Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions

Abstract

Symmetric and asymmetric GARCH models-GARCH (1,1); PARCH(1;1); EGARCH(1,1,); TARCH(1,1) and IGARCH(1,1)- were used to examine stylized facts of daily USD/UGX return series from September 1st, 2005 to August 30th, 2018. Modeling and forecasting were performed based on Gaussian, Student’s t and GED distribution densities with a view to identifying the best distribution for examining stylized facts about the volatility of returns. Initial tests of heteroscedasticity (ARCH-LM), autocorrelation and stationarity were carried out to establish specific data requirements before modeling. Results for conditional variance indicated the presence of significant asymmetries, volatility clustering, leptokurtic distribution, and leverage effects. Effectively, PARCH (1,1) under GED distribution provided highly significant results free from serial correlation and ARCH effects, thus revealing the asymmetric responsiveness and persistence to shocks. Forecasting was performed across distributions & assessed based on symmetric lost functions (RMSE, MAE, MAPE & Thiel’s U) and information criteria (AIC, SBC & Loglikelihood). The information criteria offered a preference for EGARCH (1,1) under GED distribution while symmetric lost functions provided very competitive choices with very slight precedence for GARCH (1,1) and EGARCH (1,1) under GED distribution. Following these results, it’s recommended that PARCH (1,1) and EGARCH (1,1) be respectively preferred for modeling and forecasting volatility with GED as the choice distribution. Given the asymmetric responsiveness and persistence of conditional variance, macroeconomic & fiscal adjustments in addition to stabilization of the internal political environment are advised for Uganda. Keywords: Forecasting volatility, GARCH family Models, Probability Distribution Density, Forecast accuracy.JEL Classifications: C58, C53, G17, F31DOI: https://doi.org/10.32479/ijefi.9016

Full text

International Journal of Economics and Financial
Issues
ISSN: 2146-4138
available at http: www.econjournals.com
International Journal of Economics and Financial Issues, 2020, 10(2), 268-281.
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020
268
Modeling and Forecasting USD/UGX Volatility through
GARCH Family Models: Evidence from Gaussian, T and GED
Distributions
Hatice Erkekoglu1, Aweng Peter Majok Garang2*, Adire Simon Deng3
1Department of International Trade and Logistics, Faculty of Applied Sciences, Kayseri University, Kayseri, Turkey, 2Department
of Economics, College of Social and Economic Studies, University of Juba, Juba, South Sudan, 3Department of Accounting and
Finance, School of Management Sciences, University of Juba, Juba, South Sudan. *Email: geologistcamp@aol.com
Received: 24 November 2019 Accepted: 15 February 2020 DOI: https://doi.org/10.32479/ije.9016
ABSTRACT
Symmetric and asymmetric GARCH models-GARCH (1,1), PARCH (1,1), EGARCH (1,1), TARCH (1,1) and IGARCH (1,1) were used to examine
stylized facts of daily USD/UGX return series from September 01, 2005 to August 30, 2018. Modeling and forecasting were performed based on
Gaussian, Student’s t and GED distribution densities to identify the best distribution for examining stylized facts about the volatility of returns. Initial
tests of heteroscedasticity (ARCH-LM), autocorrelation and stationarity were carried out to establish specic data requirements before modeling.
Results for conditional variance indicated the presence of signicant asymmetries, volatility clustering, leptokurtic distribution, and leverage eects.
Eectively, PARCH (1,1) under GED distribution provided highly signicant results free from serial correlation and ARCH eects, thus revealing the
asymmetric responsiveness and persistence to shocks. Forecasting was performed across distributions and assessed based on symmetric lost functions
(RMSE, MAE, MAPE and Thiel’s U) and information criteria (AIC, SBC and Loglikelihood). Information criteria oered preference for EGARCH
(1,1) under GED distribution while symmetric lost functions provided very competitive choices with very slight precedence for GARCH (1,1) and
EGARCH (1,1) under GED distribution. Following these results, we recommend PARCH (1,1) and EGARCH (1,1) for modeling and forecasting
volatility with preference to GED distribution. Given the asymmetric responsiveness and persistence of conditional variance, macroeconomic scal
adjustments in addition to stabilization of the internal political environment are advised for Uganda.
Keywords: Forecasting Volatility, GARCH Family Models, Probability Distribution Density, Forecast Accuracy
JEL Classications: C58, C53, G17, F31
1. INTRODUCTION
International nancial cash ows tend to be hugely aected by
uncertainties due to uctuations in key economic markets such as
foreign exchange and stock markets, which results into the decline
of exports and imports, which in turn aect welfare as suggested
by (Twamugize et al., 2017). As such, understanding volatility
has become very necessary especially now that foreign exchange
markets account for the largest trade volumes and liquidities in
the world.
Inuential attempts aimed at modeling volatility were introduced
into literature through the seminal work of Engle (1982) in which
he proposed conditionalizing variance in an autoregressive
heteroscedastic process by introducing the autoregressive
conditional heteroscedastic (ARCH) model. This model, however,
posed challenges due to huge lag specications. This weakness
motivated Bollerslev (1986) to introduce a generalized form of
ARCH called the GARCH model which improved on the former
by providing exibility to the lag structure. Over time, the literature
on these models has grown to capture stylized facts of nancial
This Journal is licensed under a Creative Commons Attribution 4.0 International License

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020 269
series such as leverage, volatility clustering, fat-fail distributions
(Kipkoech, 2014). Consequently, the following models have been
developed to capture dierent scenarios: Exponential GARCH
to unconditionally model asymmetries (Nelson, 1991); Power
ARCH to model nonlinearities by specifying some restrictions
(Ding et al., 1993); Threshold GARCH to analyze leverage eects
based on news and Integrated GARCH to model non-stationarities.
These models among others form the GARCH family models and
they capture dierent dynamics of nancial series.
The performance of the stated GARCH family models is
unquestionably helpful, however, issues relating to high data
frequency and increased kurtosis posit a levy distribution on return
series (Mandelbrot, 1963). Modeling and forecasting of volatility
should, therefore, consider the distribution of innovations that
should not be assumed to be normal as opposed to conventional
normality assumptions.
In recognition of these inherent gaps, this study makes attempts
at introducing alternative distributions across models to capture
dierent stylized facts of return series through careful selection
of mean variances that consider the potential weakness of
time series models such as the issues of autocorrelations and
non-stationarities. The contributions of this paper to academic
literature are twofold: The rst, to the best our knowledge we
pioneer the application of GARCH family models in Uganda’s
exchange markets, thus augmenting Namugaya et al. (2014) who
focused on stock markets. Following Coe’s (2015) contributions
on distribution densities of innovations and issues relating to
(a)symmetries, this paper provides extensional contributions
through the introduction of PARCH and IGARCH that capture
dierent stylized facts of returns across dierent distributions in
both symmetric and asymmetric frameworks.
A vast literature is available and continues to grow for GARCH
family models that have been used under dierent specications
in various disciplines to analyze volatility and stylized facts
related to forex and stock markets. Musa et al. (2014) examined
the performance of GARCH models using data on Naira/USD
for Nigeria between the periods June 2000 and July 2011. Their
ndings showed that GJR GARCH provides better performances
over other GARCH family models. Also, they found evidence for
the existence of signicant asymmetric eects. Using symmetric
lost functions (MAE, RMAE, MAPE and Thiel’s U), their results
further showed that TGARCH provided accurate forecasts. Omari
et al. (2017) used data on daily returns of KES/USD between 2003
and 2015 to investigate stylized facts about exchange rates in
both symmetric and asymmetric sets of models. They specically
investigated GARCH (1,1) and GARCH-M (1,1) for symmetric
models and EGARCH, GJR-GARCH (1,1) and APARCH (1,1)
for the asymmetric set under dierent distributions. Their results
indicated that APARCH, GJR-GARCH model and EGARCH
models better modeled volatility with t-distribution. Coe (2015)
considers the relative performance of dierent GARCH family
models along with dierent distributions across markets. His
interest was specically modeling and forecasting both asymmetric
and symmetric models in Botswana and Namibia stock markets
using normal, t and GED distributions. His ndings revealed less
persistence of shocks and asymmetry of news in both markets
and the existence of reverse volatility with models with fatter
tails providing better performance. In contrast, Abdullah et al.
(2017) adopts a similar strategy but only for a single market
by analyzing the volatility of the Bangladesh taka against the
USD with a series of daily returns running from January 2008 to
April 2015. By adopting multiple mean equations to overcome
diagnostics problems in GARCH, APARCH, EGARCH,
TGARCH and IGARCH models, their ndings revealed that
student’s t-distribution provided better performance over normal
distribution.
GARCH family models are not limited to forex markets, but
also extend to stock markets in which they are used to model
stylized facts similar to those of forex markets. With a similar
specic focus on Kenya, Maqsood et al. (2017) delved into
the analysis of GARCH family models to model and forecast
volatility by employing daily returns for the Nairobi Securities
Exchange using the NSE 20 share index. His ndings revealed the
persistence of volatility and clustering eect, leverage eect and
asymmetric response to external shocks. It was on such a basis
that he concluded that NSE is an inecient market exhibiting
stylized facts of nancial markets. Ahmed and Suliman (2011)
Undertook similar motivations to model GARCH family models
by applying daily returns of the Khartoum stock exchange (KSE)
from January 2006 to November 2010. By considering both
symmetric and asymmetric models, their empirical results revealed
that conditional variance is highly persistent (explosive process)
and provides evidence on the existence of risk premium for the
KSE index return series which supports the positive correlation
hypothesis between volatility and the expected stock returns.
Besides, they found that asymmetric models provide a better t
than the symmetric models, conrming the existence of leverage
eect.
Alternative modern approaches have been advanced to model
volatility. This includes the use of neural networks and ARIMA
that predate GARCH family models. For instance, Ou and Wang
(2011) aimed at modeling and predicting nancial volatility, but
on a Gaussian probabilistic process based on GARCH, EGARCH
and GJR-GARCH models by training dierent kernels to train
each of the models. Their ndings revealed the prowess of hybrid
models in capturing symmetric and asymmetric eects of news on
volatility than the classic GARCH, EGARCH, and GJR GARCH
methods. In similar veins, Kipkoech (2014) examined the volatility
of the Kenyan shillings (KES) against the United States dollars
to analyze the predictive performance of EGARCH models by
comparing two distributions: Gaussian and Student’s. He used
the maximum likelihood estimator and his results revealed that
student’s distribution provides better performance over other
specications of EGARCH models.
Based on a comprehensive review of the literature relating to
modeling and forecasting of exchange rates and GARCH models
especially in East Africa and Uganda particularly, it’s evident that
there exists a very huge gap. To the best of our knowledge, there
have been no absolute attempts to model exchange rates using
GARCH under any specication. Limited close cases include Etuk

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020
270
and Natamba (2015) who applied ARIMA to forecast exchange
rates and recommended the adoption of the SARIMA (0,1,1)
in modeling and forecasting based on the UGX/USD exchange
rates in Uganda between August 2014 and February 2015.
Namugaya et al. (2014) used daily closing prices between January
2015. They estimated both symmetric and asymmetric GARCH
models to examine specific stylized facts such as volatility
clustering and leverage. They employed a quasi maximum
likelihood method for estimation and AIC and BIC for model
selection. Their results indicated that the GARCH (1,1) model
outperformed the other competing models in modeling volatility
while EGARCH (1,1) performed best in forecasting volatility of
USE returns based on MAE and MSE.
The next sections present the models and the materials. Results,
discussions, and conclusions are presented in the last three sections
of the paper.
2. MATERIALS AND METHODS
This section discusses data, mean equation specication, the
GARCH family models, distributions densities and forecast
evaluation methodologies.
2.1. Data
The data used in this study consists of daily foreign exchange rate
series of the US Dollar (USD) against the Uganda shillings (UGX)
ranging from September 05, 2005 to August 30, 2018, constituting
a total of 3738 observations. The data was sourced from the ocial
website of the Bank of Uganda under the statistics section (www.
bou.or.ug/bou/rates_statistics/statistics.html).
Data transformation was performed to obtain log-returns of the
exchange rates series to overcome the diculties of modeling
with non-stationary data in time series. The following formula
was used to obtain log returns:
rlog USDUGX
USDUGX
t=−




×
/
/()1 100
Where USD/UGX is the daily observation for the USD against
UGX while USD/UGX (−1) is the lag of the same on day t. Table 1
in the appendices illustrates the line plots of the two series. It can
be observed that the USD/UGX series has a trend suggesting
that it’s nonstationary while that of the return series reverts to its
constant mean reecting that it’s a stationary process. Additionally,
volatility clustering can be deduced from the plot of the return
series since it’s easily observable that periods of low volatility
are followed by periods of low volatility while those of higher
volatility are followed by the same over a lengthy period. Figure 1
is a plot of the return series at both levels and rst dierence.
2.2. Mean Equation Specication
To address autocorrelation problems in the models, we rst
tested four mean equations starting from the constant to AR (4)
models to ascertain appropriate models. Two models (the constant
and AR [1]) were signicant and, therefore, we used these two
specications to model various GARCH family models. The two
models take the following functional expressions.
Constant Mean equation: rt=µ+ԑi
AR (1) Mean equation:
 
∑q
t i-1 i
i=1
r = + r +
According to Alexander (1961) and Andersen and Bollerslev
(1998) the variance was modeled for the above two models on
dierent GARCH models to test dierent issues across three
dierent distributions to observe various sensitivities based on
dierent tail distributions and kurtosis assumptions of nancial
series. Variance equation is given by:
tt
tt
iid
hv wherev=,~(
,)
01
2.3. The GARCH Family Models
Volatility is a crucial element of investment whose understanding
has very important implications for an economy that aspires to
grow. Earlier methods for modeling volatility were always focused
on variance and standard deviation, while ignoring conditionality
and very important aspects of nancial time series data such as
leverage eects, heavy-tailed distribution, and volatility among
others. In response to these shortcomings, Engle (1982) presented
a time-varying model that conditionalizes variances of past
innovations called the ARCH. This idea was further improved by
Bollerslev (1986) by adding past conditional variance to overcome
the huge lags specication of the ARCH model. This model was
named generalized ARCH (GARCH). Other models have since
been developed to model symmetries and asymmetries to all form
the GARCH family discussed below.
2.3.1. ARCH
This model is attributed to Engle (1982) based on a seminal
work in which he suggested that time-varying conditional
heteroscedasticity be modeled by applying past innovations to
estimate variance as follows:
h
t
i
q
it
=+
=
−
∑
ηα
ε
0
1
1
2
where 
()
t−1
2 represents the ARCH 1 process.
2.3.2. Generalized ARCH
Bollerslev (1986) advanced the ideas of Engle (1982) by suggesting
a generalized form of ARCH which overcomes the diculty of
huge lags specications. Precisely, he proposed a model in which
heteroscedasticity is determined by past innovations and past
conditional variance as a set of regressors represented as a higher-
order ARCH represented as follows:
hh
t
i
q
it
j
p
ii j
=+ +
=
−
=
−
∑∑
ηα
εβ
0
1
1
2
1
Table 1: Summary statistics for the USD/UGX returns series
Observations 3238
Mean −0.000225
Std. Dev. 0.004874
Skewness 0.362888
Kurtosis 20.79384
Jarque-bera 42788.50***
Source: Authors’ calculations

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020 271
where
()
t−1
2 is the ARCH term while is the GARCH term with
restrictions imposed as ղ0>0, αi≥0 and βi≥0. The sum of ARCH
and GARCH coecients determines the persistence of shocks
αi+β<1 to ensure that εt is stationary with positive variance.
2.3.3. Exponential GARCH
This is an exponential model developed by Nelson (1991) to
model asymmetric tendencies in volatility. This model relaxes the
non-negativity constraint restrictions placed on Alpha and Beta
in the GARCH model. This model takes conditional variance as
a function of lagged innovations as illustrated below:
ln lnhhh h
t
i
q
i
t
ti
q
i
t
ti
q
i
()
=+ +





+
=
−
−=
−
−=
∑∑ ∑
ηα
ελεβ
0
1
1
11
1
11
tt −
()
1
In this model, the leverage eect is exponential irrespective of
the sign on the coecients. λi<0 stands for negative parameters
implying negativity shocks will have higher eects on expected
volatility than positive shocks of the same magnitude. Here, λi, αi
and βi represent leverage eect, shocks magnitude and persistence,
respectively. Nelson and Cao (1992) suggest that this model gives
freedom to positive and negative shocks to determine volatility
and lets large shocks have a superior inuence on volatility.
2.3.4. Threshold GARCH (TGARCH)
This model is also known as the GJR-GARCH model named after
Glosten et al. (1993). The model introduces the aspects of good news
and bad news with dierent eects on the conditional variance.
The model is just an augmentation of the standard GARCH with
additional ARCH term conditional on past disturbances.
ln
()hI
h
t
i
q
it it t
j
p
ii j
()
=+
++
=
−−−
=
−
∑∑
ηαελ
εβ
0
1
1
2
1
2
1
1
λi measures leverage eects and I is a dummy equal to 1 where ԑi is
negative. The good news is ԑt–i>0 and bad news is ԑt–i>0 good news
inuences conditional variance by αi while bad news inuences
conditional variance by αi+λi When λi>0 bad news increases
volatility and it implies an increase in leverage eects. Glosten
et al. (1993) notes that if λi≠0 the bad news impact is asymmetric.
2.3.5. Power GARCH (PARCH) model
This model was developed by Ding et al. (1993) to model
nonlinearities by asymmetric power ARCH of (p, q) order
presented as follows:
hh
t
i
q
itiiti
j
p
it j
δδ
δ
ααελ
εβ
=+ −+
=
−−
=
−
∑∑
0
11
()
where αi βi are where αi and βi are the standard ARCH and GARCH
parameters. λi and δ represent leverage eects and power terms,
respectively. Restrictions here are that δ>0 and
| |1≤
i
.
2.3.6. Integrated GARCH (IGARCH)
This is a special form of GARCH developed to deal with series that
have a unit root. It was rst introduced by Engle and Bollerslev
(1986). The model integrates the series to achieve stationarity.
The parameters of the GARCH are restricted to a sum equal to
1 and the constant is ignored to transform a standard GARCH
model into IGARCH.
hh
ti
tiij
=+−
−−
αε α
1
21
()
Here, additional constraints are {α+(1–αi)}=1 and 0<αi<1.
2.4. Distributions Densities
In modeling GARCH family models, variance is assumed to
be stochastic although there is variance in GARCH. GARCH
structures generate heavy-tailed outputs even for returns.
Therefore, leptokurtic returns can be compatible with normal
standardized errors. This study, therefore, considers these
distributions associated with the GARCH family.
2.4.1. The Gaussian distribution
The Gaussian distribution is also known as the normal distribution
and best represented as follows:
L= ln 2+ln +Z
gaussian t
2
t
-12
([
])πσ
[]
=
∑
i
T
1
where T is the number of observations, σ is the standard deviation
and π is the constant pi.
2.4.2. The student’s t-distribution
This is also called the T distribution and it’s almost identical to
the normal distribution curve, only that it’s a bit shorter and fatter.
Under this distribution, the log-likelihood is computed as follows;
Student's =lnv+ -lnv-lnv-
-ln+
T
t
ΓΓ
1
22
12
(2
12(2











[]
π
σ111
2
2
+v ln +z
v-
t
i=1
T
[]








∑
2.4.3. The generalized error distribution
This is a generalized form of the normal distribution that has
a natural multivariate form with an unbounded top parametric
kurtosis. It has cases similar to the Gaussian and character which
controls for kurtosis. This can be represented as follows:
L=
ln vZ-+v
ln -ln1
v
GEDv
t
v
v
(-0.5 1
2 0.5
-1
λλ
















Γlln t
i=1
T
σ2)




∑
Where
λ
v=−
Γ
Γ
()
()
/
1
2
3
2
v
v
v
2.5. Forecast Evaluation Methodologies
Forecast accuracy in this study is evaluated based on the mean
square error (MSE), root mean square error (RMSE), mean
absolute percentage error (MAPE), Thiel’s U1 and Thiel’s U2
statistics and models are selected based on information criteria.
2.5.1. MAE
It measures the deviation from the original values. The closer the
MAE to zero, the better the goodness of t and thus the better the
forecast. MAE is represented by the following equation.

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020
272
MAEne
t
n
t
=
=
∑
1
1
||
2.5.2. MSE
The closer the MSE of a model to zero, the preferable the model.
MSE takes the following representation:
MSEne
t
n
t
=
=
∑
1
1
2
2.5.3. RMSE
This has similar interpretations as the MSE in the choice of the
most preferable model.
MSE
2.5.4. MAPE
This criterion is a relative measure of MAE which provides
relatives performances of dierent forecast items. The lower the
percentage MAPE, the better the forecast model.
MAEn
e
y
t
t
t
n
=×
=
∑
1100
1
|| %
2.5.5. Theil’s U
Thiel’s U is a normalized measure of forecast accuracy. There are
two types of this Thiel’s U. The rst (U1) is a measure of forecast
accuracy (Theil, 1958. p. 31-42); The second (U2) is a measure
of forecast quality.
U
PA
A
i
n
ii
i
n
i
1
1
212
1
212
=−
=
=
∑∑
[( )]
[]
/
/
U
nAP
n
A
n
P
i
n
ii
i
n
ii
n
i
2
1
2
1
2
1
2
1
2
1
2
1
2
1
11
=−
(


+
=
==
∑
∑∑
[)
[]
[]
Where Ai represents the actual observations and Pi is the
corresponding predictions for the case of U1 whereas in Thiel
U2 they represent proposed U2 a pair of predicted and observed
changes. Perfect forecasts are those with U1 closer to the 0
bound while worst forecasts are those closer to 1. U2 can be
interpreted as the RMSE of the proposed forecasting model
divided by the RMSE of a no-change model. U2 values lower
than 1.0 show an improvement over the simple no-change
forecast.
3. RESULTS
This section presents the description of the data and results
obtained from the estimation of the various GARCH models.
3.1. Descriptive Statistics of the USD/UGX Return
Series
Descriptive statistics of the series provide a general glimpse into
the behavior of the return series. The following summary statistics
are obtained from the series.
Table 1 above provides descriptive statistics for the USD/UGX
return series from which a negative mean can be observed,
reflecting the depreciation of the exchange rates over time.
The returns exhibit a non-symmetric distribution since they are
positively skewed thus indicating a depreciation of the Shillings.
A look at the coecient of kurtosis is suggestive of the leptokurtic
(fail-tail) distribution of the returns series. The series doesn’t also
follow a normal distribution as justied by a Jarqua-Bera test
which rejects normality assumption at a 1% level of signicance.
The standard deviation is positive, indicative of high uctuations
in the exchange market.
A further graphical examination of the distribution characteristics
of the return series using Q-Q and histogram as in Figure 2 in the
appendices shows that observations are scattered in an S-shape
pattern far away from the 45-degree line conrming Ahmed and
Suliman (2011) who argue that such a scatter pattern is evidence
for non-normal distribution in series, which is also conrmed by
the histogram below the Q-Q plot. An examination of the incidence
of serial correlation shows that the series is autocorrelated except
at the rst and second lags.
3.2. Stationarity
Unit root analysis is a prerequisite in time series modeling and as
such, preliminary unit root tests were examined on the USD/UGX
series. The series exhibited trend just from the basic graphical
examination and we couldn’t reject the null hypothesis from the
ADF test of unit root implying that the series wasn’t stationary.
Following these results, we obtained returns from the USD/UGX
through a log transformation of the rst lag in a process explained
above. Displayed below in Table 2 are results of the ADF unit
root tests.
3.3. Tests for Heteroscedasticity: ARCH Eects
Modeling GARCH models requires securing certainty over the
presence or absence of the ARCH eects. This can be eectively
executed through the Lagrange multiplier test for ARCH (Engle,
1982). The LM proposes that given ԑt=rt–µ as the residual for the
mean equation
t
2
is then used to test conditional heteroscedasticity,
known as the ARCH-LM effect. The AR (1) model for the
conditional mean was estimated and the ARCH-LM test was done
for the rst lag. From the results of the test in Table 3 below, the
null hypothesis of No ARCH eects is rejected at 95% condence
interval. This indicates the presence of ARCH eects implying
that there is uctuating variance in the return series.
Table 2: Unit roots for the return series
Tests at dierent
levels
ADF test PP test
t-stat. P-value Lag t-stat. P-value Lag
Intercept −38.04378 0.0000*** 1−37.91004 0.0000*** 1
Intercept and trend −38.04626 0.0000*** 1−37.90714 0.0000*** 1
None −37.96551 0.0000*** 1−38.15968 0.0000*** 1
Source: Authors’ calculations

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020 273
3.4. Results of Estimation of GARCH Models
The results reported in Tables 4-6 below are estimates of
intercept, variance equations and diagnostics of both symmetric
and asymmetric GARCH models under Gaussian, Student’s T
and GED distributions. The tables report estimations for the two
mean equations: The intercept (1) and the AR (1). Table 7 above
presents the volatility forecast performance of the models across
distributions on the basis of information criteria using the whole
sample. Additionally, Table 8, shows the forecast accuracy of
competing models examined using out-sample data between the
periods September 01, 2005 and June 029, 2018.
4. DISCUSSIONS
Having established the presence of volatility from the ARCH-LM
test, initial GARCH models were estimated to model variance
dynamics and asymmetric eects of volatility through GARCH
(1,1) and PARCH (1,1), respectively under normal distribution,
as reported in Table 4. The results of GARCH (1,1) in the mean
equation indicate that the intercept µ for both equations is
signicant at 1%. The AR (1) coecient of the dependent variable
(φ) is also signicant at 1%. Volatility equation for the GARCH
(1,1) reveals that the constant (η), ARCH (α) and GARCH (β)
terms are all positive and statistically signicant at 1%, with the
sums of α and β exceeding 1, implying indenite variance due
to non-stationarity in the residuals. Diagnostic tests for serial
correlation using the Ljung-Box Q-statistics for standardized
residuals (Q1) and their squared values (Q2) on the 4th and 8th lags
as advanced by Tse (1998) revealed the presence of autocorrelation
as Q1 statistics were signicant at 5% level.
Although the AR (1) equation provides evidence of no ARCH
eects, the F-statistic for rst the mean equation was signicant at
5% justifying the presence of ARCH eects. Given the shortcomings
of the Gaussian distribution manifested by skewness and excess
kurtosis of return series as shown in Figure 2 in the appendices,
other distribution assumptions were tested. Specically, student’s
t and GED were examined following Bollerslev (1987). The
results under normal distribution are similar to the results under
the assumptions of T and GED distributions except that the ARCH
eect is eliminated under the two distributions. However, the
problem of autocorrelation is persistent as reported in the GARCH
(1,1) models in Tables 5 and 6.
The results of PARCH (1,1) tabulated in Table 4 indicate that
AR (1) is positive and signicant and so are ARCH and GARCH
coecients at 1% levels which reect the responsiveness and
persistence to shocks, respectively. In this model, α captures the
response of conditional volatility to appreciation or depreciation
of the Ugandan shillings while β captures its persistence to market
shocks. Since their sum goes beyond 1, evidence of an innite
nonstationary variance is established. The leverage coecient (λ)
is positive and signicant implying a higher inuence of negative
past innovations on volatility than positive values of the similar
magnitudes on conditional variance. The results of a negative
coecient on leverage would produce interpretations that are
vice versa to the ones realized above. These results established
that appreciation or depreciation of the Ugandan shillings
against the dollar doesn’t have a directional eect on volatility.
However, the model is not a standard GARCH model given the
positive value and signicance of δ, which is <2 in the model,
Ding et al. (1993). The statistics for the diagnostics indicate the
presence of autocorrelation and the absence of ARCH eects as
shown by signicant Q1 statistics and insignicant F-statistic.
Similar ndings are realized when the distribution is changed
to student’s T or GED under PARCH (1,1) in Tables 5 and 6
respectively. Under T distribution, ARCH eects are no longer
a problem although serial autocorrelation continues to aect the
model as manifested by highly signicant Q statistics. However,
better results are observed under the GED assumption, in which
both autocorrelation and ARCH eects are eliminated in the AR
(1) mean equation.
Under the EGARCH (1,1) model in Table 4 with the assumption
of the normal distribution, the AR (1) coecient is positive and
signicant just like α and β coecients which are the asymmetric
and size parameters, respectively. This model considers
nonnegative restrictions placed on variance. The asymmetric term
(λ) is negative and signicant at 5% revealing the presence of
asymmetric eects implying that negative shocks to the Ugandan
Shillings will have a higher inuence on expected volatility than
positive shocks of the same magnitude. The coecients for the
intercept and AR (1) mean equations are signicant at 1%. There
are no ARCH eects although there is evidence of autocorrelation
in the residuals as indicated by very signicant Q statistics.
A change of distribution to student’s t for the EGARCH (1,1)
as reported in Table 5 provides no improvement either in terms
of diagnostics, although both mean, and variance coecients
remain similar. However, a further change of distribution to GED
in Table 6 deteriorates the model. Both the ARCH and GARCH
parameters remain positive and signicant, but the asymmetric
parameter becomes insignicant, although it’s negative.
Starting with the Gaussian distribution assumption in Table 4,
the results of TARCH (1,1) provide a highly signicant level
for the AR (1) coecient in the mean equation. In the variance
equation, The ARCH (α) and GARCH (β) coecients are
also largely significant. The asymmetric parameter (λ) is
negative and signicant with no evidence of ARCH eects
Table 3: Estimation of dierent conditional means and
testing ARCH eect
Variable (1) (2)
Dependent variables Return c Returns c returns (−1)
C−0.000225*** −0.000142**
(8.57E-05) (8.01E-05)
AR (1) 0.357624***
(0.016414)
ARCH eects
Constant 1.70E-05*** 1.66E-05***
(1.83E-06) (1.71E-06)
Lag error squared 0.282662*** 0.199036***
(0.016865) (0.017233)
Ho: No. Arch eect
F-stat. 280.9180 133.4015
Prob. 0.0000 0.0000
Source: Authors’ calculation

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020
274
Table 4: Estimations of GARCH family models under Gaussian distribution
Parameters GARCH EGARCH IGARCH TARCH PARCH
1 2 1 2 1 2 1 2 1 2
Mean equation
μ−0.000102*** −8.43E-05*** −5.66E-05*** −7.38E-05*** −0.000242*** −0.000174*** −0.000112*** −0.000102*** −5.97E-05 7.92E-05***
3.05E-05 2.60E-05 2.44E-05 −2.62E-05 1.23E-05 1.04E-05 3.65E-05 3.15E-05 3.02E-05 3.00E-05
φ0.398706*** 0.400208*** 0.392804*** 0.400919*** 0.395934***
6.01E-09 0.013949 0.010348 0.014245 0.013161
Variance equation
η1.29E-07*** 4.69E-08*** −0.950869*** −0.835380*** 1.28E-07*** 4.37E-08*** 3.08E-05*** 3.03E-05***
9.54E-09 0.014443 0.038413 0.036404 9.58E-09 5.94E-09 1.27E-05 1.44E-05
α0.376879*** 0.315093*** 0.536711*** 0.506496*** 0.139184*** 0.127316*** 0.358307*** 0.271238*** 0.337945*** 0.284962***
0.011893 0.007609 0.012467 0.010119 0.002021 0.002012 0.018045 0.012732 0.012073 0.008936
λ −0.015890** −0.031287*** 0.030578 0.066002*** 0.026995** 0.068309***
0.008182 (0.008239) 0.022196 0.018015 0.016621 0.018716
δ0.747058* 0.794211***
0.007401 0.005686
β0.724456*** 0.775484*** 0.949741*** 0.958378*** 0.725986*** 0.781830*** 1.207540*** 1.115557***
0.006340 0.003784 0.002805 0.002758 0.006539 0.003931 0.061608 0.070556
1-α 0.860816*** 0.872684***
0.002021 0.002012
Diagnostics
Q1(4) 422.59*** 47.076*** 414.90*** 41.671*** 545.45*** 64.827*** 423.19*** 46.648*** 419.05 44.432***
Q2(8) 458.16*** 62.287*** 445.94*** 54.156*** 582.84*** 79.819*** 448.42*** 61.184*** 452.53 57.858***
Q1(4) 8.4613*** 4.0447 9.3195** 5.6006 39.314*** 16.786*** 1.0063 4.4682 10.852 7.3788
Q2(8) 12.989*** 9.2877 13.705*** 11.390 42.779*** 19.498*** 1.8773 9.9280 14.256 12.202
LLK 14021.22 14221.47 14058.89 14265.72 14058.89 14086.35 195.4121 14224.45 14036.95 14239.77
F stat. 3.211059 1.962559 4.450274 2.421528 39.04113 14.85014 0.019050 2.055045 0.944095 4.804863
Prob. 0.0732 0.161336 0.0350 0.119776 0.0000 0.000119 0.8909 0.151799 0.336790 0.028451
1 and 2 represent the rst and second mean equations, respectively. On the other hand, ***, ** and * indicate signicance at 1%, 5% and 10% levels, respectively

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020 275
Table 5: Estimations of GARCH family models under student’s T distribution
Parameters GARCH EGARCH IGARCH TARCH PARCH
1 2 1 2 1 2 1 2 1 2
Mean equation
μ–1.91E-05 –1.58E-05 −2.33E-05 2.78E-05 −9.25E-06 1.71E-05 −2.10E-05 −2.08E-05 −2.47E-05*** −2.31E-05***
2.03E-05 1.88E-05 2.27E-05 2.09E-05 4.48E-05 1.57E-05 2.08E-05 1.91E-05 2.23E-05 2.01E-05
φ0.393164*** 0.393164*** 0.397789*** 0.394403*** 0.390179***
0.015968 0.015968 0.013094 0.015875 0.015591
Variance equation
η3.37E-08*** 1.73E-08*** −1.031161*** −0.782068*** 3.33E-08*** 1.65E-08*** 8.63E-05 2.03E-06
1.17E-08 7.12E-09 0.082024 0.067923 1.16E-08 6.94E-09 6.58E-05 2.39E-06
α0.780954*** 0.620822*** 0.735059*** 0.640842*** −0.000248*** 0.179547*** 0.708341*** 0.511020*** 0.492357*** 0.470335***
0.110541 0.093901 0.049095 0.048374 1.42E-05 0.007095 0.107410 0.085020 0.048577 0.058621
λ −0.039806** −0.053556*** 0.111542 0.154882** 0.050256 0.076312***
0.021815 0.022998 0.081354 0.076833 0.032041 0.035901
δ0.697083*** 0.749793***
0.015599 0.015120
β0.644117*** 0.708269*** 0.950712*** 0.966872*** 0.647639*** 0.716939*** 1.001369*** 1.395130***
0.014521 0.014091 0.006303 0.005097 0.014484 0.013931 0.110981 0.159095
1-α 1.000248*** 0.820453***
1.42E-05 0.0070595
Diagnostics
Q1(4) 353.88*** 50.866*** 370.65*** 370.65*** 438.72*** 61.333*** 353.84*** 49.997*** 361.75*** 50.895***
Q2(8) 385.45*** 65.202*** 399.51*** 399.51*** 449.29*** 75.331*** 384.90*** 63.469*** 392.62*** 64.126***
Q1(4) 2.8225 2.8428 6.0577*** 45.706*** 659.70*** 8.3659** 2.5957 2.8632 3.0180 2.8503
Q2(8) 4.3051 6.3962 9.9458*** 57.915*** 822.79*** 14.316** 3.9807 6.5203 4.4438 6.6199
LLK 14337.35 14569.37 14351.29 14580.91 13583.71 14502.41 14338.39 14571.83 14348.21 14574.99
F stat. 0.001667 0.006002 0.184422 0.370255 244.6195 7.343403 0.001469 0.015467 0.400793 0.292373
Prob. 0.9674 0.9383 0.667629 0.5428 0.000000 0.0068 0.969424 0.901032 0.526725 0.588741
Source: Authors’ calculations. 1 and 2 represent the rst and second mean equations, respectively. On the other hand, ***, ** and * indicate signicance at 1%, 5% and 10% levels, respectively

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020
276
Table 6: Estimations of GARCH family models under GED distribution
Parameters GARCH EGARCH IGARCH TGARCH PARCH
1 2 1 2 1 2 1 2 1 2
Mean equation
μ −1.11E-11 −9.41E-07 −2.75E-06 −3.22E-06 −2.78E-06 −3.80E-06 −2.75E-06 −3.22E-06 2.23E-15 −3.21E-06
1.31E-05 1.53E-05 1.65E-05 1.67E-05 1.33E-05 1.19E-05 1.56E-05 1.55E-05 1.22E-05 1.61E
φ0.386534*** 0.385706*** −0.388213*** 0.385531*** 0.384398***
0.014026 0.014018 0.011831 0.014002 0.013750
Variance equation
η3.83E-08*** 2.16E-08*** −1.027212*** −0.826357*** 3.81E-08*** 2.01E-08 5.12E-05 8.12E-06
1.10E-08 6.66E-09 0.087902 0.073586 1.09E-08 6.48E-09 4.48E-05 9.10E-06
α0.556302*** 0427962*** 0.660658*** 0.569740*** 0.180368*** 0.160132*** 0.520853*** 0.369534 0.423816*** 0.357691***
0.046206 0.033710 0.034928 0.030372 0.006460 0.006178 0.056031 0.041026 0.035286 0.027671
λ −0.026899 −0.035270*** 0.056365 0.086660*** 0.036938 0.061937
0.021629 0.020794 0.064529 0.053602 0.034860 0.038192***
δ0.710847*** 0.762656***
0.016632 0.013875
β0.664658*** 0.723195*** 0.950623*** 0.963362*** 0.666799*** 0.729562*** 1.068744*** 1.230320
0.015506 0.012996 0.006528 0.005386 0.015518 0.012811 0.125338 0.153480
1-α 0.819632*** 0.839868***
0.006460 0.006178
Diagnostics
Q1(4) 372.15*** 52.882*** 377.32*** 48.237*** 508.72*** 67.175*** 371.93*** 53.137*** 379.42 51.792***
Q2(8) 404.83*** 68.026*** 406.66*** 61.066** 544.15*** 82.016*** 404.12*** 67.547*** 411.20 65.594***
Q1(4) 0.458 3.0450 6.5281 4.9009 24.370*** 11.387*** 3.4715 3.1485 4.2372 3.5061
Q2(8) 0.680 7.1814 10.793 11.595 30.984*** 16.513*** 5.4727 7.3877 6.2422 7.7638
LLK 14367.00 14573.01 14382.05 14588.06 14293.68 14513.56 14367.45 14574.53 14376.80 14580.39
F stat. 0.054687 0.067899 0.400706 0.504495 22.25495 10.59354 0.052041 0.093782 0.925759 0.860690
Prob. 0.815115 0.7944 0.526769 0.477583 0.000002 0.0012 0.8195 0.7594 0.336039 0.3536
Source: Authors’ calculations. 1 and 2 represent the rst and second mean equations, respectively. On the other hand, ***, ** and * indicate signicance at 1%, 5% and 10% levels, respectively

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020 277
further establishing the evidence for asymmetric eects. The
model, however, suers from autocorrelation. Unlike results
from the Gaussian assumption, the asymmetric parameter is
insignicant in the AR (1) type mean equation, the parameter
is positive under both mean equations although diagnostics
don’t support the model when distribution assumption is made
as a student’s t or GED as can been seen in the TARCH (1,1)
tabulated in Tables 5 and 6 respectively. No ARCH eects, but
autocorrelation persists. Variance also continues not to be well
behaved given the fact that the persistence parameters are still
higher than the unitary value.
The autoregressive coecient for the lag dependent variable under
IGARCH (1,1) indicates a positive and signicant parameter φ
from the results under normal distribution in Table 4. This model
imposes the persistence parameter, to sum up to unit restrictions on
the model. Here, the restrictions placed on the model are positive
and signicant. However, they don’t overcome the diagnostics
problems of autocorrelation and ARCH eects established by
signicant Q and F statistics. Alteration of distributions doesn’t
change anything either since similar results are obtained under
both T and GED distributions as observed from the IGARCH
(1,1) results in Tables 5 and 6 respectively.
Table 7 above presents the volatility forecast performance of the
models across distributions compared based on information criteria
using the whole sample. Log-likelihood (LLK), Schwartz Bayesian
(SBC) and Akaike information criteria (AIC) are used. It can be
observed that although student’s t distribution out-performed
Gaussian, the general performance of the models improved when
GED distribution was used since the LLK increased while AIC
and SBC decreased. Specically, an examination of the models
under GED distribution reveals that EGARCH (1,1) provided the
best t among the models used.
Forecast accuracy of competing models was examined using
out-sample data between the periods September 01, 2005 and
June 29, 2018 as presented in Table 8 and Figure 3. The models
were estimated and used to make a month ahead forecast of
exchange rates starting from July 02, 2018 to August 30, 2018.
The performance was compared based on symmetric lost functions
which include: RMSE, MAE and Thiel’s inequality. A comparison
was made across distributions and as it can be observed, similar
observations to comparisons based on symmetric lost functions
can be made. Better forecasts are realized when student’s t and
GED are used. However, GED provides much better results with
lesser values of RMSE, MAPE and Thiel’s U. All models produced
very slightly superior performances overall in dierent criteria.
However, overly; EGARCH (1,1) and GARCH (1,1) have shown
plausible consistency in minimizing RMSE, MAE and providing
better t based on Thiel’s U.
5. CONCLUSION
This study drew motivations from the knowledge that volatility has
important economic implications for international investment, risk
management, remittances, and stock pricing among others. This
is particularly relevant for developing economies such as Uganda
seeking to expand their share of international trade and realize
stable trade balances. We model volatility using GARCH family
models by taking into consideration the assumptions on distribution
Table 8: Forecast evaluation for GARCH family models
based symmetric lost functions
Measures
of
Accuracy
GARCH PARCH EGARCH TARCH IGARCH
Normal distribution
RMSE 0.004299 0.004302 0.004290 0.004302 0.004327
MAE 0.002973 0.002976 0.002964 0.002976 0.003014
MAPE 95.97247 95.51312 97.78375 95.51312 101.4370
Theil’s U1 0.979852 0.977650 0.988710 0.977650 0.957424
Thiel’s U2 1.008709 1.009915 1.004672 1.009915 1.025187
Student’s T distribution
RMSE 0.004283 0.004283 0.004283 0.004283 0.004281
MAE 0.002956 0.002957 0.002957 0.002957 0.002954
MAPE 99.23984 99.07129 99.07102 99.15368 99.64783
Theil’s U1 0.996071 0.995208 0.995307 0.995629 0.998172
Thiel’s U2 1.002243 1.002487 1.002487 1.002366 1.001694
GED distribution
RMSE 0.004279 0.004280 0.004280 0.004280 0.004280
MAE 0.002953 0.002953 0.002953 0.002953 0.002956
MAPE 100.0000 99.89787 99.89788 99.89788 99.89697
Theil’s U1 1.000000 0.999469 0.999469 0.999469 0.999464
Thiel’s U2 1.001267 1.001387 1.001387 1.001387 1.001388
Source: Author’s calculations
Table 7: Forecast models evaluation based on information criteria
Criteria GARCH TARCH EGARCH IGARCH PARCH
Normal distribution
LLK 14221.47 14224.45 14265.72 14086.35 14239.77
AIC −8.783735 −8.784954 −8.810456 −8.701483 −8.79380
SBC −8.774340 −8.773680 −8.799182 −8.695846 −8.780649
Student t-distribution
LLK 14569.37 14571.83 14580.91 14502.41 14574.99
AIC −8.998067 −8.998968 −9.004578 −8.957931 −9.000301
SBC −8.986793 −8.985815 −8.991425 −8.950415 −8.985269
GED distribution
LLK 14573.01 14574.53 14588.06 14513.56 14580.39
AIC −9.000316 −9.000634 −9.008998 −8.964817 −9.003637
SBC −8.989042 −8.987481 −8.995844 −8.957301 −8.988605
Source: Authors’ calculations

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020
278
densities and making forecasts to determine appropriate models.
Findings established various stylized facts of returns such as
volatility clustering, leverage and leptokurtic nature of the series.
Specically, although all models showed phenomenal performance
in establishing symmetries (GARCH [1,1] and asymmetries
PARCH [1,1], EGARCH [1,1]), TGARCH (1,1); only PARCH
(1,1) under the student’s t distribution assumption was able to
overcome diagnostics issues of autocorrelation and ARCH eects.
Since the leverage coecient (λ) was positive and signicant in
PARCH (1,1), it could easily be established that negative past
innovations inuenced volatility than positive values of similar
magnitudes. This implies that the appreciation or depreciation of the
Ugandan shillings against the dollar didn’t have directional eect
on volatility. Coecients capturing responsiveness and persistence
provided evidence that the Ugandan shillings was responsive to
shocks and that conditional variance is most likely to persist. This
is demonstrated by conditional variance in Figure 4. In modeling
volatility, student’s t distribution provided better performance
since loglikelihood increased compared to Gaussian and GED
distributions. On the other hand, forecasting was done under
two strands: the entire sample and in-sample strands evaluated
based on information criteria and symmetric lost function criteria
respectively. EGARCH (1,1) provided the best goodness of t
when AIC, SBC, and LLK were used because it produced smaller
AIC and SBC values and higher LLK values compared to other
models. All models were very competitive when symmetric lost
functions were used, however, GARCH (1,1) and EGARCH (1,1)
produced slightly better results based on RMSE, MAPE, MSE and
Thiel’s U which were on average observed to be slightly lower.
All models improved when distribution was changed to student’s
t or GED, although GED provided the overall best performance
in forecasting. Useful conclusions are that the volatility in the
Ugandan shillings is very responsive to shocks (explosive)
which have been established to be persistent and whose forecast
is expected to persist. Countermeasures are advised through
macroeconomic and scal adjustments in addition to enhancing a
stable political environment. Finally, this study provides a ground
for subsequent studies to approach GARCH volatility modeling
using other methods such as non-parametric Bayesian methods in
machine learning and neural networks. Plausible contributions will
have been made should future studies consider stock markets and
new variables such as ination and interest rates across distributions
in a comprehensive scope of both symmetric and asymmetric
GARCH models, and in a panel of several markets/countries.
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Modeling and forecasting exchange rate volatility in Bangladesh
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Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020 279
APPENDICES
Figure 1: Line plots of USD/UGX and return series
Source: Authors’ illustrations
Figure 2: Empirical quantiles and histogram tests for normality
Source: Authors’ illustrations

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020
280
Figure 3: Returns and variance forecast graphs for the GARCH family models
Source: Authors’ illustrations

Erkekoglu, et al.: Modeling and Forecasting USD/UGX Volatility through GARCH Family Models: Evidence from Gaussian, T and GED Distributions
International Journal of Economics and Financial Issues | Vol 10 • Issue 2 • 2020 281
Source: Authors’ illustrations
Figure 4: Conditional variance and standard deviation for the return series