterion (AIC and BIC) of different GARCH models with three different distribution error for the models 

Information
criterion 

Model 
AIC 
BIC 
SGARCH (1,1) Norm 
6.0658 
6.0577 
SGARCH (1,1) Std 
6.1784 
6.1645 
SGARCH (1,1) Ged 
6.2178 
6.2039 
APARCH (1,1) Norm 
6.6031 
6.0465 
APARCH (1,1) Std 
6.1769 
6.1575 
APARCH (1,1) Ged 
6.1871 
6.1677 
GjrGARCH (1,1) Norm 
6.0688 
6.0549 
GjrGARCH (1,1) Std 
6.1778 
6.1611 
GjrGARCH (1,1) Ged 
6.1705 
6.1538 
IGARCH (1,1) Norm 
6.059 
6.0451 
IGARCH (1,1) Std 
6.1793 
6.1682 
IGARCH (1,1) Ged 
6.2187 
6.2076 
EGARCH (1,1) Norm 
6.0564 
6.0425 
EGARCH (1,1) Std 
6.1714 
6.1548 
EGARCH (1,1) Ged 
6.2129 
6.1963 
Akaike information criteria (AIC) and Bayes information criteria (BIC) for GARCH family models for some values of orders p and q. The table shows that the optimal model is obtained by using the principle of parsimony. The principle of parsimony considers GARCH family models with the least optimal AIC. Now using akaike information criterion (AIC), we make a comparison among the different GARCH family models with the three different distribution error. In this case ARMA (0,0)  EGARCH (1,1) model with Student’s t Distribution has least optimal for daily log return UBA stock exchange rate.
Table 8 Estimate of parameter of ARMA (0,0) EGARCH (1,1) model with the student’s t Distribution 

Parameter 
Estimate 
Std
Error 
tvalue 
Pr(>t) 

0.000074 
0.000189 
0.3887 
0.697497 

1.188998 
0.276515 
4.29994 
0.000017 

0.020313 
0.029225 
0.69506 
0.48702 

0.863507 
0.031497 
27.41556 
0 

0.521023 
0.065793 
7.91908 
0 
show that
all the parameters ( of this model are all significant different
from zero at 5% level, except. The parameter of the shape is
significant. The parameters estimate () of ARMA (0,0)EGARCH (1,1) model are
0.000074, 1.188998, 0.020313,0.863507, and 0.51023).
The
results indicate for ARMA (0,0)EGARCH (1, 1) model the leverage effect
term is negative which shows that there is an
existence of the leverage effect in future returns. When,, indicates the news impact is
asymmetric, supporting the use of skewed Studentt distribution for return.
Since (persistence in conditional volatility) is 0.8635072 which is close to 1,
implies that volatility will take long to die in the UBA.
4.6. Model Adequacy (Diagnostic) checking of the estimate model
Table 9 Show LungBox test on standardized residuals of EGARCH (1,1) model with std 

Number
of lags 
Lag1 
Lag2 
Lag4 
Statistic 
18.04 
18.23 
19.03 
pvalue 
2.167× 10^{−05} 
1.294× 10^{−05} 
3.578× 10^{−05} 
The standardized residual test of ARMA (0,0)EGARCH (1,1) with Student t distribution. By looking at Ljung Box test on residual, if the pvalue is less than the chosen level of significance. Table 9 presents the results of LjungBox test. The test fails to reject the null hypothesis and conclude that there is evidence of serial autocorrelation in the standardized s residuals
4.7. Test for heteroscedasticity for fitted model
Table 10 Show Test for ARCH effect of fitted estimate model 

Number of lags 
ARCH lag3 
ARCH lag 5 
ARCH lag 7 

Statistic 
2.881 
3.103 
3.783 

pvalue 
0.08965 
0.27514 
0.37887 
Table 10: present the ARCH LM Tests. Test for null hypothesis, since the pvalues>0.05 and it has been failed to reject the null hypothesis and therefore, we conclude that there is no ARCH effect in ARMA (0,0)EGARCH (1,1) model with Studentt distribution. This confirms that the residuals behave as a white noise process.
4.8. Forecasting
Daily UBA stock exchange price in Nigeria it has been chosen optimal model EGARCH (1,1) model with the std. The Daily UBA log return prices, which include 2225 observations for ten years, out sample data is 200. By rolling forecast method, it has been fixed length of the insample period 2025 observations.
Table 11 Show the out sample of EGARCH (1,1) 

Period 
Series 
Sigma 
T+1 
7.36E05 
0.006802 
T+2 
7.36E05 
0.007418 
T+3 
7.36E05 
0.007995 
T+4 
7.36E05 
0.008528 
T+5 
7.36E05 
0.009018 
T+6 
7.36E05 
0.009463 
T+7 
7.36E05 
0.009865 
T+8 
7.36E05 
0.010226 
T+9 
7.36E05 
0.010548 
T+10 
7.36E05 
0.010835 

Figure 5 
Figure 5 present the forecast plot of the ARMA (0,0)EGARCH (1,1) model. From the forecast shows that there is no sign of fluctuation in forecast plot (ie since all the forecast plots along with the origin). There is no of volatility and clustering
5. CONCLUSION
This study examined the behavior of the UBA daily stock exchange prices in Nigeria for period of 20122021 The data was found to be with fat tail, but not stationary. Non stationarity of the data was detected using the ADF test. The data was transformed to become stationary, after the log data and also the ARCH effect test was carried out using langrage multiplier, there was heteroscedasticity. The GARCH family models were fitted to the daily log return with student tdistribution. The ARMA (0,0)EGARCH (1,1) model showed the best optimal model because it achieves the least AIC. It further indicated volatility clustering and leverage effect in United Bank of African (UBA) stock exchange return in Nigeria for period time. Based on the finding of this study, other time series packages should be compared with GARCH models, data should be making available for easy access and investors should be encouraged to invest in United Bank for Africa (UBA, Nigeria).
REFERENCES
Abdalla, S and Winker, P (2012) modeled and estimated Stock market volatility in Khartoum Stock Exchange KSE and Cairo, Alexandria Stock Exchange, CASE from Egypt.
Akaike, H. (1974). A newlook at statistical model identification. IEEE Transactions on Automatic Control, AC19, 716723 Retrieved from https://doi.org/10.1109/TAC.1974.1100705
Aliyu, S. U. R. (2011). Reactions of stock market to monetary policy shocks during the global financial crisis. The Nigerian Case Munich personal Repec Archive. MPRA paper No. 35581. Available online at. Retrieved from http://mpra.ub.uni.muenchen.de/35581
Bollerslev, T. (1986). Generalised Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31 :307327 Retrieved from https://doi.org/10.1016/03044076(86)900631
Chang, S. (2010). Application of EGARCH Model to Estmate Financial Volatility of Daily Returns : The Empirical Case of China. School of Business, Economics and Law, University of Gothenburg Retrieved from https://gupea.ub.gu.se/handle/2077/22593
Dikko, H.G., Isah. A and Chinyere S. E (2015). Modeling the Impact of Crude Oil Price Shocks on Some macroeconomic Variables in Nigeria Using Generalized Autoregressive Conditional Heteroskedastic and Vector Autoregressive. American Journal of Theoretical and Applied Statistics., 6(4),1223 Retrieved from https://doi.org/10.11648/j.ajtas.20150405.16
Ding, Z., Granger, C.W.and Engle, R.F. (1993). A long memory property of stock market returns and a new model. Journal of empirical finance, 1(1), p.83106. Retrieved from https://doi.org/10.1016/09275398(93)90006D
Engle R.F (1982). "Autoregressive Conditional Heteroskidaticity with Estimates of the Various of UK Inflation" Economical Vol. 50 No. 4 PP. 987 1008. Retrieved from https://doi.org/10.2307/1912773
Foley, S.D., (2014). The impact of regulation on market quality.
Franses, P.H. and Van Dijk, D. (1996). Forecasting Stock Market Volatility Using (Non Linear) GARCH Models. Journal of Forecasting, 15(3), 229235. Retrieved from https://doi.org/10.1002/(SICI)1099131X(199604)15:3<229::AIDFOR620>3.0.CO;23
Glosten, L.R., Jagannathan, R. and Runkle, D.E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5) : 17791801. Retrieved from https://doi.org/10.1111/j.15406261.1993.tb05128.x
Jarque, C. M., and Bera, A. K. (1980). Efficient test for normality, heteroskedasticity and serial independence of regression residuals. Econometric Letters, 6, 255259. Retrieved from https://doi.org/10.1016/01651765(80)900245
Jayasuriya, S. (2002). Does Stock Market Libralisation Affect the Volatility of Stock Returns ? Evidence from Emerging Market Economies. Georgetown University Discussion Series.
Kearns and A.R. Pagan (1991). Journal of Financial and Quantitative Analysis, 25 (1990), pp. 203214. Australian stock market volatility Retrieved from https://doi.org/10.2307/2330824
Kwiatkowski, D., Phillips, P.C., Schmidt, P. and Shin, Y., (1992). Testing the null hypothesis of stationarity against the alternative of a Limit root : How sure are we that economic time series have à unit root ? Journal of econometrics, 54(13), pp.159178. Retrieved from https://doi.org/10.1016/03044076(92)90104Y
Miron, D. and Tudor, C. (2010). Asymmetric Conditional Volatility Models: Empirical Estimation and Comparison of Forecasting Accuracy. Romanian Journal of Economic Forecasting, 13(3), 7492 Retrieved from https://d1wqtxts1xzle7.cloudfront.net/5236676/rjef3_10_4withcoverpagev2.pdf?Expires=1641618253&Signature=Kq97KDAkUFeQf5O6jDs~PEqFvIS7gAJh7Msq66g~auWBztyBGM4wxnOAnufiJqvgb1DXVU7PN9akdcSfs8wfpHivBN2y8xMiL6o39eC1CgLaPngaM8llLn5LLOOUgu2Et57sGBHORciEaMWyQmZeIcJglX1Q4OVsrcwB6HistTP4S0jEGftVQOkR1A2c3W~E2Jh~5Y3TFcTnb0CtGQ3LZuMT7Ug9zLi2lAtcWoY7Q0jn9SCbmxQ29XkHwMSjQap7etpoLHFATexcvT4OYO7GafzTB~MOm5~Km4bw5dBkOmiQqpUHaufIbVrtaNR2NddUIQ72YpQOLIbxdHEWZQ__&KeyPairId=APKAJLOHF5GGSLRBV4ZA
Nelson, D. (1991). "Conditional Heteroskedasticity in Asset Returns: A New Approach", Econometrica, 59(2):347370. Retrieved from https://doi.org/10.2307/2938260
Ogum, G. Beer, F and Nouyrigat, G. (2005). Emerging Equity Market Volatility: An Empirical Investigation of Markets in Kenya and Nigeria. Journal of African Business 6 (1/2) :139 154. Retrieved from https://doi.org/10.1300/J156v06n01_08
Raza, M.A., Arshad, I.A., Ali, N. and Muna War, S. (2015). Empirical analysis of stock returns using symmetric and asymmetric GARH models : evidence from Karachi stock exchange. Science International, 27(1), p.795801.
Schwert, G. W. (1989). Why does market volatility change over time ? Journal of Finance, 44 :11151153. Retrieved from https://doi.org/10.1111/j.15406261.1989.tb02647.x
Sharmiri, A. and Isa Z. (2009). Modeling and forecasting volatility of the Malaysian Stock Markets, 2(3), 234240 Retrieved from https://doi.org/10.3844/jmssp.2009.234.240
Spiro, P. S. (1990). Stock market overreaction to bad news in good times : A rational expectation equilibrium model. Review of Financial Studies 12 :9751007. Retrieved from https://doi.org/10.1093/rfs/12.5.975
Tabajara, P. J., Faliano, G. L. and Luiz, E. G. (2014). Volatility behavior of BRIC capital markets in the 2008 international financial crisis. African Journal of Business management, 8(11), 373381 Retrieved from https://doi.org/10.5897/AJBM2013.7162
Vitor, G. (2015). Sensitivity, persistence and asymmetric effects in international stock market volatility during the global financial crisis. Revista de Metodos Cuantitativos Para la Economiayla Empresa,19, 4265. Retrieved from https://www.econstor.eu/handle/10419/113885
Zakoian, J.M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and control, 18(5) :93 1955. Retrieved from https://doi.org/10.1016/01651889(94)900396
Zhang, H.K. (2016). ValueatRisk estimation of metal market : A GED based TGARCH EVT Approach
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