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Iraqi Journal of Statistical Sciences, Vol. 20, No. 2, 2023, Pp (249-262)
249
Iraqi Journal of Statistical Sciences
www.stats.mosuljournals.com
The Comparison Between VAR and ARIMAX Time Series Models in
Forecasting
Esraa Awni Haydier Nasradeen Haj Salih Albarwari and Taha Hussein Ali
Salahaddin University, College of Administration and Economics, Department of Statistics and Informatics. Erbil,
Iraq
Duhok University, College of Administration and Economics, Department of Statistics and Informatics, Duhok,
Iraq
Article information
Abstract
Article history:
Received October 11, 2023
Accepted November 20, 2023
Available online Decmber 1, 2023
The research presented a comparative study in time series analysis and forecasting
using VAR models, which depend on the existence of a significant relationship between
the studied variables, and ARIMAX models, which depend on the linear effect of the
independent variables (model input) on the dependent variable (model output). The
models were analyzed using time series data for the Iraqi general budget for the period
(2004-2020), which represents foreign reserves and government spending. Time series of
government expenditure was forecast for the years (2021-2024) and a comparison was
made between the efficiency of the models estimated through the mean square error
(MSE) criterion. The analysis was carried out using the MATLAB program, and the
results of the analysis concluded that the VAR model was more efficient than the
ARIMAX model for this data and the increase in foreign reserves and government
spending for the Iraqi will continue during the coming period (2021-2024).
Keywords:
Time Series,
VAR,
ARIMAX,
Forecasting,
Iraqi budget.
Correspondence:
Esraa Awni Haydier
esraa.haydier@su.edu.krd
DOI: 10.33899/ iqioss .2023.181260 , ©Authors, 2023, College of Computer and Mathematical Science, University of Mosul.
This is an open access article under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Time series is a statistical and data analysis technique used to study and analyze data points collected or
recorded over time. It involves the analysis of data points ordered chronologically, typically at uniform
time intervals, such as daily, monthly, or yearly. Time series data can be found in various fields, including
economics, finance, environmental science, engineering, and many others. Overall, time series analysis is a
powerful tool for understanding and making predictions based on temporal data, allowing businesses and
researchers to make informed decisions and respond to changing trends and patterns over time.
VAR, or Vector Autoregression, is a statistical modelling technique commonly used in time series
analysis, especially in the context of multivariate time series data. It is an extension of the autoregressive
(AR) model, which focuses on the relationship between a single time series and its past values. In VAR
models, you analyze the relationships between multiple time series variables and their past values
simultaneously. VAR models are beneficial when you want to capture the dynamic interactions between
multiple variables over time. They are commonly used in macroeconomics, finance, and other fields where
understanding the joint behaviour of multiple time series is crucial for decision-making and policy analysis
[22].
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ARIMAX, which stands for Autoregressive Integrated Moving Average with exogenous inputs, is a
time series analysis and forecasting model that extends the traditional ARIMA (autoregressive Integrated
Moving Average) model by incorporating exogenous or external predictor variables. ARIMAX is used for
modelling and forecasting time series data when the behaviour of the series is influenced by not only its
past values but also by external factors. ARIMAX modelling is a powerful tool when you need to account
for the impact of external factors or exogenous variables in your time series analysis and forecasting. It
allows for more accurate predictions by considering not only the series' history but also the influence of
other relevant variables [1].
Forecasting in time series analysis is the process of making predictions or estimates about future data
points based on historical observations. Time series forecasting is a critical component in various fields,
including finance, economics, meteorology, and operations management, among others. To perform
accurate forecasting, several methods and techniques are available, depending on the characteristics of the
time series data. The choice of forecasting method depends on the nature of the time series data, including
its stationarity, seasonality, and the presence of exogenous variables. It's often recommended to experiment
with different methods and evaluate their performance to select the most appropriate approach for a given
dataset. Additionally, time series forecasting often requires continuous monitoring and reevaluation as new
data becomes available, allowing for model updates and improvements [4].
The Iraqi budget, like the budget of any other country, is a financial plan that outlines the government's
expected revenues, expenditures, and allocations for a fiscal year. Here, the time series of foreign reserves
and government spending will be analyzed through VAR and ARIMAX models, and then compared
between them to obtain the best model used in forecasting.
2. Theoretical Aspect
The theoretical aspect presented some basic concepts on the subject of research from the statistical side,
as shown in the following paragraphs.
2.1. Time Series
Time series forecasting is widely used across diverse fields, including statistics, inventory management,
and economics. Various forecasting models are available, ranging from simpler techniques like moving
averages and linear regression to more complex approaches like ARIMA and neural networks [21].
Time series analysis is a statistical method used to analyze and model data that is collected and ordered
over time. The information you provided offers a fundamental understanding of time series and its
components [16].
Time Series Definition: A time series is an ordered sequence of observations. These observations are
usually taken at equally spaced intervals over time.
The objective of Time Series Modeling: The primary objective of time series modelling is to study and
analyze historical data from a time series to develop appropriate models that can be used to make
predictions or forecasts for future values of the series [17]. Time series analysis helps uncover patterns,
trends, and relationships within the data.
Components of Time Series: Time series data can be decomposed into four main components, which are
essential for understanding its underlying structure:
Trend (T): The trend component represents the long-term direction or movement in the data. It captures
gradual changes or trends in the data over time, such as an increasing or decreasing pattern.
Periodic (C): The periodic component accounts for regular, repeating patterns in the data, which may not
necessarily follow a linear trend. These patterns can have different time intervals [1].
Seasonal (S): Seasonal patterns are similar to periodic patterns but occur at fixed and known intervals, such
as daily, monthly, or yearly. Seasonal effects are often associated with calendar-related events, like
holidays or weather.
Irregular (I): The irregular component, also referred to as noise or residual, represents the unexplained or
random fluctuations in the data that cannot be attributed to the other components. It includes any random
variations, outliers, or noise in the time series.
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2.2. Stationary Time Series
Stationarity holds a central position in time series, as it acts as a fundamental prerequisite for various
statistical and mathematical techniques. The core principles of stationarity entail maintaining consistent
statistical characteristics over time, which encompass an unchanging mean (indicating the absence of a
trend), a uniform variance, and a stable autocorrelation structure [19].
In practical terms, numerous time series datasets exhibit non-stationary behaviour, wherein their statistical
properties, such as means, variances, or trends, evolve [20]. This non-stationary nature can pose challenges
when applying traditional time series analysis methods, which often assume stationarity [18].
To address this challenge, non-stationary time series data can be converted into a stationary format through
two primary methods:
Differencing: This technique involves computing the differences between sequential data points to
eliminate any trend and achieve a constant mean.
Transformation: In this approach, mathematical operations, such as taking the natural logarithm or square
root, are used to stabilize the variance [2].
The primary objective in ensuring stationarity is to simplify the analysis of time series data. Several
statistical models, including ARIMA (Autoregressive Integrated Moving Average), rely on the assumption
of stationarity. By transforming a time series into a stationary one, it becomes possible to obtain more
accurate forecasts and model estimates [13].
2.3. VAR Model
A VAR (Vector Autoregression) model is a type of statistical model used in time series analysis and
econometrics. It's a multivariate time series model that describes the relationship between multiple time
series variables [14]. In a VAR model, each variable is modelled as a linear combination of its past values
and the past values of all other variables in the system. For example, the system of equations for a VAR (1)
model with two-time series (variables `y1` and `y2`) is as follows:
Key features of a VAR model include: Unlike univariate time series models, which analyze a single
variable in isolation, VAR models consider multiple variables simultaneously. This is particularly useful
when you want to capture and model the dynamic interactions and feedback loops among different
variables [22].
A VAR model is typically specified with an order, denoted as VAR(p), where "p" represents the number of
lagged time points considered in the model. The choice of order is an important part of VAR modelling and
affects the complexity of the model. Just like with univariate time series models, stationarity is important
for VAR models. In a VAR(p) model, the variables should be stationary after differencing at least p times.
Estimating the parameters of a VAR model is usually done through techniques like the method of least
squares or maximum likelihood estimation. VAR models are often used for forecasting and assessing the
impact of shocks on the variables within the system. You can calculate impulse response functions to see
how a shock to one variable affects the others over time [15].
VAR models can be extended to VARMA (Vector Autoregressive Moving Average) models to account for
moving average components, and they are also a component of the more comprehensive VARMAX
models, which incorporate exogenous variables. These models can be useful for various tasks, such as
economic forecasting, policy analysis, and risk assessment in financial markets.
2.4. ARIMAX Model
ARIMAX is a time series forecasting model that combines ARIMA principles with exogenous variables. It
extends ARIMA by including external predictors (denoted as X) to improve forecasting accuracy. This
model involves specifying AR, I, and MA components, along with the exogenous variables, estimating
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model parameters, and making forecasts [3]. It's commonly used in economics, finance, and environmental
science to account for external influences on time series data. The ARMAX(p,q,r) model equation
There represents exogenous variables, their coefficients [12], and r is the
number of exogenous variables. The provided information describes the ARIMAX and ARMAX models,
which are commonly used in time series analysis and forecasting when exogenous variables are involved.
AR Terms: The ϕ1, ϕ2, …, ϕp terms are autoregressive terms, indicating the relationship between the
dependent variable and its past values. MA Terms: The et, θ1e(t−1), θ2e(t−2), …, θqe(t−q) terms are moving
average terms, which account for the influence of past errors in the model.
Application of ARIMAX: ARIMAX models are applied in various fields, including economics, agriculture,
and engineering, to improve predictive performance compared to the basic ARIMA model. The d
parameter in the ARIMAX model indicates the number of times differencing is applied to the time series to
make it stationary. This is often necessary when dealing with non-stationary data [4].
2.5. Efficiency criteria for estimated models
AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are two commonly used
statistical measures for model selection and evaluation in the context of regression models, including linear
regression and time series models like ARIMA and VAR. They are used to assess the trade-off between
model complexity and goodness of fit. Lower values of AIC and BIC indicate better-fitting and more
parsimonious models [10]. Here's an explanation of each:
The Akaike Information Criterion (AIC) is a statistical measure used for model selection and evaluation.
AIC is commonly used in the context of various statistical models, including linear regression, time series
models, and more [9]. Its primary purpose is to assess the trade-off between the goodness of fit of a model
and its complexity.
The AIC value for a given model is calculated using the following formula [5]:
AIC = -2 log-likelihood + 2 k (4)
Here's what each component of the formula represents:
1. -2 log-likelihood: This term measures how well the model fits the observed data. The log-likelihood is a
measure of how probable the observed data is under the model [8]. The negative sign indicates that you're
trying to maximize the likelihood (i.e., find the model that best fits the data).
2. 2 k: This term represents a penalty for model complexity. "k" is the number of parameters in the model.
The penalty term encourages the selection of simpler models, as adding parameters will increase AIC [11].
Model Selection: AIC is used to compare different models. When comparing models, you typically choose
the model with the lowest AIC value because it represents the best trade-off between goodness of fit and
model complexity [7].
Bayesian Information Criterion (BIC): It is a statistical criterion used for model selection among a finite set
of models. BIC balances the goodness of fit of the model with the number of parameters, penalizing models
with more parameters to avoid overfitting [6].
BIC = -2 log-likelihood + k log (n) (5)
To compare two different models, the mean square error (MSE) is used [23]:
3. Application Aspect
The practical aspect dealt with the two correlated time series that represent two variables, namely foreign
reserves (x1) and government spending (x2) in the general budget of Iraq (in the appendix). There is a
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general trend in the time series of foreign reserves and government spending, as shown in the following
Figure 1.
The linear correlation coefficient between foreign reserves and government spending amounted to 91.4%,
which is positive and significant because the p-value is equal to zero, which is less than the significance
level (0.05). Figure 2 explains the cross-correlation between foreign reserves and government spending.
Special tests are used to ascertain whether the stationarity of the time series exists or not, and one of these
tests is the Augmented Dickey-Fuller Test (ADF) to indicate whether the time series is stable around an
average or linear trend or it is not stable due to the unit root, which tests the following hypothesis:
Null Hypothesis: The data contains a unit root
Alternative Hypothesis: The data no contains a unit root
Figure 1. Time series for the foreign reserves and government spending
Figure 2. Cross-Correlation between Foreign reserves and Government spending
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Table 1. Augmented Dickey-Fuller Test
Null Rejected
Critical Value
p-value
Test Statistics
The Variables
False
-1.956
0.673
0.1051
Foreign Reserves
True
-1.956
0.019
-2.4182
Foreign Reserves Detrend
false
-1.956
0.474
-0.4427
Government Spending
True
-1.956
0.022
-2.3499
Government Spending Detrend
It is clear from Table 1 that the results of the ADF test indicate that the time series of Foreign Reserves and
Government Spending is stationary at the first difference since the value of the absolute test statistic is
(2.4182, 2.3499), which is greater than the absolute critical value (1.956) and p-value (0.019, 0.022) is less
than significant level (0.05). Figure 3, and 4 explains the sample Autocorrelation and sample Partial
Autocorrelation Function for the variable (x1) and (x2), respectively.
Figure 3. ACF, and PACF for the series Foreign Reserves
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Figure 4. ACF, and PACF for the series Government Spending
3.1. VAR Models for time series data:
As mentioned in the theoretical aspect, the VAR models represent an extended version of the AR model that
includes two variables that allow taking advantage of the cross-correlation that may be present in the VAR
model. Now we will find a suitable model to predict the x1 and x2 together. The AR-stationary 3-
dimensional VAR {3) model with linear time trend (VAR) was chosen depending on the lowest values of
the criteria AIC and BIC. The best model is selected by comparing the three estimated models based on
criteria AIC and BIC as shown in Table 2:
Table 2. VAR models efficiency criteria
Model
AIC
BIC
VRA {1}
1137.2
1143.4
VRA {2}
1068.9
1077.4
VRA {3}
996.58
1006.8
Table 2 shows that the VAR {3) model was the best because the values of criterion AIC and BIC are less
than their value in the other models, so the following third model was relied upon:
Table 3 shows the estimation results for the VAR {3} model:
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Table 3. VRA {3} Model Estimation Results
Parameter
Value
Standard Error
t Statistic
P-Value
Constant (1)
38263745.4077
7743689.4858
4.9413
7.7611e-07
Constant (2)
29188085.2295
17475124.0686
1.6703
0.094867
AR {1} (1,1)
0.36293
0.36353
0.99836
0.3181
AR {1} (2,1)
0.84245
0.82037
1.0269
0.30446
AR {1} (1,2)
0.1729
0.21943
0.78794
0.43073
AR {1} (2,2)
0.28585
0.49518
0.57727
0.56376
AR {2} (1,1)
0.066633
0.36065
0.18476
0.85342
AR {2} (2,1)
-0.45441
0.81387
-0.55833
0.57662
AR {2} (1,2)
-0.31267
0.19895
-1.5716
0.11603
AR {2} (2,2)
-0.56248
0.44896
-1.2528
0.21026
AR {3} (1,1)
0.79275
0.31034
2.5545
0.010634
AR {3} (2,1)
1.7824
0.70033
2.5451
0.010924
AR {3} (1,2)
-0.64906
0.2134
-3.0416
0.0023534
AR {3} (2,2)
-0.9322
0.48157
-1.9358
0.052896
Trend (1)
1551051.2794
641607.91
2.4174
0.01563
Trend (2)
1251583.3558
1447911.5998
0.86441
0.38736
Table 3 clearly shows the statistical significance of some estimated parameters, the trend 1 parameter,
which supports the strength of this model. Therefore, the model is
1
23
2
0.36293 0.17290 0.06663 0.31267 0.79275 0.64906
10.84245 0.28585 0.4544 0.56248 1.7824 0.93220
38263745.4077 1551051.2794
29188085.2295 1251583.3558
t
t
y
L L L y
Figure 5 shows the model fit for both variables. Figure 6 shows that the residuals fluctuate around the zero
line. Also, all values of the autocorrelation coefficients fall within the confidence interval for both
variables. Furthermore, most of the residual values fall within the standard curve. After passing all the
necessary tests, the VAR {3} model is ready to forecast future values.
The best model estimated above VAR {3} with MSE equal to (114.5) was used to forecast the Foreign
Reserves and Government Spending for Iraq for the four years (2021-2024), and are summarized in Table
4:
Table 4. Forecasting the Foreign Reserves and Government Spending VAR {3}
Year
Foreign Reserves
Government Spending
2021
65210000
73450000
2022
74940000
90700000
2023
107890000
146650000
2024
111350000
147220000
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Figure 5. Models fit for the VAR {3}
Figure 6. Residual Sample Autocorrelation Function for VAR {3} model
Table 4 shows that there is an expected increase in the coming years in Foreign Reserves and Government
Spending, with the balance of Government Spending remaining higher than Foreign Reserves, which
constitutes the continuation of the general deficit in the budget of Iraq in the coming years, as shown in
Figure 7.
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Figure 7. The time series data with forecasting
3.2. ARIMAX Models for time series data:
As mentioned in the theoretical aspect, the ARIMAX models represent an extended version of the ARIMA
model that includes other independent (predictive) exogenous variables. ARIMAX models are similar to
multiple regression models except that allow taking advantage of the autocorrelation that may be present in
the regression residuals. Now we will find a suitable model to predict the y with the presence of an external
variable x. Out of a total of (32) possible models, the ARIMAX (1,1,0), (2,1,0), and (3,1,0) models were
chosen depending on the significance of the estimated model parameters and the lowest values of the
criteria AIC and BIC as shown in Table 5:
Table 5. ARIMAX models efficiency criteria
Model
AIC
BIC
ARIMAX (1,1,0)
556.6420
559.4742
ARIMAX (2,1,0)
522.1963
525.3915
ARIMAX (3,1,0)
516.0204
519.4101
Table 5 shows that the ARIMAX (3,1,0) model was the best because the values of criterion AIC and BIC
are less than their value in the other models, so the following third model was relied upon:
23
1 0.82468 0.77908 0.39604 1 182311541.8493 2.7722
tt
L L L L y x
Table 6 shows the estimation results for the classical ARIMAX (3,1,0) model:
Table 6. ARIMAX (3,1,0) Model Estimation Results
Parameter
Value
Standard Error
t Statistic
P-Value
Constant
-182311541.8493
5.1437e-09
-3.544342513864502e+16
0
AR {1}
-0.82468
0.27898
-2.956
0.0031162
AR {2}
-0.77908
0.244
-3.193
0.0014082
AR {3}
-0.39604
0.23394
-1.6929
0.0404660
Beta (x)
2.7722
0.077775
35.6431
3.0159e-278
Table 6 clearly shows the statistical significance of the estimated parameters and the regression parameter,
Figure 8 shows the model fit.
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Figure 8. Model fit for the ARIMAX (3,1,0) model
Figure 9. Residual Sample Autocorrelation Function for ARIMAX (3,1,0) model
Figure 9 shows that the residuals fluctuate around the zero line. Also, all values of the autocorrelation
coefficients fall within the confidence interval for both variables. Furthermore, most of the residual values
fall within the standard curve. After passing all the necessary tests, the ARIMAX (3,1,0) model is ready to
forecast future values with MSE equal to (223.38). Forecast the government spending for Iraq for the four
years (2021-2024), and are summarized in Table 7:
Table 7. Forecasting the Government Spending ARIMAX (3,1,0)
Year
Government Spending
2021
95662138.267428
2022
111978320.796619
2023
136363227.811626
2024
221399943.396227
Table 7 shows that there is an expected increase in the coming years in government spending, with the
balance of government spending remaining higher than foreign reserves, which constitutes the continuation
of the general deficit in the budget of Iraq in the coming years, as shown in Figure 10.
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Figure 10. The time series of government spending with forecasting
3.3. Comparison between VAR and ARIMAX models:
To compare the VAR and ARIMAX estimated models, the MSE criterion was used, and preference was
given to the VAR model because the value of MSE is equal to (114.5), which is less than its value (223.38)
for the ARIMAX Model. Therefore, the VAR can be relied upon in future forecasting and planning of the
Iraqi budget on its basis.
4. Conclusions & Recommendations
The following main conclusions and recommendations were summarized:
4.1. Conclusions
1. The VAR model is better than the ARIMAX model for this data depending on the MSE criterion.
2. There is a simple linear correlation (and correlation) between foreign reserves and government
spending to 91.4%, which is positive and significant.
3. The forecast for the period (2021-2024) shows an increase in government spending.
4. There is a general trend in the time series of foreign reserves and government spending indicating
a significant increase in the sustainable deficit of the general budget in Iraq.
4.2. Recommendations
1. Approval of the VAR (3) estimated model with detrend and forecasting values for the coming years
to draw plans.
2. Develop financial and economic stability in Iraq to achieve financial sustainability and reduce the
general budget deficit.
3. Conducting a prospective study based on another time series analysis using Wavelet Shrinkage for
foreign reserves and government spending data.
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Appendix
General budget in Iraq
Government Spending (x2)
Foreign Reserves (x1)
Year
32117491
13652193
2004
26375175
19901327
2005
38806679
27763676
2006
39031232
38217704
2007
59403375
58718278
2008
52567025
51872810
2009
70134201
59252271
2010
78757666
71410911
2011
105139575
82001306
2012
119128000
90648557
2013
113473517
77363120
2014
70397515
62810373
2015
67067437
52617915
2016
75490115
58364993
2017
104158183
76481186
2018
111723523
80383896
2019
67082443
78888824
2020
VARARIMAX
VAR
ARIMAX
MSE
MATLABARIMAXVAR
