Page 1270
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
Assessing Models Behaviors and the Forecasting Performance of Arima
and Garch Models: An Empirical Study
Olusola S. Oluwabunmi
1
& Nasiru M. Olakorede
2
1,2
Department of Statistics, University of Abuja, Abuja, Nigeria
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150400110
Received: 10 April 2026; Accepted: 16 April 2026; Published: 19 May 2026
ABSTRACT
This study investigates the statistical properties, volatility dynamics, and forecasting performance of crude oil
returns using a comprehensive time series modeling approach. Descriptive analyses reveal that returns are
centered around zero with small negative averages, exhibiting pronounced volatility clustering and episodes of
extreme deviations driven by geopolitical and macroeconomic shocks. Diagnostic tests confirm the presence of
nonlinear dependence and heteroskedasticity, making models such as GARCH suitable for capturing the
persistent volatility patterns. Stationarity of the return series supports the application of ARIMA and GARCH-
based models, which effectively accommodate the complex features observed in the data. Model selection across
varying sample sizes consistently favors parsimonious ARIMA(0,0,q) structures with no autoregressive terms,
while GARCH(1,1) captures the high volatility persistence evident in the market. Forecasting evaluations
demonstrate that model accuracy improves with larger datasets, with combined ARIMA-GARCH models,
especially those with higher-order ARIMA specifications, outperforming simpler models in larger samples. The
findings underscore the importance of incorporating volatility modeling and selecting appropriate model
complexity based on data availability to enhance forecast precision. Overall, the results provide valuable insights
into the dynamics of crude oil prices and offer robust guidance for modeling and forecasting in volatile energy
markets.
Keywortds: ARIMA, GARCH, Forecasting, Volatility, Parsimonious, Financial, Crude Oil.
INTRODUCTION
Forecasting in statistics is concerned with predicting future outcomes using historical observations arranged in
time order. A time series differs from other datasets because its values are recorded at regular intervals, making
it possible to detect systematic patterns such as upward trends, cyclical fluctuations, or random disturbances.
Understanding these dynamics is essential in domains like agriculture, business, and finance where planning
depends on reliable projections.
The Autoregressive Integrated Moving Average (ARIMA) framework is one of the most established approaches
for such analysis. It combines two fundamental ideas: the autoregressive (AR) model, which relates current
observations to previous values, and the moving average (MA) model, which expresses present outcomes in
terms of past forecast errors. When data are non-stationary, the integrated (I) component applies differencing to
stabilize the mean and variance before modeling. This makes ARIMA particularly useful for data that exhibit
trends over time.
Since its development and popularization by Box and Jenkins (1976), ARIMA has become a versatile method
for short-term prediction of linear time series. It can be implemented in two major forms: a non-seasonal ARIMA
model, which captures trends and dependencies without seasonal adjustment, and a seasonal ARIMA (SARIMA)
model, which incorporates recurring cycles such as monthly or yearly effects. These variations allow ARIMA
to be adapted across contexts, from analyzing financial returns to modelling agricultural production levels.
Financial markets are often characterized by irregular fluctuations in asset prices, where periods of calm are
interrupted by episodes of extreme turbulence. This feature, known as volatility clustering, implies that returns
are not constant over time and that traditional models assuming fixed variance are inadequate. Understanding
Page 1271
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
and forecasting this variability is central to decision-making in risk management, asset pricing, and portfolio
allocation.
The early attempt to capture changing variance was made by Engle (1982) through the Autoregressive
Conditional Heteroskedasticity (ARCH) model, which demonstrated that the conditional variance of a return
series could be linked to past error terms. Although successful, the ARCH model often required a large number
of lagged terms to fit financial data adequately. To address this, Bollerslev (1986) proposed the Generalized
ARCH (GARCH) model, which extended Engle’s approach by allowing the conditional variance to depend both
on lagged squared errors and on past variances. This development made the model more flexible and
computationally efficient.
Since its introduction, the GARCH family of models has become a dominant tool in financial econometrics.
They are widely applied to estimate Value-at-Risk (VaR), study exchange rate dynamics, analyze stock market
returns, and support regulatory capital requirements. The core strength of GARCH lies in its ability to capture
persistence in volatility and to reflect the clustering commonly observed in empirical financial data. In addition,
several extensions such as Exponential GARCH (EGARCH), Threshold GARCH (TGARCH), and GJR-
GARCH have been introduced to address asymmetry in volatility, where negative shocks tend to have a stronger
effect than positive ones.
Recently, there have been increased attention in the literature towards analysis of volatility data using different
approaches.
Mbonigaba et al. (2025) employed GARCH models for predicting market volatility in high-frequency trading.
They studied data from 2020 to 2024. The research compared traditional GARCH models with newer versions
like EGARCH and TGARCH. They found that while traditional GARCH models are a good starting point for
forecasting volatility, they aren't accurate enough in high-frequency trading without improvements. The study
showed that the newer GARCH versions work better at capturing how volatility changes in response to market
events. To improve predictions in high-frequency trading, the authors suggest combining GARCH models with
machine learning, using methods to reduce noise, and creating ways to adjust models in real time for better
results.
Di-Giorgi et.al. (2025) proposed a hybrid approach to volatility prediction by combining GARCH models with
deep neural networks (GRU, LSTM, BiLSTM). Their method also introduces an optimal sliding window
mechanism. Through simulations and applications to real financial data (Chile’s IPSA, S&P 500, ASX200), the
approach showed improved accuracy in forecasting stock return volatility, particularly one week ahead.
Sirisha et al. (2024) Conducted a comparative analysis of time series forecasting methods using both statistical
and deep learning approaches. Specifically, they applied the Autoregressive Integrated Moving Average
(ARIMA), Seasonal ARIMA (SARIMA), and the deep learning-based Long Short-Term Memory (LSTM)
neural network models to forecast profit data. To ensure the appropriateness of the statistical models, the dataset
used was transformed to stationarity for ARIMA, while no such transformation was performed for SARIMA and
LSTM. The models were trained and evaluated using test data to assess forecasting performance. Their findings
revealed that the LSTM model outperformed both ARIMA and SARIMA, demonstrating superior predictive
accuracy and highlighting the potential of deep learning methods in complex time series forecasting tasks.
Bunnag (2024) analyzed how gold acts as a safe-haven asset when the economy is uncertain. They used two
models, ARIMA (2,1,3) and ARIMA (2,1,3)-GARCH (1,1), to predict gold prices. When they compared the
models, the ARIMA (2,1,3)-GARCH (1,1) one performed better, with lower MAE (80.1371) and RMSE
(96.8299), showing it's more accurate. The model predicts that gold will cost about 1942.094 USD per troy ounce
by the end of 2024, which shows how well it can forecast trends in the gold market.
Hasanov et al. (2024) examined the empirical significance of structural changes in exchange rate data by
replicating and extending the volatility forecasting study of Rapach and Strauss (2008). Using generalized
autoregressive conditional heteroskedasticity (GARCH) models and out-of-sample tests, they incorporated
recent U.S. dollar daily exchange rate data, employed alternative software, and applied updated forecast accuracy
Page 1272
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
tests and loss metrics. Their objective was to achieve scientific replication in a broad sense while assessing
robustness. The findings broadly confirmed the original study, with additional evidence showing that models
accounting for structural breaks consistently outperformed those that ignored instabilities across all forecast
horizons and loss functions.
Jatinder et al. (2023) highlights the prominence of the ARIMA model as a widely accepted and robust time series
forecasting method. They conducted an extensive analysis of ARIMA and hybrid models incorporating ARIMA,
spanning fields such as environmental, health, atmospheric, and other disciplines. Their findings suggest that
hybrid models, which integrate ARIMA with other approaches, outperform standalone ARIMA models by
effectively capturing a broader range of patterns within time series data. Consequently, the adoption of hybrid
modelling has become a standard practice, leveraging the combined strengths of multiple methods to enhance
forecasting accuracy and robustness.
Quang et al. (2024) ground their study in financial time-series and volatility modeling theory, which explains
Bitcoin price behavior through historical price movements and time-varying volatility. The framework assumes
that Bitcoin prices exhibit non-stationarity, autocorrelation, and volatility clustering. Bitcoin price returns are
treated as the dependent variable, with the ARIMA model capturing short-term return dynamics in the mean
equation, while the GARCH model explains conditional volatility in the variance equation. The framework
acknowledges the influence of macroeconomic and financial factors on Bitcoin prices but excludes them from
the ARIMA–GARCH structure, indicating a limitation and direction for future research. Model specification
and frequent recalibration are incorporated as moderating elements due to Bitcoin’s high volatility. Overall, the
framework highlights the effectiveness of the ARIMA–GARCH integration for short-term Bitcoin price
forecasting while emphasizing the need for adaptive and extended modeling approaches for improved long-term
accuracy.
The aim of this study is to evaluate and compare the forecasting accuracy of ARIMA, GARCH and ARIME-
GARCH models through real-world data applications, thereby providing insights into their respective
effectiveness in modeling and forecasting time series data exhibiting volatility dynamics.
The objectives were to;
i. Examine the trends and patterns of crude oil prices returns;
ii. Fit the crude oil prices returns to both ARIMA and GARCH models; and
iii. evaluate and compare the forecasting accuracy of ARIMA, GARCH and ARIMA-GARCH models.
METHODOLOGY
Autoregressive Integrated Moving Average (ARIMA) Model
The autoregressive integrated moving average (ARIMA) model is a sophisticated statistical tool that synthesizes
autoregressive (AR) and moving average (MA) components with a differencing process to effectively model
time series data. The differencing step is pivotal in transforming non-stationary data into a stationary form,
thereby facilitating reliable statistical analysis. This integrated approach, combining AR, MA, and differencing,
is termed ARIMA. This section elucidates the mathematical derivation of the autoregressive (AR) component,
establishing the groundwork for its integration with MA and differencing processes. An autoregressive (AR)
model is a representation of a type of random process. It is used to describe certain time-varying processes in
time series data.
The ARIMA Model
ARIMA models are used for time series data that either already have a constant mean and variance over time
or can be made to have these properties through a process called differencing. The model is written as ARIMA
(p, d, q),
where:
Page 1273
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
p: Order of the Autoregressive (AR) component.
d: Degree of differencing required to make the time series stationary.
q: Order of the Moving Average (MA) component.
Component of ARIMA
Autoregressive (AR): The AR part of the model looks at how an observation connects to past observations. It
believes that the current value of the time series is made up of a mix of earlier values and some random error.
Mathematical Representation:
2.1
Where:
: Value of the time series at time (t)
(c): Constant term.
: Parameters (coefficients) of the lagged terms.
: Random error (white noise) at time (t).
(p): Number of lagged observations (order of the AR model).
Integrated (I): The "I" in ARIMA stands for integration, which refers to differencing the time series to make
it stationary. A stationary time series has a constant mean, variance, and autocorrelation structure over time.
Differencing removes trends and stabilizes the mean.
Differencing:
First-order differencing:
Second-order differencing (if needed):
(d): Number of times the data is differenced to achieve stationarity.
Moving Average (MA): The MA component shows how the current observation is connected to previous
forecast errors. An MA(q) model shows the current value as a mix of past error terms.
Mathematical Representation:
2.2
Where:
: The observed time series value at time (t).
c: The mean of the time series (often assumed to be 0 for simplicity in a stationary series)
: The white noise error term at time (t), assumed to be independent and identically distributed with mean
0 and variance
: The moving average coefficients, which measure the impact of past errors on the current
observation.
(q): The order of the MA model, indicating the number of lagged error terms included.
Hence, The ARIMA(p,d,q) model can be written as:
2.3
Where:
: The observed time series value at time (t).
Page 1274
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
: The autoregressive (AR) operator, defined as
where
are the AR coefficients, and (B) is the backshift operator.
The differencing operator is applied (d) times to make the series stationary by removing trends.
The moving average (MA) operator, defined as
Where
These are the MA coefficients.
: The error term (white noise) at time (t), assumed to be independent and identically distributed with mean 0
and variance
Forecasting in the ARIMA Model
In ARIMA modelling, forecasting is a crucial application for predicting future values in a time series. Moreover,
we often want to predict the future value of a series, Y
t
, based on past observations. Two common forecasting
strategies are One-step-ahead forecasting and multi-step-ahead forecasting.
One-Step Ahead Forecasting in ARIMA
One-step ahead forecasting predicts the next value in the time series, which is the observation at time t+1, by
using all the data available up to time t. It calculates the expected value of the time series based on the structure
of the ARIMA model.
Mathematically,
2.4
Where;
: forecast of
given data up to t.
autoregressive coefficients.
forecast error at time t.
Multistep Ahead Forecasting in ARIMA
Multistep ahead forecasting predicts values beyond one period (
. Since future observations
and errors are unknown, the forecast uses expected values, setting future errors to zero (E [
and
replacing future observations with their forecasts.
The forecast equation becomes recursive:
2.5
Where:
: forecast
given data up to t.
For h>1, the past values and errors used are themselves forecasts rather than actual observations.
: forecast error at step h, which is assumed to be zero for h>1
Generalized Autoregressive Conditional Heteroskedasticity (GARCH) Model
The GARCH model is commonly used in time series analysis to understand and predict volatility, especially in
financial and economic data. Volatility often shows clustering, meaning big changes tend to follow big changes,
and small changes follow small ones. Unlike ARIMA, which focuses on the average value of a time series,
GARCH looks at how the variability changes over time. It builds on the ARCH model by adding lagged variance
terms, making it more adaptable and efficient. The GARCH(p,q) model includes two key equations: one for the
Page 1275
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
average and one for the variance, where p represents the number of lagged variances and q represents the number
of lagged squared errors.
Mathematically,
A time series
can be represented as:
2.6
Where:
: conditional mean (can be modelled as ARMA, ARIMA, etc.)
: error term,
iid (0,1) Standardize White Noise
: Conditional Variance at time t.
Hence, the GARCH (p,q) model for the conditional variance is:
2.7
: conditional variance (the forecasted volatility)
: constant term (baseline variance)
≥ 0: ARCH parameters
≥ 0: GARCH parameters (long-run persistence of past variance).
: ensures stationarity (volatility does not explode)
Forecasting in the GARCH Model
Forecasting in GARCH focuses on predicting future conditional variances
for horizon h= 1,2... which is
crucial for risk management, Value-at-Risk (VaR) estimation, option pricing, and portfolio optimisation. Unlike
ARIMA models, which forecast the level of the series, GARCH captures volatility clustering, Value, and time-
varying heteroskedasticity. ARCH forecasts are generated after estimating the model parameters using maximum
likelihood estimation (MLE) on historical data. The process involves two components: the mean equation (often
a simple constant or ARMA for returns) and the variance equation. Forecasting methods are recursive and rely
on the autoregressive nature of the model. Below are the forecasting methods, along with their mathematical
representations for the general GARCH(p,q) model and the commonly used GARCH(1,1) variant.
One-Step Ahead Forecasting
One-step ahead forecasting predicts the next period's volatility
using all information up to time (t ). This is
the most accurate short-term forecast since it incorporates the latest observed residual
and variance
For h =1
2.8
: constant term (long-run average variance).
: ARCH parameter: measures how much yesterday’s shock (
) affects today’s volatility.
Page 1276
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
: GARCH parameter: measures how much yesterday’s variance (
) carries over into today’s variance.
: the error/residual from the mean equation at time t.
: the error/residual from the mean equation at time t.
Multi-Step Ahead Forecast
For horizons h≥2, forecasts are recursive because future residuals
(k>0) are unknown. The
approach sets future shocks to their expectation (
≈ 0) and uses previously forecasted
variances. This leads to forecasts that decay toward the unconditional variance as (h ) increases, reflecting mean-
reversion in volatility.
For GARCH (1,1):
) 2.9
This means:
As h→∞,
Where:
ω: constant (baseline variance)
ɑ: ARCH coefficient
β: GARCH coefficient
: most recent conditional variance at time t.
h: forecast horizon (number of steps ahead).
: unconditional (long-run) variance. 2.10
ARMA- GARCH Model
The hybrid ARIMA-GARCH model combines an ARIMA(p, d, q) process for the conditional mean of the time
series
with a GARCH(r, s) process for the conditional variance (volatility) of the residuals
. The Mean equation (ARIMA p,d,q Component), is:
2.11
Expanded form:
2.12
Where
is the innovation (residual) term that is not white noise.
The Variance Equation GARCH component
i.i.d. (0,1)
Page 1277
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
2.13
The most widely used specification in practice is GARCH (1,1), i.e., r = 1, s = 1.
Where:
Observed value of the time series at time ( t )
= Constant mean (drift) of the process
= 1
AR (autoregressive) polynomial of order ( p )
1-
MA (moving average) polynomial of order ( q )
= AR coefficients
MA coefficients
Residual/innovation at time ( t )
Conditional variance (volatility) at time ( t )
(B) = Backshift (lag) operator, where
( d ): Order of differencing (integration) to achieve stationarity
Constant term in the variance equation
ARCH coefficients (impact of past squared shocks)
GARCH coefficients (impact of past conditional variances)
Standardized white-noise innovation
( r ): Order of ARCH terms in GARCH
( s ): Order of GARCH terms
The onstraints for GARCH stationarity:
> 0,
,
Forecasting Model of ARIMA-GARCH
The hybrid model produces point forecasts for the mean and volatility forecasts for uncertainty.
h-step-ahead forecast for the mean (
Obtained directly from the fitted ARIMA(p, d, q) equation
using the recursive ARMA forecasting formula on the differenced series, then integrated back (undo
differencing).
h-step-ahead forecast for conditional variance
Computed recursively from the GARCH equation
h = 1
=
2.14
Page 1278
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
Since
E
=
for future periods)
The full forecast interval for
is typically:
Where:
= is the critical value from the distribution of
= h-step-ahead point forecast of the original time series
( h ): Forecast horizon (number of steps ahead)
= h-step-ahead forecast of the conditional variance
Constant (intercept) term in the GARCH variance equation.
= ARCH coefficients
GARCH coefficients
Residual (innovation) from the ARIMA mean equation at time t
Conditional variance at time t
( r ): Order of the ARCH terms in the GARCH model
( s ): Order of the GARCH terms in the GARCH model
Critical value from the distribution of the standardized innovation
ANALYSIS AND RESULT
Crude Oil Prices Returns
Crude oil prices data use for this study consist of daily time series observations extracted from the Central Bank
of Nigeria (CBN) website, spanning the period from 25
th
October 2021 to 11
th
November 2025. The dataset
captures the day-to-day movement in crude oil prices over the study period, making it appropriate for time series
analysis because it preserves the chronological ordering and short-run fluctuations inherent in oil price dynamics.
As a daily frequency dataset, it is particularly useful for examining patterns such as volatility, trends, shocks,
and possible structural changes in crude oil prices over time. The use of data sourced from CBN enhances the
credibility and reliability of the study given the institution’s role as an official repository of key macroeconomic
and financial statistics in Nigeria. The analysis of the crude oil return data reveals several key insights relevant
to real-world applications in financial econometrics and commodity market analysis.
Table 3.1: Descriptive Statistics and Preliminary Tests of the Crude Oil Price Returns
Sample Size
50
100
250
500
1000
Descriptive Statistics
Mean
-0.0016355
-0.002128
-0.00075
-0.000740
-0.000276
Min.
-0.0430339
-0.094261
-0.09426
-0.09426
-1.086203
Page 1279
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
Max.
0.0515610
0.051561
0.06921
0.06921
1.102399
SD
0.0195205
0.019979
0.01993
0.01942
0.05408
Preliminary Tests
ADF Test
I(0) 0.0131*
I(0) 0.01*
I(0) 0.01*
I(0) 0.01*
I(0) 0.01*
BDS
0.0098*
0.01*
0.0326*
0.0097*
0.00002*
ARCH Test
0.7538
0.9095
0.0145*
0.0031*
2.20E-16*
Note: * denotes significant at 0.05 level
Source: Researchers’ Compilations from R-Output
Figure 3.1. Time Series Plots of Crude Oil Price Returns
-1.5
-1
-0.5
0
0.5
1
1.5
11/11/2025
13/08/2025
09/05/2025
03/02/2025
01/11/2024
02/08/2024
30/04/2024
25/01/2024
25/10/2023
27/07/2023
20/04/2023
18/01/2023
18/10/2022
18/07/2022
11/04/2022
12/01/2022
Crude Oil Price Returns
n=1000
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
11/11/2025
22/10/2025
02/10/2025
12/09/2025
22/08/2025
04/08/2025
14/07/2025
24/06/2025
30/05/2025
09/05/2025
15/04/2025
24/03/2025
04/03/2025
12/02/2025
23/01/2025
03/01/2025
11/12/2024
21/11/2024
Crude Oil Price Returns n=250
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
11/11/2025
31/10/2025
22/10/2025
13/10/2025
02/10/2025
23/09/2025
12/09/2025
03/09/2025
22/08/2025
13/08/2025
04/08/2025
24/07/2025
14/07/2025
03/07/2025
24/06/2025
Crude Oil Price Returns n=100
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
11/11/2025
05/11/2025
30/10/2025
24/10/2025
20/10/2025
14/10/2025
08/10/2025
02/10/2025
26/09/2025
22/09/2025
16/09/2025
10/09/2025
04/09/2025
Crude Oil Price Returns n=50
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
11/11/2025
30/09/2025
18/08/2025
04/07/2025
19/05/2025
28/03/2025
14/02/2025
03/01/2025
19/11/2024
08/10/2024
23/08/2024
12/07/2024
29/05/2024
12/04/2024
23/02/2024
12/01/2024
28/11/2023
Crude Oil Price Returns n=500
Page 1280
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
Table 3.2: Estimation of Appropriate ARIMA Models across the Crude Oil Price Returns Sample Sizes
50
100
250
500
1000
ARIMA(0,0,0)
ARIMA(0,0,0)
ARIMA(0,0,2)
ARIMA(0,0,3)
ARIMA(0,0,1)
log-likelihood
97.47
log-likelihood
233.73
ma1
-0.0593*
mar1
-0.0244*
ma1
0.5436*
AIC
-192.94
AIC
-465.45
ma2
0.1575*
ma2
0.1501
log-likelihood
1607.16
-
-
log-likelihood
599.85
ma3
0.0322*
AIC
3210.31
-
-
AIC
-1193.7
log-likelihood
1234.67
-
-
-
-
AIC
2467.33
-
Note: * denotes significant at 0.05 level.
Source: Researchers’ Compilations from R-Output
Table 3.3: Estimation of GARCH(1,1) Model across the Crude Oil Price Returns Sample Sizes
Sample Size
50
100
250
500
1000
Mu
-0.001774
-0.001184
-0.000650
-0.00061
-0.000225
Omega
0.00001
0.00000
0.000094*
0.00004
0.000093
alpha1
0.00000
0.00000
0.123083*
0.10441*
0.164807
beta1
0.99899*
0.99900*
0.64626*
0.79678*
0.659608
logLikelihood
97.6143
233.9206
603.2151
1249.119
2285.302
AIC
-4.6807
-5.1093
-4.9935
-5.0821
-4.6087
Note: * denotes significant at 0.05 level
Source: Researchers’ Compilations from R-Output
Table3.4: Estimation of ARIMAGARCH Models across the Crude Oil Price Returns Sample Sizes
50
100
250
500
1000
ARIMA(0,0,0)
GARCH(1,1)
ARIMA(0,0,0)
GARCH(1,1)
ARIMA(0,0,2)
GARCH(1,1)
ARIMA(0,0,3)
GARCH(1,1)
ARIMA(0,0,1)
GARCH(1,1)
Mu
-0.0018
Mu
-0.001184
Mu
-0.000588
Mu
0.000342
Mu
-0.0002
Omega
Omega
ma1
mar1
ma1
Page 1281
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
0.000001
0.000000
-0.083115
-0.00243
-0.0236
alpha1
0.0000
alpha1
0.000000
ma2
0.125316
ma2
0.06254*
Omega
0.00009
beta1
0.99899*
beta1
0.999000*
Omega
0.000100
ma3
0.003292*
alpha1
0.16221
Loglikelihood
97.6143
LogLikelihood
233.9206
alpha1
0.121281
Omega
0.00002
beta1
0.66789
AIC
-4.6807
AIC
-5.1093
beta1
0.626822*
alpha1
0.08695
logLikelihood
2285.52
-
-
logLikelihood
605.44
beta1
0.84859*
AIC
-4.582
-
-
AIC
-4.9954
logLikelihood
1259.123
-
-
-
-
AIC
-5.1811
-
Note: * denotes significant at 0.05 level. Source: Researchers’ Compilations from R-Output
Table3.5: Comparison of Models’ Forecasting Performance across the Crude Oil Price Returns At Different
Sample Sizes
Sample
Size
Models
Fitting Performance
Forecast Accuracy
Loglikelihood
AIC
RMSE
MAE
50
ARIMA(0,0,0)
97.47
-
0.000405
0.000390
-
192.24
-
-
GARCH(1,1)
97.6143
-
0.000404
0.000389
-
4.6807
-
-
ARIMA(0,1,0)+GARCH(1,1)
97.6143
-
0.000403
0.000389
-
46807
-
-
100
ARIMA(0,1,1)
233.73
-
0.002724
0.001139
-
465.45
-
-
GARCH(1,1)
233.9206
-
0.002725
0.001138
-
5.1093
-
-
ARIMA(0,1,1)+GARCH(1,1)
233.9206
-
0.002725
0.001138
-
5.1093
-
-
250
ARIMA(1,0,0)
599.85
-
0.000329
0.0003167
-
1193.7
-
-
GARCH(1,1)
603.2151
-
0.000300
0.000290
Page 1282
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
4.9935
ARIMA(1,0,0)+GARCH(1,1)
605.44
-
0.000299
0.000289
4.9954
-
-
500
ARIMA(1,0,2)
-
-
0.0004689
0.000363
1234.67
2467.3
-
-
GARCH(1,1)
1249.119
-
0.0008069
0.000786
-
5.0821
-
-
ARIMA(1,0,2)+GARCH(1,1)
1259.123
-
0.0000867
0.000079
-
5.1811
-
-
10000
ARIMA(4,0,4)
1607.16
-
0.0021442
0.002129
-
3210.3
-
-
GARCH(1,1)
2285.302
-
0.000399
0.000389
-
4.6087
-
-
ARIMA(4,0,4)+GARCH(1,1)
2285.52
-4.582
0.0003957
0.000387
Source: Researchers’ Compilations from R-Output
DISCUSSION OF RESULTS
Table3.1 depicts the descriptive statistics and preliminary test of the crude oil price returns.
Firstly, the descriptive statistics indicate that crude oil returns are centered around a near-zero mean, with small
negative average returns observed across different sample sizes. This aligns with the typical behavior of financial
asset returns, which generally fluctuate around zero due to the efficient and unpredictable nature of market prices.
The small drift observed may reflect gradual market trends or macroeconomic factors, but overall, the data
exhibit a high degree of unpredictability.
Secondly, the volatility characteristics of crude oil returns demonstrate strong clustering behavior, a hallmark of
financial time series affected by market shocks, geopolitical events, and macroeconomic fluctuations. The time
series plots visually confirm periods of elevated volatility interspersed with calmer intervals, illustrating
persistent volatility that can last over extended periods. These patterns are driven by exogenous shocks such as
geopolitical conflicts, supply disruptions, OPEC decisions, and macroeconomic policy changes, which induce
abrupt and significant price movements.
Thirdly, the diagnostic tests reinforce the presence of nonlinear dependence and heteroskedasticity in the data.
The ARCH LM tests indicate significant conditional heteroskedasticity, especially in larger samples, confirming
that volatility in crude oil markets is not constant but evolves over time. The BDS tests reveal strong nonlinear
dependence, further suggesting that simple linear models like ARIMA are insufficient for capturing the complex
dynamics of crude oil returns.
The stationarity tests consistently classify the return series as stationary (I(0)), which is an important property
for modeling and forecasting. This stationarity implies that the statistical properties of returns do not change
over time, making models like GARCH appropriate for capturing the persistent volatility patterns. It also
supports the application of volatility models that focus on the conditional variance without requiring differencing
or detrending.
The time series visualizations reinforce these statistical findings, with crude oil returns displaying irregular
spikes corresponding to real-world shocks. Unlike simulated data, where shocks are generated within a
controlled process, the spikes in actual market data are motivated by tangible geopolitical and macroeconomic
Page 1283
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
events. These irregular and often extreme deviations highlight the importance of models that can accommodate
fat tails and asymmetric shocks.
Furthermore, the parameter estimates from GARCH(1,1) models consistently show high volatility persistence,
with parameters close to unity. This indicates that shocks to volatility tend to persist over time, reflecting the
structural nature of market uncertainty in crude oil trading. Such high persistence underscores the challenges in
forecasting crude oil volatility and emphasizes the need for models that can adapt to prolonged periods of market
turbulence.
Figure 3.1 presents the time series plots of the crude oil returns. The time series plot of crude oil returns, albeit
with more pronounced spikes attributable to real market shocks. The plot displays clear periods of heightened
volatility corresponding to global disruptions such as geopolitical conflicts, OPEC policy shifts, or
macroeconomic crises.Volatility clustering is evident, reinforcing the presence of persistent volatility typical of
commodity markets. The visual pattern demonstrates that the crude oil market behaves like other financial assets,
with returns fluctuating around zero, interspersed with extreme deviations. These characteristics justify the use
of volatility models such as GARCH and ARIMA–GARCH for modelling and forecasting such data.
Table3.2 examines the evolution of optimal ARIMA model structures as a function of sample size in the context
of crude oil price return series. The model selection process, based on likelihood functions and the Akaike
Information Criterion (AIC), reveals notable trends that inform understanding of the autocorrelation dynamics
inherent in the data.
At all examined sample sizes, the selected models consistently feature an autoregressive (AR) parameter of zero,
resulting in models of the form ARIMA(0,0,q). This consistent absence of autoregressive terms suggests that the
return series exhibits minimal to no evidence of autocorrelation at prior lags, which aligns with the efficient
market hypothesis where returns are often considered serially uncorrelated. The lack of AR terms indicates that
past return values do not significantly predict future returns, and any autocorrelation present is better captured
through moving average components rather than autoregressive structures.
At the smallest sample sizes (n=50 and n=100), the data favor a simple ARIMA(0,0,0) model, indicative of a
white noise process with no significant autocorrelation detected. This suggests that limited data constrains the
capacity to identify underlying temporal dependencies, resulting in a preference for minimal model complexity.
As the sample size increases to 250, the optimal model shifts to an ARIMA(0,0,2), incorporating MA(1) and
MA(2) components. This transition indicates that larger datasets enhance the statistical power to detect short-
term autocorrelation structures, which manifest as significant moving average terms. The inclusion of these
additional parameters reflects a more nuanced understanding of the autocorrelation pattern, likely attributable to
increased data-driven signal detection.
Further expansion of the sample size to 500 observations favors an ARIMA(0,0,3), suggesting the
autocorrelation extends further into recent lags. This progression underscores the capacity of larger datasets to
uncover more complex short-term dependencies within the return series.
Interestingly, at the largest sample size examined (n=1000), the optimal model reverts to the simpler
ARIMA(0,0,1). This reversion may indicate a refinement of the autocorrelation structure, where the data support
a more parsimonious model, or it may reflect overfitting concerns associated with higher-order models in smaller
samples. The result underscores the importance of model parsimony and the potential for over-parameterization
when datasets are overly complex relative to their true underlying structure.
Table3.3 investigates the stability and characteristics of volatility modeling in crude oil returns through the
estimation of a GARCH(1,1) model across different sample sizes: 50, 100, 250, 500, and 1000 observations. The
parameters estimated, mean return (Mu), constant variance (Omega), ARCH effect (alpha1), and GARCH effect
(beta1), along with model fit indices such as log-likelihood and AIC, provide insights into the evolving dynamics
of volatility as the dataset expands.
Page 1284
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
The estimated mean return remains negative across all sample sizes, decreasing in magnitude from
approximately -0.00177 at n=50 to -0.000225 at n=1000. This consistent negative mean aligns with the typical
characteristics of crude oil returns, which often exhibit slight negative drift over time. The diminishing
magnitude suggests that as the sample size increases, the average return tends toward zero, reflecting a more
stable estimate of the underlying return process.
Omega estimates are near zero for the smaller samples (n=50, 100), with a significant positive estimate at n=250
(p<0.05). For larger samples (n=500, 1000), Omega again hovers around 0.00009 but is not statistically
significant. The initial significance at n=250 indicates a persistent baseline level of volatility that is more
detectable with moderate sample sizes, while at larger sizes, the variance estimate stabilizes and becomes less
distinguishable from zero, suggesting that the model attributes most of the volatility clustering to the ARCH and
GARCH effects rather than a constant component.
The ARCH parameter (alpha1) is negligible and statistically insignificant in the smallest samples (n=50, 100),
implying that short-term shocks do not significantly influence volatility in these datasets. As sample size
increases to 250 and above, alpha1 becomes statistically significant (p<0.05), with estimates ranging from
approximately 0.123 to 0.165. This indicates that recent shocks have a meaningful impact on current volatility,
and the strength of this effect appears to grow with larger samples, reflecting the model's capacity to capture
more pronounced volatility clustering in extensive datasets.
The GARCH parameter (beta1) is highly significant and close to unity across all sample sizes, ranging from
approximately 0.646 to 0.999. Notably, at smaller sample sizes (n=50, 100), beta1 is extremely close to 1
(≈0.999), suggesting persistent volatility shocks that decay very slowly over time. As the sample size increases,
beta1 decreases somewhat (to around 0.646 at n=250 and 0.660 at n=1000), indicating a somewhat faster mean
reversion of volatility in larger datasets. The high significance of beta1 across all samples emphasizes the
presence of strong volatility persistence in crude oil returns.
Log-likelihood values increase substantially with sample size, from 97.6143 at n=50 to 2285.302 at n=1000,
reflecting improved model estimation with more data. Correspondingly, AIC scores improve (become more
negative), indicating better model fit in larger samples. The AIC decreases from about -4.6807 at n=50 to
approximately -5.1093 at n=100, then stabilizes around -4.6 to -5.08, suggesting that the model captures volatility
dynamics more effectively as the sample size grows.
Both the ARIMA and GARCH(1,1) models demonstrate valuable insights into the dynamics of crude oil returns,
yet each exhibits distinct strengths in capturing different aspects of the data. The ARIMA models consistently
favor simpler structures with no autoregressive terms across all sample sizes, indicating that crude oil returns
exhibit minimal autocorrelation and are predominantly driven by short-term shocks rather than persistent
autoregressive effects. In contrast, the GARCH(1,1) models reveal strong volatility clustering and persistence,
with high and significant GARCH parameters close to unity across all sample sizes, emphasizing the importance
of modeling volatility dynamics explicitly. While ARIMA models effectively capture the mean process and
short-term dependencies, the GARCH models excel in modeling the heteroskedastic nature of returns, which is
critical for risk management and derivative pricing.
Table3.4 presents an extensive comparison of ARIMAGARCH models applied to crude oil price returns across
varying sample sizes (50, 100, 250, 500, and 1000 observations), incorporating different ARIMA(0,0,0), (0,0,1),
(0,0,2), and (0,0,3) specifications with GARCH(1,1) volatility modeling. The results highlight that models with
minimal or no autoregressive components in the mean equation—particularly ARIMA(0,0,0)—yield high
likelihoods and strong GARCH parameter estimates, with beta1 consistently close to unity (≈0.999), indicating
persistent volatility. For smaller samples (n=50, 100), the models exhibit very high log-likelihoods and superior
AIC scores, emphasizing their effectiveness in capturing the volatility clustering characteristic of crude oil
returns. The inclusion of MA components (e.g., ma1, ma2, ma3) in the mean equation introduces additional
dynamics, but their estimated coefficients are generally small or insignificant, suggesting limited impact on the
mean process. Compared to the standalone ARIMA and GARCH models, the combined ARIMAGARCH
models demonstrate enhanced flexibility by simultaneously modeling mean and volatility dynamics, with the
best fit observed in models with no autoregressive terms and a simple GARCH(1,1) structure. Overall, the
Page 1285
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
ARIMAGARCH framework, particularly with minimal mean model complexity, outperforms the previous
models in capturing the persistent volatility and subtle mean dynamics of crude oil returns, providing a more
comprehensive and effective approach for modeling such financial time series. Consequently, for comprehensive
modeling of crude oil, a combined approach that incorporates both ARIMA for mean dynamics and GARCH for
volatility clustering provide the most robust framework, with the GARCH component clearly demonstrating
superior performance in capturing the persistent volatility characteristic of crude oil markets.
Table3.5 provides a comprehensive evaluation of the forecasting performance of various time series models
applied to crude oil price returns across differing sample sizes. The comparison encompasses both in-sample
fitting metrics; log-likelihood and Akaike Information Criterion (AIC), and out-of-sample forecast accuracy
measures; Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). The results elucidate how model
efficacy evolves with increasing data availability, offering valuable insights into model selection strategies for
financial time series forecasting.
Across all sample sizes, the models demonstrate consistent in-sample fitting capabilities, with higher log-
likelihood values indicating better fit. Notably, the ARIMA(1,0,2)+GARCH(1,1) model exhibits superior in-
sample likelihood at larger sample sizes (e.g., 250 and 500 observations), suggesting its enhanced capacity to
capture underlying data dynamics as the dataset grows. The AIC values, align with these findings, favoring
models that balance fit and parsimony.
At the minimal sample size of 50, models achieve remarkably low RMSE (~0.0004) and MAE (~0.00039),
indicating high forecast precision in short-term predictions with limited data. The ARIMA(0,1,0)+GARCH(1,1)
model slightly outperforms others, highlighting its robustness in sparse data environments.
As the sample size increases, forecast errors decrease notably, with the ARIMA(1,0,2)+GARCH(1,1) model
attaining the lowest RMSE (~0.0000867) and MAE (~0.000079) at 500 observations. This trend underscores the
advantage of larger datasets in enhancing model accuracy, especially when volatility modeling via GARCH
components is incorporated.
At the largest sample size examined, the ARIMA(4,0,4)+GARCH(1,1) model demonstrates superior forecast
accuracy, evidenced by the lowest MAE (~0.000387). The results suggest that, with ample data, models capable
of capturing complex dynamics, such as higher-order ARIMA components combined with GARCH, offer the
most precise forecasts. Additionally, the substantial increase in in-sample likelihoods corroborates the improved
fit and predictive power.
The findings highlight the importance of model complexity in relation to sample size. While simpler models like
ARIMA(0,1,0)+GARCH(1,1) perform adequately with limited data, more sophisticated models with higher
orders (e.g., ARIMA(4,0,4)+GARCH(1,1)) outperform in large-sample contexts, benefiting from their capacity
to model intricate data structures. Incorporating volatility dynamics via GARCH consistently enhances forecast
accuracy across different sample sizes, reaffirming the significance of volatility modeling in financial return
series.
Overall, the analysis demonstrates that model performance in forecasting crude oil returns improves with
increasing sample size, with complex models gaining prominence in larger datasets. The integration of GARCH
components consistently contributes to reducing forecast errors, emphasizing the importance of capturing
volatility in financial time series modeling. These insights provide a valuable framework for practitioners and
researchers in selecting appropriate models based on data availability and the underlying data-generating
process.
SUMMARY OF FINDINGS
This analysis presents a comprehensive examination of the statistical properties and modeling strategies for crude
oil return series. Descriptive statistics reveal that returns are centered near zero with small negative averages,
consistent with typical financial asset behavior characterized by high unpredictability. Visual inspections of the
Page 1286
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
time series indicate pronounced volatility clustering, with periods of elevated market turbulence often linked to
geopolitical events, supply disruptions, and macroeconomic shocks. Diagnostic tests confirm the presence of
nonlinear dependence and heteroskedasticity, notably through significant ARCH effects and nonlinear dynamics
identified by the BDS test. Stationarity assessments verify that the return series are stationary (I(0)), supporting
the suitability of volatility models such as GARCH.
Model selection procedures across varying sample sizes show a consistent preference for models with minimal
autoregressive components, specifically ARIMA(0,0,q), indicating limited autocorrelation in returns’ a finding
aligned with the efficient market hypothesis. As sample size increases, models incorporating short-term moving
average terms (e.g., ARIMA(0,0,2) and ARIMA(0,0,3)) become optimal, reflecting the improved detection of
subtle autocorrelation structures with more data. Volatility modeling via GARCH(1,1) consistently captures
persistent volatility clusters, with high GARCH parameters close to unity across all sample sizes, underscoring
the structural and enduring nature of market uncertainty.
Furthermore, combined ARIMA-GARCH models demonstrate superior performance in both in-sample fit and
out-of-sample forecasting accuracy, especially when model complexity is balanced to prevent overfitting. The
results indicate that while simple models effectively capture the mean process, the integration of GARCH
components is crucial for accurately modeling volatility dynamics and risk estimation. Notably, larger datasets
enable the use of more complex models; such as ARIMA(4,0,4)+GARCH(1,1); which outperform simpler
specifications in forecasting accuracy, as evidenced by lower RMSE and MAE values.
Overall, the findings suggest that for modeling crude oil returns, a parsimonious mean model combined with a
GARCH(1,1) volatility component offers an effective framework. As data availability increases, employing
higher-order ARIMA models alongside GARCH structures enhances predictive performance, capturing both the
short-term dependencies and persistent volatility characteristic of crude oil markets. These insights provide
valuable guidance for practitioners and researchers aiming to develop robust forecasting models in the context
of volatile commodity markets.
CONCLUSION
Based on the comprehensive analysis conducted, the study provides valuable insights into the modeling and
forecasting of crude oil returns, aligning with the stated objectives. The descriptive statistics and preliminary
tests confirmed that crude oil returns are centered around zero, exhibit strong volatility clustering, and
demonstrate nonlinear dependence and heteroskedasticity—characteristics typical of financial time series
affected by exogenous shocks such as geopolitical events and macroeconomic fluctuations. These findings
underscore the importance of employing models capable of capturing complex volatility dynamics.
In terms of model fitting, the results reveal that simple ARIMA models, particularly those with no autoregressive
components, are sufficient to capture the mean dynamics of crude oil returns, consistent with the efficient market
hypothesis. Conversely, volatility modeling through GARCH(1,1) consistently captures persistent volatility
clustering and high volatility persistence, highlighting the necessity of incorporating heteroskedasticity in the
modeling framework.
The comparison of forecasting performances indicates that model accuracy improves with increasing sample
size. While simpler models perform adequately with limited data, more sophisticated models, particularly the
ARIMA-GARCH combinations with higher-order ARIMA components offer superior forecasting accuracy in
larger datasets. Notably, the ARIMA(4,0,4)+GARCH(1,1) model consistently outperforms others in terms of
forecast precision, as evidenced by the lowest out-of-sample RMSE and MAE at larger sample sizes. The
integration of volatility modeling via GARCH enhances predictive performance across all models, emphasizing
its critical role in capturing the persistent volatility characteristic of crude oil markets.
In conclusion, the findings demonstrate that the effectiveness of time series models in forecasting crude oil
returns depends significantly on data size and complexity. The combination of ARIMA for mean dynamics and
GARCH for volatility provides a robust and flexible framework capable of accurately modeling and forecasting
the complex behavior of crude oil prices. This study affirms that incorporating volatility models such as GARCH
Page 1287
www.rsisinternational.org
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue IV, April 2026
into traditional ARIMA structures enhances forecast accuracy, thereby fulfilling the research objectives of
evaluating and comparing the forecasting capabilities of these models in real-world data applications.
Implication and Future Research Directions
The findings have important policy and practical implications. The persistent volatility clustering and high
volatility persistence highlight the need for market participants to adopt advanced volatility forecasting models,
such as ARIMA-GARCH, for risk management and hedging. The minimal autocorrelation supports market
efficiency, suggesting that policies promoting transparency can further stabilize markets. Recognizing the impact
of geopolitical and macroeconomic shocks on volatility can inform policymakers in designing strategic
interventions to mitigate market turbulence. For investors, incorporating volatility forecasts can enhance asset
allocation and risk assessment.
Future research should explore models incorporating exogenous variables (e.g., macroeconomic indicators,
geopolitical risk), asymmetric GARCH models to capture leverage effects, and regime-switching frameworks to
account for structural breaks. Additionally, analyzing high-frequency data and comparing alternative volatility
models, including machine learning approaches, could improve real-time forecasting accuracy and robustness.
These avenues will deepen understanding of crude oil market dynamics and enhance predictive capabilities.
REFERENCE
1. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of
Econometrics, 31(3), 307–327. doi:10.1016/0304-4076(86)90063-1
2. Box, G.E.P. and Jenkins, G.M. (1976) Time Series Analysis: Forecasting and Control. Revised Edition,
Holden Day, San Francisco.
3. Bunnag, T. (2024). The importance of gold’s effect on investment and predicting the world gold price
using the ARIMA and ARIMA-GARCH model. Ekonomikalia Journal of Economics, 2(1), 38–52.
doi:10.60084/eje.v2i1.155
4. Di-Giorgi, G., Salas, R., Avaria, R., Ubal, C., Rosas, H., & Torres, R. (2025). Volatility forecasting using
deep recurrent neural networks as GARCH models. Computational Statistics, 40(6), 3229–3255.
doi:10.1007/s00180-023-01349-1
5. Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of
United Kingdom inflation. Econometrica, 50(4), 987–1007. doi:10.2307/1912773
6. Hasanov, A. S., Brooks, R., Abrorov, S., & Burkhanov, A. U. (2024). Structural breaks and GARCH
models of exchange rate volatility: Re-examination and extension. Journal of Applied Econometrics,
39(7), 1403–1407. doi:10.1002/jae.3091
7. Kaur, J., Parmar, K. S., & Singh, S. (2023). Autoregressive models in environmental forecasting time
series: A theoretical and application review. Environmental Science and Pollution Research, 30, 19617–
19641. doi:10.1007/s11356-023-25148-9
8. Mbonigaba, C., Vasuki, M., Kumar, A. D., & Asamoah, P. J. (2025). Applications of GARCH models
for volatility forecasting in high-frequency trading environments. International Journal of Applied and
Advanced Scientific Research, 10(1), 12–21. doi:10.5281/zenodo.14904200
9. Phung Duy, Q., Nguyen Thi, O., Le Thi, P. H., Pham Hoang, H. D., Luong, K. L., & Nguyen Thi, K. N.
(2024). Estimating and forecasting Bitcoin daily prices using ARIMA-GARCH models. Business
Analyst Journal, 45(1), 11–23. doi:10.1108/BAJ-05-2024-0027
10. Rapach, D. E., & Strauss, J. K. (2008). Structural breaks and GARCH models of exchange rate volatility.
Journal of Applied Econometrics, 23(1), 65–90. doi:10.1002/jae.976
11. Sirisha, U. M., Belavagi, M. C., & Attigeri, G. (2024). Profit prediction using ARIMA, SARIMA and
LSTM models in time series forecasting: A comparison. IEEE Access, 10, 124715–124727.
doi:10.1109/ACCESS.2022.3224938
