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American Journal of  Applied 
Statistics and Economics (AJASE)

Forecasting Nigeria’s Oil Price Volatility: A Comparative Analysis of  GARCH Models 
and Heston’s Stochastic Models

Omorogbe Joseph Asemota1, Mustapha Bello2, Samuel Olorunfemi Adams2*

Volume 4 Issue 1, Year 2025
ISSN: 2992-927X (Online)

DOI: https://doi.org/10.54536/ajase.v4i1.4693
https://journals.e-palli.com/home/index.php/ajase

Article Information ABSTRACT

Received: March 10, 2025

Accepted: April 16, 2025

Published: June 04, 2025

Modeling the volatility of  crude oil prices is essential because it gives substantial influence to 
the oil producing countries. Nigeria, the biggest oil producer in Africa and a major participant 
in the world oil market, has significant economic difficulties changes in oil prices. This study 
uses 14 years of  crude oil price data (2010–2023) to assess and compare the forecasting 
effectiveness of  the Heston stochastic volatility model and GARCH-type models (GARCH, 
EGARCH, IGARCH, TGARCH, and FIGARCH). According to the analysis, GARCH-type 
models with Student’s t-distribution perform better than models with typical innovation. 
With a log-likelihood value of  12022.3, an AIC of  -4.7012, a mean error (ME) of  0.0254, 
and a root mean square error (RMSE) of  0.0534, the EGARCH model outperformed the 
others. Nonetheless, the Heston model outperformed all GARCH-type models in terms of  
forecast accuracy, achieving the smallest error (0.000564) and successfully capturing fat-tail 
characteristics in daily return distributions. The study indicates that the Heston model offers 
a better fit and more accurate forecast than GARCH-type models using data from January 
to December 2023 for out-of-sample forecasting. These results provide stakeholders and 
policymakers with important information for controlling the volatility of  Nigeria’s crude 
oil market.

Keywords

EGARCH, FIGARCH, 
GARCH, Heston Model, 
IGARCH, Symmetric, TGARCH

1 National Institute for Legislative Studies, Abuja, Nigeria
2 Department of  Statistics, Faculty of  Science, University of  Abuja, Abuja, Nigeria
* Corresponding author’s e-mail: samuel.adams@uniabuja.edu.ng

INTRODUCTION
One of  the most important energy sources in the world is 
petroleum, a fossil fuel that was created over millions of  
years from the remains of  marine plants and animals. It is 
known as “crude oil” in its natural state. Crucially, crude 
oil serves as a raw material and a necessary energy source 
(Sekati et al., 2020; Yi et al., 2021; Dunn & Holloway, 
2012). Common forms of  oil that power cars, ships, and 
airplanes include heating oil, diesel, motor gasoline, and 
jet fuel. Oil is used in the production and transportation 
of  many commonplace goods, and the industry that 
produces it has a big impact on other industries. 
Production expenses and the state of  the economy as a 
whole are significantly impacted by changes in the price 
of  petroleum products (Fondo et al., 2021). Changes in oil 
prices impact many aspects of  society, including 
household appliances, detergents, prescription 
medications, and food supplies. Like any commodity, oil 
prices fluctuate in response to supply and demand, which 
can have a favorable or negative effect on a number of  
economic sectors (Gasper & Mbwambo, 2023). Crude oil 
is therefore still a major economic issue and a hot topic in 
discussions about international economic policy. It is 
clear from the last three decades that the housing bubble-
related financial crisis had a negative effect on oil prices, 
which fell from US$133.88 per barrel in June 2008 to less 
than $40 per barrel in the months after the disaster. 
Following that, prices rose to $100 in 2014 before sharply 
falling to $30 in 2016 as a result of  an increase in the 
supply of  crude oil. The start of  the Covid-19 pandemic 
made matters worse and caused prices to drop to $16.55, 

the lowest level in 20 years, in April 2020. However, prices 
saw another increase as a result of  the events in Ukraine. 
Countries that rely significantly on crude oil are surely 
impacted by these global changes in crude oil prices 
(Rodhan, 2023). The extraction and sale of  crude oil is 
Nigeria’s main source of  revenue. Following years of  
exploration that started in 1938, Shell D’Arcy, now known 
as Shell Petroleum Company, made the first commercial 
oil discovery in 1956 near Oloibiri in Bayelsa State, 
according to the Nigerian National Petroleum 
Corporation (NNPC, 2013). With its first oil field going 
online in 1958 and producing 5,100 barrels per day, 
Nigeria’s abundance of  oil won it a prominent place in 
the world market. Foreign businesses were then allowed 
to explore for oil in Nigeria, which resulted in the oil 
industry’s steady expansion and made Nigeria a world 
leader. Nigeria currently produces the most oil in Africa, 
accounting for 33–35 percent of  the continent’s oil and 
gas reserves, or 1.347 million barrels a day. It is the fifth-
largest oil exporter to the United States of  America and 
the fifth-largest exporter in the Organization of  
Petroleum Exporting Countries (OPEC) (Mary, 2023). 
Later, in 1977, the Nigerian National Petroleum 
Corporation (NNPC) was established with the intention 
of  overseeing and taking part in the nation’s oil industry 
(Nwokeji, 2007). The nation had 36.966 billion barrels of  
oil and condensate reserves in 2023. This amounts to 
5.906 billion barrels of  condensate and 31.060 billion 
barrels of  oil. Nigeria produces more than 1.5 million 
barrels of  oil per day, with a total oil reserve of  37.064 
billion barrels (Nuprc, 2023). Nigeria ranked 11th on the 



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list of  nations with oil reserves that exceed one billion 
barrels (38.6%), with about 37 million barrels of  known 
oil reserves. Nigeria produced over 1.93 million barrels of  
oil per day and exported 85% of  its oil production, 
placing it 15th in the globe on the list of  nations that 
produce oil that year (John, 2023). In order to diversify 
the economy, increase the domestic market, and lessen an 
excessive reliance on crude oil exports, Nigeria has taken 
a number of  actions to reform the energy sector, including 
the Petroleum Industry Act in 2021 and the elimination 
of  subsidies in 2023. The Nigerian government raised the 
price of  petrol at the pump from the regulated price of  
N185 to more than N700 per liter in May 2023 and 
announced the elimination of  the fuel subsidy. An 
unusual protest against the government’s decision to raise 
the price of  fuel pumps was sparked by this rise. The 
elimination of  the Premium Motor Spirit (PMS) gasoline 
subsidy is one of  the most controversial topics in Nigeria 
right now (Odewale, 2023). The subsidy is a type of  price 
manipulation in which the government sets the pump 
price for consumers to purchase and reimburses the store 
for the difference between the official or regulated price 
per liter and the actual market price. Subsidies, in my 
opinion, are the additional funds that the government 
spent or incurred in order to lower the price of  PMS 
pumps for those with lower incomes. Nigeria’s level of  
living will be significantly impacted by the elimination of  
the petroleum subsidy, particularly for the already 
underprivileged populations. The price of  gasoline 
pumps has increased as a result of  the elimination of  the 
petroleum subsidies, raising the expense of  food, 
transportation, and other necessities. When compared 
year-over-year, the food inflation rate in May 2023 was 
24.82%, 5.33% higher than the rate in May 2022 (19.50%). 
The transportation and storage industry, which 
contributes roughly 0.89% of  the GDP, saw the biggest 
fall of  any sector in the second quarter of  2023, 
contracting by an astounding 50.64%. The average fare 
passengers paid for bus trips within the city each drop 
rose by 97.88% from N649.59 in May 2023 to N1,285.41 
in June 2023, according to data from the NBS 
Transportation Watch. It increased by 120.63% year over 
year from N583 in June 2022 (NBS, 2023). For Nigerians, 
floating the country’s currency has socioeconomic 
ramifications as well, hurting residents’ purchasing power 
and their capacity to pay for necessities. Strong social 
safety nets, focused interventions, and transition-
protective legislation would be necessary to mitigate these 
consequences. Low-income families and individuals’ 
budgets are strained as a result, which lowers their level 
of  living. Additionally, rising gasoline prices have caused 
inflationary pressures throughout the economy, raising 
the cost of  products and services. People with fixed 
incomes are disproportionately affected by inflation, 
which makes poverty worse. Low-income people bear a 
disproportionate amount of  the burden of  rising living 
expenses and petroleum pump prices, which exacerbates 
the nation’s income inequality and widens the income gap 

(Yakubu et al., 2023). It is commonly acknowledged that 
the fluctuations in oil prices have a substantial impact on 
economic activity. Commodity market price fluctuations 
are frequently influenced by changes in oil prices, which 
can cause economic slowdowns and price swings for 
other commodities when oil prices abruptly rise or fall. 
Accordingly, predicting the price of  crude oil is an 
important field of  study, although it faces inherent 
challenges including excessive volatility (Wang et al., 
2004). While limited liquidity and occasional trade in 
imperfect markets may cause a delay in responding to 
new information, oil prices may not always react 
immediately to it (Monoyios & Sarno 2002). According to 
this viewpoint, a substantial body of  research has been 
done on enhancing econometric models’ capacity to 
simulate oil prices. A portion of  the research uses 
different GARCH models to examine the trajectory of  oil 
prices. Alessandri and Mumtaz (2019), Popp and Zhang 
(2016), Adams et al. (2024), and Van Robays (2016) are 
only a few of  the research that have demonstrated the 
substantial influence of  economic factors on rising 
volatility, particularly during times of  regime transition. 
The forecasting skills of  single-regime models are 
significantly diminished by regime shifts, which are 
influenced by a variety of  economic factors. Additionally, 
by upsetting the trends shown in economic time series, 
economic considerations have a major impact on business 
cycles. For example, the performance of  econometric 
models was significantly impacted by the oil crises of  
1974 and 1979. Because of  this, conventional volatility 
models that do not consider regime-switching features, 
including those brought on by oil shocks, are no longer 
sufficient to simulate volatility in gasoline prices. 
GARCH-type models (Ahmed & Shabri, 2014; Wacuka 
Ng’ang’a & Oleche, 2022; Adams & Bello, 2022; Deebom 
& Essi, 2017), Support Vector Machines (SVM), which 
forecast data with high volatility (Okasha, 2014), and 
Autoregressive Integrated Moving Average (ARIMA), 
also known as the Box-Jenkins Methodology 
(Shambulingappa et al., 2020; Awujola et al., 2015; Rodhan 
& Jaaz, 2022) are some of  the analytical techniques that 
have gained a lot of  attention recently in crude oil 
forecasting. Since the symmetric models of  the GARCH 
family are better at predicting the price of  crude oil, they 
are regarded as fitted models (Haque et al., 2021; Arachchi, 
2018; Herrera et al., 2018). They are seen as crucial for 
figuring out how volatile various commodities are 
(Charles & Darné, 2021). By taking into consideration the 
effects of  leverage, volatility clustering, and leptokurtosis 
in the time series analysis, the GJR-GARCH model 
further sets itself  apart from other forecasting models. 
Furthermore, it is discovered that both symmetric and 
asymmetric models are successful in capturing volatility 
(Ekong & Onye, 2017). A crucial part of  many financial 
decision-making procedures is the examination of  
financial time series volatility. Building less risky portfolios, 
maximizing asset allocation, and increasing returns all 
depend on accurate volatility forecasts. As a result, 



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accurate volatility forecasting and analysis have become 
more crucial in recent years. The best choice of  volatility 
models, however, is hotly debated, which presents a 
problem for researchers because the choice has an 
immediate effect on their findings. In order to solve the 
problem of  volatility forecasting, this paper suggests 
using stochastic models and GARCH-type models. This 
study intends to add to the continuing discussion on the 
best techniques for volatility analysis in financial time 
series by assessing several risk models. A comparison of  
the forecasting capabilities of  various GARCH-type 
models and the Heston stochastic volatility model was 
also presented in the study. 

LITERATURE REVIEW
 Numerous econometric models have been used to forecast 
the volatility of  crude oil prices. With an emphasis on 
GARCH-type models, regime-switching models, ARIMA 
models, and the macroeconomic ramifications of  oil 
price volatility, this study looks at the main approaches 
and conclusions from earlier studies. 

GARCH-Type Models for Forecasting Oil Price 
Volatility
Crude oil price volatility has been widely modeled and 
predicted using GARCH models and its extensions. Saltik 
et al. (2016) used the GARCH, IGARCH, GJR-GARCH, 
EGARCH, FIGARCH, and FIAPARCH models to 
examine the return volatility of  Henry Hub natural gas 
and WTI crude oil over various time periods. According 
to their research, asymmetric and integrated GARCH 
models outperformed ordinary GARCH models in terms 
of  forecast accuracy. In particular, according to Mean 
Square Error (MSE) and Mean Absolute Error (MAE) 
criteria, FIGARCH under skew Student-t performed best 
for one period, whereas EGARCH under the generalized 
error distribution was optimal for another. Similarly, 
Herrera et al. (2018) discovered that because of  the strong 
kurtosis in oil returns, models with a Student-t distribution 
outperformed those with a normal distribution. They 
came to the conclusion that EGARCH(1,1) was better 
for medium-term forecasting, whereas GARCH(1,1) 
and RiskMetrics models performed best for short-term 
projections. Kutu and Ngalawa (2017) confirmed a 
significant negative influence of  oil price shocks on the 
South African currency rate using the EGARCH (1,1) 
model. Several autoregressive models were used in other 
research, including Agnolucci (2009) and Ramzan et al. 
(2012), to confirm the persistence of  volatility in oil 
return series. The necessity of  proactive monetary policy 
interventions was highlighted by Fasanya and Adekoya 
(2017), who evaluated symmetric (GARCH, GARCH-M) 
and asymmetric (EGARCH, TGARCH) models and 
concluded that EGARCH was the most suitable for 
simulating inflation volatility. 

Regime-Switching and Alternative Volatility Models
GARCH models have been investigated in a number of  

studies to account for structural shifts in the volatility 
of  oil prices. Zhang et al. (2019) looked at both regime-
switching and single-regime GARCH models and found 
that simpler single-regime models frequently performed 
better than more intricate regime-switching models. To 
account for long-range dependence in financial time series, 
Li et al. (2013) developed the Mixture Memory GARCH 
(MM-GARCH) model, which combined the conventional 
GARCH and FIGARCH models. According to Klein and 
Walther (2016), MM-GARCH models performed better 
in variance and value-at-risk forecasting than conventional 
GARCH models. An empirical investigation contrasting 
the MRS-GARCH (Markov Regime-Switching GARCH) 
model with conventional GARCH models was carried 
out by Zhang et al. (2019). Their results indicated that 
whereas regime-switching models improved in-sample 
estimates, their out-of-sample forecasting performance 
was not always enhanced. The significance of  mean 
equation optimality was further highlighted by Hasanov 
et al. (2020), who showed that GARCH models with 
optimal mean equations generated better predictions. 

ARIMA Models in Crude Oil Price Forecasting
Forecasting has made extensive use of  ARIMA models in 
addition to GARCH-type models. When Selvi et al. (2018) 
used ARIMA to anticipate crude oil prices from 2017 to 
2021, they found that prices would continue to grow, 
highlighting the necessity of  price stability measures. 
In line with Selvi et al. (2018), Shah & Kiruthiga (2020) 
determined that ARIMA (0,1,4) was the best model for 
predicting crude oil prices. Similarly, Rodhan & Jaaz 
(2022) discovered that ARIMA (1,1,4) produced the most 
accurate forecasts after examining 375 months of  WTI 
crude oil price data. 

Macroeconomic Implications of  Oil Price Volatility
One important topic of  study has been how the volatility 
of  crude oil prices affects macroeconomic factors. Using 
the ARCH, GARCH, and EGARCH models, Sekati et al. 
(2020) investigated how South Africa’s GDP, inflation 
rate, and currency rates affected the price of  oil globally. 
According to their findings, a 1% increase in each 
indicator had a varied impact on oil prices, with GDP 
and exchange rates having a positive effect and inflation 
having a negative one. These findings, however, were 
in contrast to those of  Kutu and Ngalawa (2017), who 
discovered that shocks to the price of  crude oil had a 
negative impact on exchange rates. Numerous studies 
emphasize the detrimental consequences of  volatility in 
the price of  crude oil on economic stability, especially 
in developing nations. According to Yildirim (2017) 
and Demirer et al. (2018), living standards are adversely 
affected by ongoing changes in the price of  oil. Crude oil 
prices are more volatile than those of  other non-financial 
assets, which adds to economic uncertainty, according 
to Adelman (2000) and Lipsky (2009). In their analysis 
of  oil price volatility from an investing standpoint, Liu 
et al. (2022) concluded that no single model consistently 



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outperformed others, supporting the use of  a variety 
of  models to increase forecasting accuracy. According 
to the literature evaluation, asymmetric and integrated 
GARCH models outperform symmetric models in terms 
of  forecasting the volatility of  crude oil prices. Regime-
switching models may not always increase predicting 
accuracy, although they do show promise for in-sample 
analysis. While macroeconomic models highlight the 
wider economic ramifications of  oil price volatility, 
ARIMA models are nevertheless useful for short-term 
price forecasting.

MATERIALS AND METHODS
Data Description and Source
The Central Bank of  Nigeria website provided daily 
crude oil prices from January 2010 to December 2023 for 
modeling and forecasting purposes. Because Brent crude 
oil is regarded as the standard for crude oils in Europe 
and Africa, this information was used. Another way that 
crude oil is exchanged is either on its own or in relation 
to other forms of  crude oil. Another factor in this choice 
was the data’s accessibility during the specified time 
window. Continuously compounded daily stock returns 
are fitted to conditional variance models.
yt =100(lnkt- lnk(t-1))                 (1)
Where kt = current period of  stock market exchange, k(t-

1)= previous period stock market exchange, yt= current 
period stock returns (stock market exchange -RT), and 
Ω(t-1)= All stock returns up to the immediate past.

Model’s Description
The models used to estimate the volatility of  crude oil 
prices are introduced in this section. The features of  the 
historical crude oil spot price data are used to establish the 
modeling approach that is used. The best models of  price 
volatility are not universally agreed upon since energy 
prices have complicated characteristics. The following 
are the procedures used to model the volatility: examine 
past data to determine its characteristics; Verify if  the 
observations are normal. Verify the series’ stationarity 
and look for ARCH effects; The Lagrange multiplier 
test is used to find out whether ARCH (Autoregressive 
Integrated Moving-Average) effects are present, and the 
Augmented Dickey-Fuller (ADF) and Philips Perron 
(PP) tests are used to check for stationarity. Describe the 
estimate processes for the five GARCH type models and 
the Heston stochastic model that were utilized in this 
work to model and forecast the price and return of  crude 
oil. The best-fitting model is then selected by comparing 
the results with the predicting outcomes.

GARCH-type Models
Engle (1982) created the fundamental concept of  the 
auto regressive conditional heteroskedasticity (ARCH) 
model in his groundbreaking study. In the literature, 
the ARCH model and its later generalized versions are 
widely recognized for their capacity to capture the most 
significant stylized facts found in all volatility measures 

(e.g., squared log-returns, absolute log-returns, etc.), such 
as clustering effects, long-memory and short-memory 
effects, and asymmetric leverage effects. Five distinct 
GARCH models that were employed in this study are 
presented below.

Models of  Volatility    
The Family of  Autoregressive Conditional 
Heteroskedasticity (ARCH) Models. Every ARCH or 
GARCH family model requires two distinct specifications: 
the mean and variance equations. According to Engel, 
conditional heteroskedasticity in a return series, can 
be modeled using ARCH model expressing the mean 
equation in the form:
yt = E(t-1) (yt )+ εt                 (2)
Such that εt = φt σt
Equation 2 is the mean equation which also applies to other 
GARCH family model. E(t-1)  is expectation conditional 
on information available at time t-1, εt is error generated 
from the mean equation at time t and φt is a sequence of  
independent, identically distributed (iid) random variables 
with zero mean and unit variance. E {εt⁄Ω(t-1)}=0; and 
σ2

t=  {(ε2
t)⁄Ω(t-1)} is a nontrivial positive valued parametric 

function of   Ω(t-1). The variance equation for an ARCH 
model of  order q is given as:  
σ2

t = α0+ ∑q
(i=1) αi ε

2
(t-1) + μt               (3)

Where α0 > 0, αi ≥ 0, i=1,…,q, and αq  >0
In practical application of  ARCH (q) model, the decay 
rate is usually more rapid than what actually applies to 
financial time series data. To account for this, the order 
of  the ARCH must be at maximum, a process that is 
strenuous and more cumbersome.

The Unconditional Kurtosis of  ARCH (1)
Suppose the innovations are normal, then            
E(at

4| Ft-1) = 3[ E(at
2 | Ft-1) ]

2               (4)
= 3(α0+α1at-1

2 )2,
it follows that 
Eat

4 = 3α0
2 (1 +α1 ) / [( 1 -α1 ) ( 1 -3α1

2 )]                  (5)
and  
Eat

4 /(Eat2 )
2 = 3 ( 1 -α1

2 )/ ( 1 -3α1
2 )>3                (6)

This shows that the tail distribution of  at is heavier than 
that of  a normal distribution.

Generalized ARCH (GARCH) Model 
The conditional variance for GARCH (p, q) model is 
expressed generally as:
                       (7)

where p is the order of  the GARCH terms, and q is the 
order of  the ARCH terms, ε2. Where β0>0, αi ≥0, i=1,…, 
q-1, j=1,…, p-1 and βp, αq  >0. σ2

t is the conditional 
variance and ε2

t, disturbance term. The reduced form of  
equation 3 is the GARCH (1, 1) represented as:
σ2

t= β0 + β1 ε
2
(t-1)+ β2 σ2

(t-1)               (8)
The three parameters (β0 , β1  and β2) are nonnegative and  
β1 + β2  <1 to achieve stationartiy.   



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EGARCH Model
A different that also captures the leverage is the 
exponential GARCH MODEL OR EGARCH
                 (9)

A lower AIC means a model is considered to be closer 
to the true model. The loglikelihood is used to select the 
best model for estimation and forecasting. The higher 
the loglikelihood, the better the model. Besides this, the 
information criteria are also used to pick the model. A 
good model had the highest loglikelihood or the lowest 
information criteria. Therefore, a higher log likelihood 
translates to a low information criterion. The information 
criteria used in this study are the Akaike, Bayes, Shibata 
and Hannan-Quinn.
Forecasting performance of  the five models is analysed 
by comparing the errors i.e., comparing the forecasted 
returns with realized returns. This is done by comparing 
the mean error (ME), mean absolute error (MAE) and 
root mean square error (RMSE). The lesser the errors 
the better the more accurate the model is in forecasting 
correct return for Brent Crude Oil.
ME = 1/n ∑n

(j=1) (yi-y
*)             (19)

MAE = 1/n ∑n
(j=1) |(yi-y

*)|               (20)
RMSE = √(1/n ∑n

(j=1) ((yi-y
*))2)                 (21)

RESULTS AND DISCUSSIONS
The 5112 data observations for the oil price volatility 
from January 2010 to December 2023 are the main 
emphasis of  this section. To improve the accuracy of  the 
data analysis, some processes are looked at. As can be 
seen from the graphing of  data behavior with extensive 
mobility, the first result indicates that the series is not 
stationary.

Plot
Figure 1 shows the energy data for the daily price of  
crude oil in Nigeria plotted against time. The graphic 
shows that fluctuations in crude oil prices show clustered 
volatility with sporadic surges and spikes. The crude 
oil price plot indicates that the price of  crude oil is 
not regularly distributed during the given period. The 
recession may have caused the decline in oil prices in 
2016. The significant decline in oil prices earlier in 2020, 
particularly from March 2020 to April 2021, which was 
exacerbated by the Covid-19 outbreak, must also be 
noted. At this time, the corona virus had infected the 

Which displays the usual leverage effect if  αφ<0. The 
EGARCH model has the advantage that the logarithmic 
specification ensures that variance is always positive, but 
it has the disadvantage that the future expected variance 
beyond one period cannot be calculated analytically. 

Weekend Effect
It is always known that days that followed a weekend or a 
holiday have higher variance than average day. We can try 
the following model:
σt+1

2=ω+βσt
2+ασt

2 Zt2+γITt+1,               (10)
where ITt+1 takes value 1 if  day t+1 is a Monday, for 
example.

More General EGARCH
The exponential GARCH, or EGARCH model is
log(σt)= α0+ ∑q

(i=1) αi g(εt-1)+ ∑p
(i=1) βi log(σt-1)                (11)

where g(ϵt )= θϵt + γ{|ϵt|-E(|ϵt|)}
   
IGARCH Model
A GARCH (p, q) process is called an I-GARCH process 
if
∑q

(i=1) αi + ∑p
(i=1) βi=1                 (12)

The IGARCH processes are either non-stationary or 
have an infinite variance.

Model Selection Criteria
This section explains the model selection criteria used to 
select the model combination to use. To select the best 
fitting ARMA-GARCH models, Akaike Information 
Criteria (AIC) due to (Akaike, 1974), Bayesian 
Information Criterion (BIC) due to Schwarz Information 
Criterion (SIC) due to (Schwarz, 1978) and Hannan-
Quinn Information Criterion (HQC) due to (Hannan, 
1980) and Log likelihood are the most commonly used 
model selection criteria. These criteria are used in this 
study and are computed as follows:
AIC (K) = -2log L + 2k                (13)
BIC (K) = -2log L + (log N).d              (14)
SIC (K)= -2log L + K log T               (15)
HQC (K) = 2log [log T]K – 2log L                 (16)
Where K is the number of  independently estimated 
parameters in the model. T is the number of  observations; 
L is the maximized value of  the Log-Likelihood for the 
estimated model defined as follows:

                  (17)

                 (18)

Thus, given a set of  estimated GARCH type models for a 
given set of  data, the preferred model is the one with the 
minimum information criteria and largest log likelihood 
value. Figure 1: Plot of  Crude Oil Price from 2010 to 2023



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majority of  nations, and governments were beginning 
to enforce travel restrictions and lockdowns. The 
aforementioned chart demonstrates volatility clustering; 
rice prices rise steadily for a while before falling steadily 
for another. It is evident from Figure 2 that the sport 
prices are not distributed properly. It is clear from Figure 
3 that the financial time series share characteristics. The 
mean reversion property is demonstrated by the fact that 
variance is not constant across the figures. The charts also 
reveal volatility clustering.

Descriptive Statistics of  the Crude Oil Returns
Table 1 shows the summary statistics of  the Crude Oil 
returns, it was shown in the result that, Crude oil return has 
a negative minimum return value and a standard deviation 

of  3.501543. From the measure of  skewness, all the series 
for crude oil return are skewed to the left, exhibit positive 
excess kurtosis, which are the stylized facts observed 
in financial time series data. Based on the p-values of  
the Jarque Bera test, we reject the null hypothesis of  
normality for the differenced series of  crude oil returns. 
The series have positive means and are mean-stationary since 
the returns are concentrated around zero as indicated in 
Figure 3. These series exhibit leptokurtosis as their kurtosis 
is greater than the normal kurtosis value of  3. Time plots 
are used to determine the observable characteristics of  the 
returns as presented in Figure 3.

Figure 2: QQ Plot for Crude oil Sport Prices
Table 1: Summary Statistics
Statistics Oil Returns
Mean 0.0000009
Standard Error 0.475410198
Median 0.07831885
Standard Deviation 3.501543
Sample Variance 0.002704771
Kurtosis 70.87588
Skewness -1.537876
Jarque-Bera statistics 1072872
Minimum -0.6604506
Maximum 0.588928

Source: Author’s estimation using R 4.4.1

Figure 3: Plot Return of  Crude Oil 

Testing for Normality
To determine whether the return series is normally 
distributed, the normal QQ-plot is utilized to examine 
the distributional features. A scatter plot of  a specific 
distribution is represented by the typical QQ-plot. The 
evidence against the null hypothesis that the distribution 
is normal increases with the degree of  deviation from this 
line. With a few outliers (those that appear farther from 
the normal line) that can be interpreted as the heavier tails 

in the preceding image, Figure 4 illustrates that returns 
are generally normally distributed. This plot demonstrates 
that the returns in this study can be modeled using a 
normal distribution, but the heavy tails would not be 
addressed. In order to address the heavy tail, the student 
t distribution which is known to be capable of  capturing 
heavy tails is now taken into consideration. The majority 
of  the previously noted outliers have been eliminated 
by the plot in Figure 5. Because the T-distribution can 



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catch heavier tails, as shown by our research above, it was 
added to Table 2. At the 5% level of  significance, the high 
Jarque-Bera statistics value provides evidence that the null 
hypothesis of  normalcy can be rejected. This conclusion is 
further supported by the large excess kurtosis and negative 
skewness. Similar to the normal distribution, the student t 

distribution is bell-shaped and symmetrical. Nonetheless, 
the t-distribution is more likely to yield values that deviate 
significantly from its mean due to its thicker tails. We decided 
to use the normal distribution and t-distribution to suit 
the volatility models because we are unable to completely 
dismiss the normal distribution in this investigation.

Figure 4: QQ Plot for Normal Distribution

Figure 5: QQ Plot for Student t Distribution

Volatility Clustering
Return series fail to follow the financial stylized facts. Hence 

improving ARIMA model by using GARCH (Generalized 
Auto Regressive Heteroskedasticity) types model.

Table 2: Jarque Bera Test, Skewness, Kurtosis for Checking Normality
X-squared p-value

Jarque Bera Test 1072872 0.0000
Skewness -1.537876
Kurtosis 70.87588

Source: Author’s estimation using R 4.4.1

Figure 6: Volatility Clustering Plot



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Stationary Tests
To investigate whether the return series are stationary, the 
Augmented Dickey-Fuller (ADF) test was applied. The 
hypothesis was such that: H0: Non stationary versus H1: 
Stationary. The p-value is < 0.05. This allows the rejection 
of  the null hypothesis. To confirm the results above, we 

use the Philips Perron (PP) test which is a non-parametric 
correction to ADF to account for autocorrelation associated 
with breaks in the data. As the test values are lower than the 
critical values by choosing the 1% confidence level, it can 
be certainly confirmed that the null hypothesis is rejected 
and the Crude Oil returns series is stationary.

Table 3: Stationary Test
Augmented Dickey 
Fuller  (ADF)

Estimate t-Statistic Prob. Test critical values 
(1% level)

Intercept 4.26797 2.693 0.00768 -3.46
z.lag.1     -0.05367 -2.797 0.00567
z.diff.lag   0.30584 4.509 1.11e-05
F-Statistics 12.54
Phillips-Perron Unit Root Test
Dickey-Fuller  = -70.883 P_ value = 0.01

Source: Author’s estimation using R 4.4.1

Presence of  Volatility
We check the presence of  volatility using time series 
plot for log returns, square returns, and absolute returns. 
This part of  investigating the Crude Oil return series 
represents a special place in quantitative research, 
because useful information could be found for testing 
the stationary hypothesis, heteroskedasticity effect 
and by the descriptive statistical information about the 
average, variance, asymmetry indicators and the type 
of  distributions, before applying the desired models of  
estimating and predicting the conditional volatility of  

these returns. From this point of  view, this study aligned 
with the preferred tools of  analysis the oil return series 
used by Yildirim (2017); Zhang et al. (2019); Haque and 
Shaik (2021) or Mohammadi and Su (2010). 
The next step was to identify the presence of  ARCH 
terms, thus testing the level of  increased probability of  
ARCH effects (q). This plays an important role in the 
use of  ARCH-GARCH models and the manner in which 
the analysed time series can be estimated and fitted by 
these models (Yi et al., 2021; Sekati et al., 2020; Oyuna and 
Yasbin, 2021).

Figure 7: Plot of  Crude oil Price Volatility



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Presence of  ARCH (q) Effects
Figure 7 sows the returns (square and absolute) show high 
level of  auto-correlation. We double checked present of  
auto-correlation in square returns by applying Ljung-Box 
test on square returns and the p-value (0.0000) is < 0.05 
which indicated that the data is not independent. The null 
hypothesis was rejected which means auto correlation is 
present. The Lagrange multiplier (LM) test by ENGLE 
(1982) was applied to the residuals of  simple time series 
models. The ARCH-LM tests results provided strong 

evidence for rejecting the null hypothesis as shown in 
Table 4. Implicitly, the associated p-value, which is lower 
than the 5% confidence level indicates that ARCH effects 
exist; hence, an ARCH or GARCH model should be 
employed in modeling the return time series. This fact 
is also observed at the p-value of  lagged squared error 
terms (0.000) that is less than 5% confidence level and 
indicates the same type of  conclusion: the presence of  
ARCH effects in the oil return series.

Figure 8: Checking Autocorrelation in Return

Table 4: ARCH Effect Test
Variable Coefficient Std.Error Prob.
C 0.90826 0.12440 0.0000
μ2

(t-1) 0.35307 0.04198 0.0000
Chi-squared 761.94 2.2e-16

Source: Author’s estimation using R 4.4.1

GARCH Model
To get a precise assessment of  the volatility of  crude 
oil, this part used symmetrical and various asymmetrical 
GARCH models of  the normal and student’s t 
distribution innovation. The GARCH model (p, q) 
was used to examine two equations: the conditional 
variance and the conditional mean. With the exception 
of  mu, which is not significant, Table 5 visual 
representation suggests that all of  the coefficients are 
positive and highly statistically significant (5% p-value). 
At 11981.45, the log probability return is positive. The 
respective values for the AIC, BIC, SC, and HQ are 
-4.6856, -4.6792, -4.6856, and -4.6834. When compared 
to other models, these values are all fairly near to one 
another and serve as criteria for determining which 
model is best. Additionally, our findings demonstrated 
that volatility shocks endure, as indicated by the sum 
of  the ARCH and GARCH coefficients, which is a 
common finding in recent research (Brandt and Gao, 
2019; Miao et al., 2017; Yildirim, 2017). With regard to 
Table 6, TGARCH (1,1) model results, every parameter 

aside from mu, is positive and statistically significant 
at the 5% p-value. This indicates that the TGARCH 
(1,1) model’s requirements are met. Table 7 details the 
outcomes received for this model. When calculating 
the average return on crude oil, the TGARCH (1,1) 
model is novel since it accounts for volatility (either as 
conditional variance or conditional standard deviation). 
Thus, like earlier research (Yildirim, 2017; Kutu and 
Ngalawa, 2017; Neshat et al., 2018), the TGARCH (1,1) 
model can show how investors’ risk aversion affects the 
world’s oil price fluctuations. The individual outcomes 
of  the IGARCH (1,1) model are shown in Table 8. 
Because of  its popularity and capacity to analyze the 
asymmetric reaction to different market shocks, it is 
used to quantify the risk associated with any kind of  
financial or non-financial asset. With the exception of  
mu, which is not, all of  the coefficients are positive 
and statistically significant at the 1% level based on the 
parameters. The stability of  the IGARCH (1,1) model 
is further confirmed by the statistical significance of  
the computed parameters at the level of  conditional 



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variation (Saltik et al., 2016; Mohammadi and Su, 
2010; Herrera et al., 2018). Additionally, 11981.46 is a 
positive value for the IGARCH loglikehood. The AIC, 

which has a value of  -4.6860, was computed using the 
IGARCH (1,1) model.

GARCH (1,1)

Table 5: The main results about the estimated conditional variance by using the GARCH (1,1) models
Variables/Parameters Coefficients Std. error Prob.
Mu 0.000089 0.000247 0.71926

Omega 0.000032 0.000006 0.00000
Alpha 0.211262 0.026751 0.00000
beta1 0.787717 0.021613 0.00000
Shape 3.638566 0.189317 0.00000
Log likelihood 11981.45
Akaike information criterion (AIC) -4.6856
Bayes information criterion (AIC -4.6792
Shibata criterion (SC) -4.6856
Hannan-Quinn criterion (HQ) -4.6834

Source: Author’s estimation using R 4.4.1

EGARCH (1,1)

Table 6: The results for EGARCH (1,1) model
Variables/Parameters Coefficients Std. error Prob.
Mu 0.000089 0.000082 0.27958
Omega -0.480951 0.063635 0.00000
alpha 1 0.075537 0.011234 0.00000
beta1 0.925548 0.009301 0.00000
gamma1 0.303257 0.023440 0.00000
Log likelihood 10963.96
Akaike information criterion (AIC) -4.2875
Bayes information criterion (AIC -4.2811
Shibata criterion (SC) -4.2875
Hannan-Quinn criterion (HQ) 4.2853

Source: Author’s estimation using R 4.4.1

Table 7: TGARCH (1,1) Model
Variables/Parameters Coefficients Std. error Prob.
Mu 0.000089 0.000249 0.720996
Omega 0.000032 0.000006 0.000000
Alpha1 0.164108 0.028883 0.000000
beta1 0.786630 0.021824 0.000000
gamma1 0.096304 0.036656 0.008608
Shape 3.633869 0.190538 0.000000
Log likelihood 11985.2
Akaike information criterion (AIC) -4.6867
Bayes information criterion (AIC -4.6790
Shibata criterion (SC) -4.6867
Hannan-Quinn criterion (HQ) -4.6840

Source: Author’s estimation using R 4.4.1

The Results from TGARCH (1,1) Model



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IGARCH (1,1) Model

Table 8: The results from IGARCH (1,1) model
Variables/Parameters Coefficients Std. error Prob.
Mu 0.000089 0.000247 0.71896
Omega 0.000032 5.45374 0.00000
Alpha1 0.212413 0.021528 0.00000
beta1 0.787587 NA NA

3.629988 0.147572 0.00000
Log likelihood 11981.46
Akaike information criterion (AIC) -4.6860
Bayes information criterion (AIC -4.6809
Shibata criterion (SC) -4.6860
Hannan-Quinn criterion (HQ) -4.6842

Source: Author’s estimation using R 4.4.1

FIGARCH (1,1) Model

Table 9: The results from FIGARCH (1,1) model
Variables/Parameters Coefficients Std. error Prob.
mu 0.000089 0.000247 0.719240
omega 0.000034 0.000008 0.000036
Alpha1 0.135460 0.055987 0.015543
beta1 0.745468 0.053460 0.000000
delta 0.876198 0.067964 0.000000
shape 3.652282 0.149356 0.000000
Log likelihood 11985.19
Akaike information criterion (AIC) -4.6867
Bayes information criterion (AIC -4.6790
Shibata criterion (SC) -4.6867
Hannan-Quinn criterion (HQ) -4.6840

Source: Author’s estimation using R 4.4.1

The Investigation Analysis of  the Applied GARCH 
Type Models
The literature (Yi et al., 2021; Yildirim, 2017; Er 
and Fidan, 2013; Kulikova and Taylor, 2013; Zhang 
and Wang, 2015; Aye et al., 2014) outlines a number 
of  requirements that each model used must meet, 
beginning with the specific hypotheses in applying the 
ARCH-GARCH approaches. Accordingly, the models 
should have the fewest parameters, the highest Log 
Likelihood ratio, the lowest Schwarz Information 
Criteria, significant ARCH and GARCH parameters, 

and neither autocorrelation nor heteroskedasticity in the 
residual or errors terms. Starting from this assumption, 
we concentrated on the analysis and diagnosis of  the five 
models that were used: GARCH (1,1), IGARCH (1,1), 
EGARCH (1,1), TGARCH, and FIGARCH (1,1). The 
specific goal was to determine which model was best 
suited for estimating the conditional variance, or which 
model had the most criteria completed. The results of  
the ARCH-LM and Durbin-Waston tests, which do not 
account for heteroskedasticity or autocorrelation, are 
displayed in Table 10.

Table 10: The results of  evaluation models (Test for Residuals)
Variables/Parameters Std. error Prob.
MODEL ARCH-LM TEST DURBIN-WASTON TEST
GARCH (1,1) 0.3758 (0.5398) 1.999914
EGARCH (1,1) 0.2767 (0.5988) 1.999906
TGARCH (1,1) 0.4385 (0.5078) 1.999936
IGARCH (1,1) 0.1128 (0.7370) 1.999911
FIGARCH (1,1) 0.002673 (0.9588) 1.999983

Source: Author’s estimation using R 4.4.1



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It can be stated categorically state that the models 
(GARCH, IGARCH, TGARCH, and FIGARCH) that 
were employed have been successful in meeting the 
error series’ residual requirements. The hypothesis of  
homoskedasticity is thus validated since the probability 
related to the ARCH-LM test are higher than the 5% 
(0.05) threshold that the existence of  ARCH (q) effects in 
the residual series is denied and the errors are uniformly 
distributed. The Durbin-Waston Test results, which were 
about equivalent to 2.00, verified that there was neither 
autocorrelation nor serial correlation on the residual 
series. In addition, Table 10 shows that the EGARCH 
(1,1) model is the best accurate model for estimating the 
conditional variance of  the Crude Oil return series. This 
model is completely compliant and genuinely respects the 
primary restrictive constraints imposed by the literature in 
contrast to the GARCH (1,1), TGARCH, IGARCH (1,1), 

and FIGARCH (1,1) models. Studies by Yang et al. (2020), 
Saltik et al. (2016), and Yildirim (2017) that use asymmetrical 
and non-parametric GARCH models to evaluate crude oil 
volatility across time are positioned in a similar manner. 

The Best Estimated Model for Selection 
The best model is chosen based on its logarithm maximum 
likelihood function value. The better the model, the lower 
the values of  the information criteria. According to 
Tables 11 and 12 above, the EGARCH model has the 
lowest information criteria values for the t-distribution 
and the highest log likelihood value for the normal 
distribution. But, because it has a greater log probability 
value and fewer information criteria than the normal 
distribution, the t-distribution matches the data better. 
This demonstrates that, out of  all the models examined, 
the EGARCH-t distribution fits the data the best.

Table 11: Loglikelihood and Information Criteria values normal dist
MODEL ARCH significant? GARCH significant? LOG LIKELIHOO Akaike IC
GARCH Yes Yes 10943.67 -4.2800
EGARCH Yes Yes 10966.96 -4.2889
TGARCH Yes Yes 10966.84 -4.2887
IGARCH Yes Yes 10939.28 4.2787
FIGARCH Yes Yes 10842.81 -4.2401

Source: Author’s estimation using R 4.4.1

Table 12: Loglikelihood and Information Criteria values—student’s dist
MODEL ARCH significant? GARCH significant? LOG LIKELIHOOD Akaike IC
GARCH Yes Yes 11981.45 -4.6856
EGARCH Yes Yes 12022.3 -4.7012
TGARCH Yes Yes 11985.2 -4.6867
IGARCH Yes Yes 11981.46 -4.6860
FIGARCH Yes Yes 11985.19 -4.6867

Source: Author’s estimation using R 4.4.1

Forecast 
The forecast of  future conditional volatility of  the crude 
oil return series was calculated following the diagnostic 
analysis of  GARCH models. Demirer et al. (2018), 
Escribano and Valdes (2017), and Ahmed and Shabri 
(2014) used the data from each volatility model with the 
lowest values of  Mean Absolute Error (MAE), Mean 
Absolute Percent Error (MAPE), and Root Mean Square 
Error (RMSE) to determine the optimal forecasting 
model. In a more unpredictable, challenging, and stressful 
time brought on by the COVID-19 Pandemic, there was 
a greater need to monitor the persistence and fluctuating 
and oscillating movement of  crude oil volatility, which is 
why the 2010–2023 timeframe was chosen for analysis. 
The overall assessment of  five GARCH-type models 
using the RMS and MAE error metrics is shown in Table 
13. According to the findings, the EGARCH model 
outperforms the GARCH, IGARCH, FIGARCH, and 
TGARCH models in terms of  forecast accuracy. The Figure 9: Forecast of  Crude Oil Prices Using EGARCH



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outcome demonstrated that each of  the four models 
yields outcomes that are comparatively satisfactory. 
This suggests that none of  the models’ significance in 
predicting crude oil returns can be disputed. But the 
EGARCH model performs better than all the others, 
with the IGARCH and GARCH models coming in 
second and third, respectively. This demonstrates that the 
basic GARCH model is still useful for predicting crude 

oil prices, despite the creation of  additional GARCH 
extension models. This outcome makes it abundantly 
evident that the model that performs best when evaluating 
forecast performance is also the most appropriate for 
modeling crude oil returns in Nigeria. This is consistent 
with (Dana, 2016) results that the best model for forecasting 
is also the one that fits the data the best.

Table 13: Forecast Results (2010-2023)
MODEL RMSE MAE Rank
GARCH (1,1) 0.053350424 0.0256712 3
EGARCH (1,) 0.05335017 0.0254007 1
TGARCH (1,1) 0.05350445 0.02542721 4
IGARCH (1,1) 0.05335032 0.02540104 2
FIGARCH (1,1) 0.05337822 0.02562058 5

Source: Author’s estimation using R 4.4.1

Approximation of  Volatilities and the Model Parameters
By applying the Parkinson extreme value method, we 
approximate daily volatilities. The Parkinson extreme 
value method is a best estimator of  volatility than the 

traditional volatility measure (Parkinson, 1980). Hence, 
by using this method, we find the daily and annual crude 
oil price volatilities (σ) with respective variances (σ2) for 
each year.

Table 14: Approximation of  the Crude Oil Volatilities, 2010–2023
Time interval Number of  

Trading days
Minimum 
Price

Maximum 
price

Annual 
Volatility

Annual 
Variance

Daily 
Volatility

Daily 
Variance

4.1.2010-31.12.2010 252 64.78 91.47 0.1280 0.016 0.00006 4.228E−09
3.1.2011-29.12.2011 252 75.40 113.40 0.0226 0.001 0.00009 8.076E−09
3.1.2012-31.12.2012 252 75.40 109.39 0.1316 0.017 0.00069 4.717E−09
2.1.2013-27.12.2013 252 86.55 110.61 0.0792 0.006 0.00025 6.184E−10
3.1.2014-31.12.2014 249 53.45 107.96 0.1145 0.132 0.00053 2.773E−09
2.1.2015-30.12.2015 252 34.55 61.35 0.2030 0.041 0.00016 2.674E−08
4.1.2016-30.12.2016 252 26.19 54.01 0.2111 0.045 0.00018 3.126E−08
3.1.2017-29.12.2017 250 42.48 60.45 0.1071 0.012 0.00005 2.106E−09
2.1.2018-31.12.2018 249 44.48 77.41 0.1367 0.019 0.00008 5.638E−09
4.1.2019-09.12.2019 250 46.31 66.23 0.1483 0.022 0.00009 8.757E−09
2.1.2020–28.12.2020 252 50.57 86.07 0.1282 0.0164 0.00006 4.25212E–09
2.1.2021–30.12.2021 255 35.26 66.33 0.1768 0.0312 0.00012 1.50102E–08
4.1.2022–30.12.2022 255 27.10 54.97 0.2004 0.04012 0.00016 2.48212E–08
3.1.2023–30.12.2023 256 88.69 128.14 0.0995 0.0099 0.00004 1.58121E–09

Source: Author’s estimation using R 4.4.1

Simulations with Heston stochastic volatility model 
was performed for crude oil data by focusing on the 
logarithmic crude oil price behavior. The Heston model 
addresses well all kinds of  fat-tails properties in the 
daily price return distributions under various market 
circumstances. Initially, the parameters were computed 
from the observation from January 2010 to December 
2023. To forecast, we used the data from 02.01.2023 to 
30.10.2023. That is, for t = 1 to N is used to estimate 
the parameters and then forecast N +1. The Euler–
Maruyama numerical method is employed in this study 
to simulate the Heston model. The computed parameters 
are presented in Table 15.

Table 15: Computed parameters for Euler–Maruyama
Parameter Symbol Value
Initial price Y0 46.31
Initial volatility V0 2.3 × 10−4

Vol-volatility σ 9.0 × 10−5

long-run variance θ 8.8 × 10−9

Reversion rate β 2.95 × 10−3

Mean log-return μ 4.94 × 10−4

Source: Author’s estimation using R 4.4.1



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Table 16: The Error Analysis, The Euler-Maruyama Scheme
Step size (Δt) Error
0.0010 0.000564
0.1010 0.010153
0.2010 0.011524
0.3010 0.010771
0.4010 0.012742
0.5010 0.013142
0.6010 0.012421
0.7010 0.012101
0.8010 0.010621
0.9010 0.011283

Source: Author’s estimation using R 4.4.1

Figure 10: Forecasting using Bootstrap

Table 16 presents the error analysis for the Heston model 
by using the Euler–Maruyama method. Results obtained 
are compared with results of  GARCH-type models 
presented in Table 11. The model with smallest error 
compared implies that it is more accurate to estimate the 
model. Comparing results of  Tables 13 and 16, results 
in general show that the Heston model approximation 
presents small errors if  compared with the improved 
GARCH models. Thus, it can be concluded that the 
Heston model forecast better crude oil price volatility 
than the counterpart models.

Discussion
The performance of  the GARCH, EGARCH, IGARCH, 
TGARCH, FIGARCH, and Heston models in simulating 
and predicting the volatility of  crude oil returns using data 
from Nigeria is compared in this paper. A deviation from 
normality is shown by the return series’ positive excess 
kurtosis and negative skewness. The null hypothesis 
of  normality is rejected for every differenced series 
of  crude oil return, according to the Jarque-Bera test 
results. Additionally, as Figure 2 shows, the series exhibits 
mean stationarity, with returns centered around zero. 
The adoption of  the Student’s t-distribution to account 
for heavy tails is further supported by the presence of  
leptokurtosis, when kurtosis values are higher than the 
typical threshold of  3. Additionally, the ARCH Lagrange 
Multiplier test and visual inspections are used to confirm 
the evidence of  mean reversion and heteroscedasticity. 

A variety of  GARCH-type models were estimated in 
order to choose the best model for volatility. The Akaike 
Information Criterion (AIC), forecasting accuracy metrics, 
and log-likelihood values were adopted in choosing the 
best model. Better model fit is indicated by a higher log-
likelihood value, but more model efficiency is suggested 
by a lower AIC. The EGARCH model performed 
the best among the competing models, obtaining the 
lowest AIC (-4.7012) and the greatest log-likelihood 
value (12022.3). Additionally, the EGARCH model 
demonstrated greater prediction accuracy with the lowest 
Mean Error (0.0254007) and Root Mean Squared Error 
(RMSE) (0.05335017). Thus, among the GARCH-type 
models analyzed, the EGARCH model with Student’s 
t-distributed innovations is found to be the best model 
for predicting the volatility of  Nigeria’s crude oil return. 
The study assesses the Heston model, which successfully 
represents the fat-tailed characteristics of  daily return 
distributions under various market scenarios, in addition 
to the GARCH family models. Data from January 2010 
to December 2023 were used to estimate the model 
parameters, and data from January 2023 to December 
2023 were used to provide forecasts. The model was 
simulated using the Euler-Maruyama numerical method, 
and the calculated parameters are shown in Table 13. 
The Heston model provided results with less forecasting 
errors than the EGARCH model, according to a 
comparison of  forecasting accuracy (Tables 12 and 14). 
This result implies that the Heston model outperforms 
the EGARCH model in forecasting crude oil return 
volatility, even though the latter offers robust volatility 
modeling. As a result, the Heston model outperforms 
the GARCH-type models taken into consideration in this 
study and is the recommended option for predicting the 
volatility of  crude oil prices. 

CONCLUSION
In this study, the Heston stochastic model and GARCH 
type models were used to estimate the volatility of  
crude oil data. Based on a number of  factors, including 
the lowest AIC value of  -4.7012 and the greatest log-
likelihood value of  12022.3, it was discovered that the 
student’s t test of  the EGARCH (1,1) model is the best 
fitted model to estimate volatility of  crude oil data among 
other GARCH type models. Even though EGARCH 
was the best-fitting model in our study, this does not 
necessarily mean that it is the best model for estimate 
and volatility forecasting in other contexts. Based on 
the lowest MAE value of  0.0254007 and RMSE of  
0.05335017 for Nigeria Crude Oil data, the EGARCH 
(1,1) model likewise seems to be the most effective 
GARCH model for predicting. This demonstrates how 
crucial it is to evaluate the model’s performance at every 
level, including predicting performance and best fitted 
model, in order to select the optimal model. The Heston 
volatility model was also approximated using the Euler–
Maruyama approach. Additionally, the approximation 
results of  the modified GARCH models were compared 



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with the outcomes of  the Heston model. The findings 
showed that the Heston stochastic volatility model 
outperforms the selected GARCH models in predicting 
the volatility of  crude oil. As a result, it is concluded that, 
the Heston model in this study is the most effective for 
volatility forecasting and estimate.

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