Forecasting demand for petroleum products in ghana using time series models

Journal of Economics and Sustainable Development
ISSN 2222-1700 (Paper) ISSN 2222-2855 (Online)
Vol.4, No.17, 2013

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Forecasting Demand for Petroleum Products in Ghana using Time
Series Models
1.
2.
3.

Godfred Kwame Abledu(PhD)1*, Boakye Agyemang2,and Semevor Reubin3
School of Applied Science and Technology, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
Statistics Department, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
Applied Mathematics Department, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
* E-mail of the corresponding author: godfredabledu@gmail.com

Abstract
The objective of this study was to forecast and analyse the demand for petroleum products in Ghana using
annual data from 2000-2010. It focused on studying the feasibility forecast using nested conditional mean
(ARMA) and conditional variance (GARCH, GJR, EGARCH) family of models under such volatile market
conditions. A regression based forecast filtering simulation was proposed and studied for any improvements in
the forecast results.
Keywords: time Series models, regression model, forecast filtering, petroleum products, stationarity of time
Series data.
1.

Introduction
The demand for petroleum has increased in the last decade all over the world including the United
States, Middle Eastern nations, and other Asian nations, which has contributed in the high prices. The demand
for petroleum products in India has been increasing at a rate higher than the increase of domestic availability
(Banapurmath, et. al., 2011). At the same time, there is continuous pressure on emission control through
periodically tightened regulations particularly in metropolitan cities. Over the period 1980-2008, the price of
crude oil had fluctuated significantly, with a mean, minimum and maximum values of $ 32.31 (bbl), $ 12.72
(bbl) and $ 140 (bbl) respectively(WAMA(2008). The above statistics, in addition to a standard deviation of
17.08 over the sample period show that the prices of crude have always been characterized with severe
instability. Monthly fluctuations have in fact been more severe than these annual trends, with the price of crude
oil reaching $140 (bbl) in July 2008. Such instability in the prices of crude oil is bound to cause macroeconomic
distortions, especially in net-oil importing countries, like some ECOWAS countries (WAMA, 2008).
Ghana’s demand for crude oil and refined petroleum products has also been growing over the past
decade, and, the country’s demand for oil increased dramatically surprising many energy analysts. This growth
has been driven by socio-economic and technical factors that have influenced each category of final energy use.
These changing petroleum requirements are closely related to its high rates of growth in economic output and
personal incomes. The growth in incomes and the accompanying changes in petroleum demand are themselves
driven by an ongoing population shift from rural to urban areas. That growing urban population is demanding
new vehicles and new roads, raising the demand for energy in the transportation sector. The growth in output in
the industrial sector is driving the high demand for petrochemical feed stocks, including naphtha-based
petrochemicals, which are similar in composition to motor gasoline. Fluid catalytic cracking of heavy ends to
high-value liquid fuels is a common unit operation in oil refineries (Khan, et al., 2011). In this process, the heavy
feedstock that contains sulfur is cracked to light products
At the core of the development of every nation is petroleum. Currently, petroleum is among the most
important natural resources. Every nation uses petroleum products such as gasoline, jet fuel, liquefied petroleum
gas and diesel to run cars, trucks, aircraft and other vehicles. There is therefore the need to build stocks to meet
the seasonal demands. In the long term, blending non-petroleum additives into petroleum products such as
ethanol or other oxygenating agents into gasoline will be necessary to reduce crude oil demand. Efficient
refining capacity is a requirement to meet the demand of the nations. The past few years have witnessed an
increased impetus toward renewable energy to replace fossil fuels that has been driven both by environmental
and national security concerns (Hensel, 2011).
Many researchers analyzing the demand for petroleum products have looked at the aggregate
consumption of petroleum. Sa'ad(2009), used annual time series data for the period 1973 to 2007 in two
econometric techniques namely the structural time series model (STSM) and unrestricted error correction model
(UECM) to estimate demand for petroleum products. The results from both models revealed that the demand for
petroleum products is price inelastic. The robust optimization methodology is applied to deal with uncertainties
in the prices of saleable products, operating costs, product demand, and product yield in the context of refinery
operational planning (Leiras, et al , 2010 and Munim, et al.,2010

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MATERIALS AND METHOD
Conditional mean models were used to forecast mean while the conditional variance models were used
for forecasting variance or volatility in the demand for oil. In this study, the nesting of these two models was
used first to forecast the conditional mean and then the conditional variance was estimated to get the value of
forecast demands for oil(Shrivastava, 2009). After analysis of data for quarter 1 (Q1r) and quarter 2 (Q2r),
ARMA(2,2) models had been found most appropriate for forecasting mean, hence ARMA(2,2) and
(GARCH(1,1)/GJR(1,1) /EGARCH(1,1)) were used for forecasting oil demand for 2012 and 2013.
The conditional mean and variance models have been viewed from a linear filtering perspective, and
then the application of the iterated conditional expectations to the recursive equations was conducted, one
forecast period at a time (Dhar, et. al. , 2009). For example for forecasting demand for the second quarter (Q2),
demand data were taken of quarter 1 (Q1) and forecast using above defined method has been done. Using
observed quarter 2(Q2) demand as dependent variable and (ARMA/GARCH) forecast as independent variable
regression model was obtained. Calculation of quarter 3 demand (Q3r) using observed data of Q2 was obtained
using nested models and the regression model obtained in the previous step was used to filter quarter 3 (Q3)
demand forecast.
Conditional variance models (Shrivastava, 2009), unlike the traditional or extreme value estimators,
incorporate time varying characteristics of second moment/volatility explicitly. The following models fall into
the category of conditional volatility models(Hull, 2006):
i.
ARCH (m) Model (Auto Regressive Conditional Heteroscedasticity)
ii.
EWMA Model (Exponentially Weighted Moving Average Model)
iii.
GARCH(a,b)Model(Generalized Autoregressive Conditional Heteroscedasticity).
iv.
EGARCH Model.
The stationarity was tested using ADF test with and without drift and trend, the AR(p) was determined
using PACF and MA(q) was determined using ACF. The number of lagged terms to be included in the model
was identified based on the minimum value of AIC and SBC criteria. The ARMA model was tested for ARCH
effects using the ARCH LM test and the measures of performance were calculated for the static and dynamic
forecasts made for the out-sample period. The in-sample data constituting 80% was used for estimating the
coefficients of the parameters and 20% the out-of- sample data was forecasted.
The forecasted results from random walk model, ARMA, ARMA-GARCH, ARMA-EGARCH models
using static and dynamic forecasting were compared based on the predictive power using the three forecasting
accuracy measures: Root Mean Square Error, Mean Absolute Error and Thiel Inequality Coefficient. Theil’s U
statistic was rescaled and decomposed into 3 proportions of inequality – bias, variance and covariance – such
that bias + variance + covariance = 1 and these measures were also calculated.
1.1. Autoregressive Moving Average( ARMA )Models
The Autoregressive Moving Average (ARMA) Models have been used by many researcher for

Z

forecasting(Shrivastava, et, al. , 2010; Abu and Behrooz, 2011) . Given a time series of data t , the ARMA
model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two
parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the
ARMA (a, b) model where “a” is the order of the autoregressive part and “b” is the order of the moving average
part. The notation ARMA (a, b) refers to the model with “a” autoregressive terms and “b” moving-average
terms. This model contains the AR(a) and MA(b) models. A time series
a

Zt

follows an ARMA (1, 1) model if it

b

Z1 = k + ωt + ∑ β i Z t −i + ∑ α iωt −i

ω

i −1
i =1
, where { t } is a white noise series. The above equation
satisfies
implies that the forecast value is depended on the past value and previous shocks.
The notation MA(b) which refers to the moving average model of order b is written as
b

Z t = k + ωt + ∑ α iωt −i
i =1

and the notation AR(a) which refers to the autoregressive model of order a, is written
a

Z1 = k + ωt + ∑ β i Z t −i
as

i −1

where the

α1 ,...,αb

are the parameters of the model, µ is the expectation of Zt

ω ,...,ω

t −b are again, white noise error terms.
(often assumed to equal to 0), and the t
The Autoregressive Moving Average model(ARMA) is a method which can be used when the time
series variable is related to past values of itself. By regressing Zt on some combination of its past values, we are

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able to derive a forecasting equation. We expect the autoregressive technique to perform reasonably well for a
time series that:
1.
Is not extremely volatile and does not contain extreme amounts of random
movement.
2.
Requires “q” short-term or medium- term forecast, that is less than two years
The fact that the autoregressive procedure does not perform well on a time series is not a serious
disadvantage. Practically all forecasting techniques perform poorly in this situation. Suppose we want to predict
the values of Zt using the previous observation, we use the expanded equation:
where t takes the values = 3, 4, 5, …. The values

b0 b1
, ,

and

Z t = b0 + b1Z t −1 + b2 Z t − 2 ,

b2 are the least squares regression estimates

Zt −1

obtained from any multiple linear regression. There are two predictor variables here, the lagged variables,
and

Zt − 2

. The above equation is a second order autoregressive equation because it uses the two lagged terms. In
^
th

general, the a – order autoregressive equation is given as :

Z t = b0 + b1Z t −1 + b2 Z t − 2 + ... + ba Z t − a

, and

t = a + 1, a + 2,...
The assumption underlying the ARMA model is that the future value of a variable is a linear function of
past observations and random errors. In this model it is possible to find an adequate description of data set. This
method consists of four steps:
1. Model identification,
2. Parameter estimation,
3. Diagnostic checking and
4. Forecasting.
In the identification step, it can be seen that a model generated from an ARMA process may contain some
autocorrelation properties, so there will be some potential models that can fit the data set but the best fitted
model is selected according to AIC information criteria. Stationarity is a necessary condition in building an
ARMA model used for forecasting, so data transformation is often required to make the time series to be
stationary. In this study, the unit root test, known as the Dickey and Fuller test (Shrivastava, et, al., 2010;
Gujarati, 2006; Abu and Behrooz, 2011 ), is used to test the stationarity of the time series.
Based on the result obtained, the data set is stationary at first difference even with the existence of
structural break. Once a tentative model is obtained, estimation of the model parameters is applicable. The
parameters are estimated such that an overall measure of errors is minimized. The third step is diagnostic
checking for model adequacy. Autocorrelation and also serial correlation of the residuals are used to test the
goodness of fit of the tentatively obtained model to the original data. When the final model is approved then it
will be used for prediction of future values of the oil demand.
1.2. The ARCH/GARCH Models
The first model that provides a systematic framework for volatility modeling is the ARCH model of Engle
(Gujarati, 2006). The basic idea of the ARCH model is that the shock
uncorrected but dependent; also the dependence of
lagged

values.

2
t

2
1 t−1

Specifically,

an

ARCH

(m)

2
m t−m (Gujarati, 2006), where

model

assumes

that

at = σtεt

α0 > 0 and αi ≥ 0

for

must satisfy some regularity condition to ensure that the unconditional variance of

ε

,

{ε t } is a sequence of independent and identically

distributed (i.i.d.) random variables with mean zero and variance 1,

αi

of an asset return is serially

αt can be described by a simple quadratic function of its

σ =α0 +α a +.....+α a
coefficient

αt

i > 0 . The

at

is finite. In

practices, t is often assumed to follow the standard normal or a standardized student t distribution or a
generalized error distribution.
GARCH models are used as a successful treatment to the financial data which often demonstrate timepersistence, volatility clustering and deviation from the normal distribution. Among the earliest models is Engel
(1982) linear ARCH model, which captures the time varying
feature of the conditional variance. Bollerslev (1986) develops Generalized ARCH (GARCH) model, allowing
for persistency of the conditional variance and more efficient testing. Engle and
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Bollerslev (1986) invent the Integrated GARCH (IGARCH) model that provides consistent estimation under the
unit root condition. Engle, Lilien, and Robins (1987) design the ARCH-inMean (ARCH-M) model to allow for time varying conditional mean. Nelson’s (1990a & b) Exponential
GARCH (EGARCH) model allows asymmetric effects and negative coefficients in
the conditional variance function.
The leveraged GARCH (LGARCH) model documented in Glosten, Jagannathan and Runkle (1993)
takes into account the asymmetric effects of shocks from different directions. Since their introduction by Engle
(1982), Autoregressive Conditional Heteroskedastic (ARCH) models and their extension by Bollerslev (1986) to
generalised ARCH (GARCH) processes, GARCH models have been used widely by practitioners. At a first
glance, their structure may seem simple, but their mathematical treatment has turned out to be quite complex.
Although the ARCH is simple, it often requires many parameters to adequately describe the volatility process of
an asset return some alternative models must be sought. Shrivastava, et al. (2010) and Hull(2006) proposed a
useful extension known as the generalized ARCH (GARCH) model. An important feature of GARCH-type
models is that the unconditional volatility σ depends on the entire sample, while the conditional volatilities
are determined by model parameters and recent return observations.

σt

(ε t ) t ∈ ℤ be a sequence of independent and identically distributed (i.i.d.) random variables, and
p ∈ ℕ = {1, 2, 3,..., )
p ∈ ℕ o = ℕ ∪ {0}
α0 > 0 , α1 ,..., α p −1 ≥ 0 , α p > 0 ,
let
and
. Further, let
β1 ,..., β q −1 ≥ 0
β >0
( X t ) t ∈ℤ with volatility
and q
be non-negative parameters. A GARCH(p, q) process
(σ ) t ∈ℤ is then a solution to the equations:
process is t
Let

X t = σ tε t , t ∈ℤ

(1)

p

q

i =1

j =1

σ t2 = α t + ∑α i X t2−1 + ∑ β jσ t2−1

t ∈ℤ

,

(2)

(σ ) t ∈ℤ

(ε t ) t ∈ ℤ

t
where the process
is non-negative. The sequence
is referred to as the driving noise
sequence. GARCH (p, 0) processes are called ARCH (p) processes. The case of a GARCH (0, q) process is
excluded since in that case, the volatility equation (2) decouples from the observed process and the driving noise
sequence.

It is a desirable property that
is measurable with respect to

σ

σt

algebra generated by (ε t − h ) h ∈ ℕ . If this condition holds, we shall call the

GARCH (p, q) process causal. Then
generated by

(εt − h)h∈ℕ0 .

should depend only on the past innovations (ε t − h ) h ∈ ℕ , that is, it

Also,

( Xt ) is measurable with respect to σ
σt

is

independent of

algebra

(εt + h)h∈ℕ0 ,

and

σ (εt − h : h ∈ℕ0 ) ,

Xt

is independent of

σ (εt + h : h ∈ℕ ) , for fixed t. The requirement that all the coefficients α1,...,αp and β1,..., β q
2

σ

are non-

2

negative ensures that σ is non-negative, so that t can indeed be defined as the square root of σ .
Equation(1) is the mean equation and is specified as an AR(p) process. Equation (2) is the conditional
variance equation and it is specified as the GARCH(1, 1) process. Conditional variance models (Shrivastava,
2009), unlike the traditional or extreme value estimators, incorporate time varying characteristics of second
moment/volatility explicitly. By successively substituting for the lagged conditional variance into equation(2),
the following expression is obtained:

ht =

α0
∞
+ α1 ∑ i =1 βi −1ε t2−i
1− β

(3)
An ordinary sample variance would give each of the past squares an equal weight rather than declining
weights. Thus the GARCH variance is like a sample variance but it emphasizes the most recent observations.
Since

ht

is the one period ahead forecast variance based on past data, it is called the conditional variance. The
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squared residual is given by:

vt = ε t2 − ht

(4)
Equation (4) is by definition unpredictable based on the past. Substituting equation (4) into equation(2)
yields an alternative expression as follows:

ε t2 = ω + (α1 + β )ε t2−1 + vt − β vt −1

(5)

{a }
From the structure of the model, it is seen that large past squared shocks

m

2
t −i i =1

σ

2
t

a

imply a large

a

for the innovation t . Consequently, t tends to assume a large value (in modulus).
conditional variance
This means that, under the ARCH framework, large shock tend to be followed by another shock; because a large
variance does not necessarily produce a large realization. It only says that the probability of obtaining a large
variate is greater than that of a smaller variance. To understand the ARCH models, it pays to carefully study
ARCH (I) model

at = σtεt

,

σt2 =α0 +α1at2−1, where α0 > 0 and α

≥0

I

. The unconditional mean of

at

remains zero

because

E ( at ) = E  E ( at / Ft −1 )  = E σ t E ( ε t )  = 0




The

conditional

variance

if

at

can

be

var ( at ) = E at2 = E  E at2 / Ft −1  = E α 0 + α1at2−1 = α 0 + α1 E at2−1 .


E ( at2−1 ) = 0 var ( at ) = var ( at −1 ) = E ( at2−1 )
at

( )

(

)

(

)

( )

is a stationary process with

Because

var ( at ) = α 0 + α1 var ( at )

0 ≤ α1 ≤ 1

and

α
var ( at ) = 0 .
1 − α1 Since

(6)
as

obtained

the variance of

. In some applications, we need higher order moments of

at

at

. Therefore, we have

must be positive, we require

to exist and, hence,

α1 must also satisfy

some additional constraints. For instance, to satisfy its tail behavior, we require that the fourth moment of
finite. Under the normality assumption, we have
2

2

E at4 / Ft −1 = 3  E at2 / Ft −1  = 3 α 0 + α1at2−1


(Brockwell and Davis, 1996).
2
2
E at4 = E  E at4 / Ft −1  = 3E α 0 + α1at2−1 = 3E α 0 + 2α 0α1at2−1 + α12 at4−1


Therefore ,
at

(

If

)
(
)
(
( )
(
)

)

at

(

)

(

)

is fourth – order stationary with

m4 = E ( at4 )

, then we have

m4 = 3 α + 2α 0α1 var ( at ) + α1m4 


α 
2
= 3α 0 1 + 2 1  + 3α12 m4
1 − α1 

2
0

Consequently

m4 =

2
3α 0 (1 + α1 )

(1 − α1 ) (1 − 3α12 )

.

This result has two important implications: since the fourth moment of
must also satisfy the condition

1 − 3α13 > 0;

that is,

0 ≤ α12 ≤ 1

133

at

is positive, we see that

α1

3 ; and the unconditional Kurtosis of at is

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( )

E at4

 var ( at ) 



2

=3

α 02 (1 + α1 )

(1 − α1 ) (1 − 3α12 )

Thus, the excess of

at

×

(1 − α1 )
α 02

2

1 − α12
=3
1 − 3α12>3

is positive and the tail distribution of

(7)

at

is heavier than that of a normal

a

distribution. In other words, the shock t of a conditional Gaussian ARCH (I) model is more likely than
Gaussian white noise series to produce “outcome”. This is in agreement with the empirical finding that “outliers”
appear more often in asset returns than that implied by an iid sequence of normal random variates. These
properties continue to hold for general ARCH models, but the formula becomes more complicated for higher
order ARCH models.
The condition

αi ≥ 0 in σ t2 = α 0 + α1at2−1 + .... + α m at2− m can be related. It is a condition to ensure
σ2

that the conditional variance t is positive for all t. The model has some weakness: it assume that positive and
negative shocks have the same effects on volatility because it depends on the square of the previous shocks. In
practices it is well known that price of a financial asset responds differently to positive and negative shocks.
The ARCH model is rather restrictive. For instance,

α12 of an ARCH (I) model must be in the interval

0 1 
 3  if the series has a finite fourth moment. The constraint becomes complicated for higher order ARCH
models. In limits, the ability of ARCH models with Gaussian innovations is to capture excess kurtosis. The
ARCH model does not provide any new insight for understanding the sources of variation of a financial time
series. It merely provides a mechanical way to describe the behavior of the conditional variation. It gives no
indications of what causes such behavior to occur. ARCH models are likely to over predict the volatility because
they respond slowly to large isolated shocks to the return series(Brockwell and Davis, 1996).
1.3. The EGARCH Model
This model is used to allow for symmetric effects between positive and negative asset returns. An EGARCH
(m, s) model can be written as (Dhar, et. Al. , 2009).

at = σtεt
wher

α0

( )

In σ
,

2
t

1 + β1B + ... + βs−1Bs−1
= α0
g ( εt −1 )
1 − α1B....αm Bm

is a constant, B is the back-shift (or lag) operator such that

1 + β1B + β s − B + ...

(8)

Bg ( ε t ) = g ( ε t −1 )

are polynomials with zeros outside the unit circle and have no common factors. By outside

the unit circle, we mean that absolute values of the zeros are greater than 1. Here, it is understood that
i > m and

βj = 0

and

αi = 0

for

for j > s. The latter constraint on αi and βi implies that the unconditional variance αt is finite,

σ2

ε

whereas its conditional variance t evolves over time, and t is often assumed to be a standard normal
standardized student-t distribution or generalized error distribution:
m

m

i =1

i =1

σ t2 = α 0 + ∑ α iα t2−1 + ∑ β jσ t2− j
(9)
reduces to a pure ARCH (m) model if S=0.
The αi and βi are referred to as ARCH and GARCH parameters respectively. The unconditional mean of

In(σ t2 )

is

α 0 . It uses logged conditional variance to relax the positiveness constraint of model coefficients. The

g (ε t ) enables the model to respond asymmetrically to the positive and negative lagged values of α . The
t
θ ≠ 0 . Since negative shocks tend to have larger impacts, we expect θ to be negative.
model is nonlinear if
use of

For higher order EGARCH model, the nonlinearity becomes much more complicated. This model can be used to
obtain multistep ahead volatility forecasts.

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1.4. Fitting the Parameters of the Model
Once a model is selected and data are collected, it is the job of the researcher to estimate the parameters
of the Model. These are values that best fit the historical data. It is hypothesized that the resulting model will
provide a prediction of the future observation. It is also hypothesized that all values in a given sample are equal.
The time series model includes one or more parameters. We identify the estimated values with a hat.

⌢
β is denoted β . The procedure also provide estimates of the standard
For instance, the estimated value of

σ

deviation of the noise, ε .
1.5. Forecasting from The Model
The main purpose of modeling a time series is to make forecasts which are then used directly for making
decisions. In this analysis, we let the current time be T, and assume that the demand for periods 1 through T are

known. We now want to forecast the demand for the period (T+ ς ). The unknown demand is the random

x

ς
variable X(T+ ς )., and its realization is
(T+ ).. Our forecast for the realization is
1.6. Measuring the Accuracy of the Model
The forecast error is the difference between the realization and the forecast. Thus

eς

x

xT + ς

.

x

T +ς
ς
=
.
(T+ )...Assuming the model is correct, then we have

eς

=

E [ X T +ς ] + ε ς −

(10)

xT +ς

(11)
We investigate the probability distribution of the error by computing its mean and variance. One

desirable characteristics of the forecast

xT +ς

is that it is unbiased. For an unbiased estimate, the expected value

of the forecast is the same as the expected value of the time series. Because
zero, an unbiased forecast implies

E[ε ς ]

2

is assumed to have a mean of

. The fact that the noise in independent from one period to the next

period means that the variance of the error is:

σ ε (ς ) = σ E (ς ) + σ

εt

Var[ε t ] = Var{E[ X T +ς ] − xT +ς } + Var[εT +ς ]

and

2

2

.
(12)
2.
3. DATA ANALYSIS AND RESULTS
3.1. Data and method of analysis
The data for the study was obtained from Tema Oil refinery. The AFC and the PACF of the time series
are shown in Figure 1. The PACF shows a single spike at the first lag and the ACF shows a tapering pattern. The

φ

positive, geometrically decaying pattern of the AFC, coupled with single significant coefficient 11 strongly
suggest an AR(1){=ARMA(1,0)} process.
The time series plot(Figure 2) of the standardized residuals mostly indicates that there is no trend in the
residuals, no outliers, and in general, no changing variance across time. The ACF of the residuals shows no
significant autocorrelations, an indication of a good result. The Q-Q plot is a normal probability plot. It doesn’t
look too bad, so the assumption of normally distributed residuals looks okay. The bottom plot gives p-values for
the Ljung-Box-Pierce statistics for each lag up to 20. These statistics consider the accumulated residual
autocorrelation from lag 1 up to and including the lag on the horizontal axis. The dashed blue line is at 0.05. All
p-values are above it indicating that this is a good result
The time series data ranged from January 2000 until December 2012. The coefficient of variation (V) was
used to measure the index of instability of the time series data. The coefficient of variation (V) is defined as:

V=

σ
Y

(13)

where σ is the standard deviation and

Y=

1 n
∑ Yt
n t =1

(14)

is the mean of petroleum prices changes.

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A completely stable data has V = 1, but unstable data are characterized by a V>1 (Telesca et al., 2008).
Regression analysis was used to test whether trends and seasonal factors exist in the time series data. The
existence of linear trend factors was tested through this regression equation:
2
Y = β0 + β1T + ε ε ∼ WN (0, σ )

,
(15)
Stationarity is tested using unit root test. The stochastic time series of interest is Zt. Taking the first

∆Z = A + A + A Z + µ

∆Z

t
1
2
3 t −1
t is the first difference, t is the trend
where
difference we have the following:
variable taking on the values from 1, 2, 3, …, n. and Zt-1 is the one period lagged value of the variable Z. The
null hypothesis is that A3, the coefficient of Zt-1 is zero. That is to say that the underlying time series is
nonstationary. This is called the unit root hypothesis. We proceed to show that a3, the estimated value of A3 is
zero. The unit root test is used since we have already assumed that the time series is nonstationary. The tau test
whose critical values are tabulated by the creators on the basis on Monte Carlo simulation are used(Gujarati,
2006 ). The rule for testing the hypothesis is that if the computed t(tau) value of the estimated A3 is greater (in
absolute value) than the critical Dickey Fuller(DF) tau values, we reject the root hypothesis, that is, we conclude
that the said time series is stationary. On the other hand, if the computed tau value is smaller (in absolute values)
than the critical DF tau values, we do not reject the unit hypothesis. In that case, the time series is nonstationary.
Data in Table 1 describe the nested ARMA(2,2) and GARCH(1,1) models. The forecasts have closest
mean with respect to observed mean while EGARCH(1,1) model has shown maximum correlation with the
observed Q2 returns. In all these cases Q1 returns data is used to calculate GARCH family model’s parameters.
3.2. Empirical Results
Eight model selection criteria as suggested by Ramanathan (2002) were used to choose the best
forecasting models among ARIMA and GARCH models, while the best time series methods for forecasting
demand for petroleum products was chosen based on the values of four criteria, namely RMSE, MAE, MAPE
and U-statistics (Table 2). Finally, the selected model was used to perform short-term forecasting for the next
twelve months for demand for petroleum products starting from January 2013 until December 2013.
The results showed that the coefficient of variation (V) of the time series data was 1.312 (V>1).
Because the V value was closed to 1, it was concluded that the time series data was stable (Telesca et al., 2008).
The results of the regression analysis had shown that positive linear trend factor existed in the time series data
but seasonal factor was not. Referring to the Augmented Dickey-Fuller tests results, the time series data of the
study was not stationary. But after the first order of differencing was carried out, the time series data became
stationary.
The double exponential smoothing method was used as the regression result had shown that positive
linear trend factor exists in the time series data. Double exponential smoothing models consisted with two
parameters which were symbolized as α for the mean and β for the trend. The best model of the double
exponential smoothing was selected based on the lowest value of MSE (Mean Square Error) from the
combination of α and β with condition 0<α, β<1.
The result showed that the combination α = 0.9 and β = 0.1 was the best forecasting model of double
exponential smoothing method (Table 3). The double exponential smoothing model was written in equation

F

= a + bh = 4764.2345 + h *(−32.3465)

form, from Table 4, as T + k
All models which fulfilled the criteria of p+q≤5 have been considered and compared in this study.
There were twenty ARIMA (p, d, q) models which fulfilled the criteria(Table 5). The parameters of the models
were estimated with the least square method. Parameters which were not significant at 5% confidence level were
dropped from the model. Using the eight model selection criteria suggested by Ramanathan (2002), the ARIMA
(3, 1, 2) model was selected as the best model among the other ARIMA models. However, the parameters of AR
(1) and MA (1) were found not significant and thus dropped from the model.
Identification and estimation of GARCH (p, q) models in this study were done by following the four
steps that were ARCH effect checking, estimation, model checking and forecasting. Four GARCH (p,q) models
were selected and compared, namely GARCH (1, 1), GARCH (1, 2), GARCH (2, 1) and GARCH (2, 2). Using
the eight model selection criteria suggested by Ramanathan (2002), the GARCH (1, 1) model was selected as the
best model among the other three GARCH models(Table 6). ARCH effect which was tested by using a
regression analysis existed in the ARIMA (3, 1, 2) model. What this meant was that the ARIMA (3, 1, 2) model
could be mixed with the best GARCH model (i.e., GARCH(1, 1)). Four model selection criteria were used to
select the best forecasting model from the four different types of time series methods. Based on the results of the
ex-post forecasting (starting from January until December 2013), the ARIMA (3, 1, 2)/GARCH (1, 1) model was
the best short-term forecasting model for the demand for petroleum products (Table 9).
A linear relationship between Q2r and GARCH family forecast for different combinations was also

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obtained. R-sq values in table of models gave the percentage of variations which the regression was able to
explain. It was clear that relationship (7) best explains the variations in the actual returns and forecasted returns,
while relationship (5) was the second best in explaining the variations while relationship (3) was the third best in
Table 8.
To determine whether there is significant difference for the mean demand and the standard deviation
values of the observed and predicted data for each month, a z-test (for means) and F-test (for standard
deviations) were applied (Haan, 1977; Devore and Peck, 1993). Since monthly mean values from observed and
predicted data is between z-critical table values (± 1.96 for 2 tailed at the 5% significance level), the data support
the claim that there is no difference between the mean values of observed and predicted data. Similarly, monthly
standard deviation values from observed and predicted data were smaller than F- critical table values at the 5%
significance level. Furthermore, these results show that the predicted data preserve the basic statistical properties
of the observed series.
The coefficient of correlation (R), which measures the strength of the association between 2 variables,
and the significance level (Rsig) related to the R of regression shows that there is a statistically significant linear
relationship between the observed and predicted data. On the other hand, the coefficient of determination (Rsquare), which is interpreted as the proportionate reduction of total variation associated with the use of the
predictor variable (the observed data in this study), and adjusted R-square measure, which presents the sample
response of the population for each regression, were close to one. In addition, the results (F-value and FSig)
concerning tests applied for determining whether the estimated regression functions adequately fit the data
emphasize that the association between the observed and predicted monthly data sequences is linear. Based on
these results, it is concluded that the selected best ARIMA model for each station can make accurate estimates.
CONCLUSION
Seven multiple regression relationships for different combination of nested ARMA / GARCH were
used to filter their Q3 demand forecast. Filtered result analysis shows improvements in the correlation coefficient
of the forecast demands and observed Q3 demands. Correlation coefficient is positive in some simulations,
which were always negative with GARCH family model’s forecast. Regression filtered results follow market
trend better, while other descriptive parameters like variance, skewness and kurtosis become more comparable to
actual Q3 demand. Therefore the proposed simulation framework under given observations to some extent has
improved nested conditional mean and variance models forecast of Q3 forecast for petroleum products under
such market conditions of 2013. However, it is not generally possible to get a definite relationship between
observed and forecasted result.
This study also investigated four different types of univariate time series methods, namely exponential
smoothing, ARIMA, GARCH and the mixed ARIMA/GARCH. The results showed that the mixed
ARIMA/GARCH model outperformed the exponential smoothing, ARIMA and GARCH for forecasting the
demand for petroleum products.
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Tables
Table 1: Descriptive Statistics of ARMA/GARCH forecast and Regression filtered forecast
Variables
Q3r

Mean
-0.0049

StDev
0.0398

Variance
0.0016

Minimum
-0.1160

Maximum
0.0790

Range
0.1950

Skewness
-0.0600

Kurtosis
-0.0500

Q3g2211f

-0.0067

0.0178

0.0003

-0.0056

0.0065

0.0121

1.0100

5.7100

Q3j2211f

0.0088

0.0707

0.0050

-0.0088

0.0110

0.0198

0.0400

-1.5100

Q3e2211f

-0.0072

0.0141

0.0002

0.0637

-0.0358

-0.0996

0.1111

-0.0767

Q3g2211rf

-0.0010

0.0283

0.0008

0.1173

-0.0698

-0.1872

0.0044

-0.8067

Q3j2211rf

0.0001

0.0566

0.0032

0.1709

-0.1038

-0.2748

0.0122

-1.5367

Q3e2211rf

-0.0112

0.1131

0.0128

0.2245

-0.1378

-0.3624

0.0005

-2.2667

Q3gjrf

0.0201

0.2262

0.0512

0.2781

-0.1718

-0.4500

0.0013

-2.9967

Q3jerf

-0.0313

0.4524

0.2047

0.3317

-0.2058

-0.5376

0.0134

-3.7267

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Vol.4, No.17, 2013

Table 2: Criteria for Assessing Forecast Accuracy
Double
Exponential
Smoothing
Criteria
Formula

RSME

GARCH(1, 1)

ARIMA(3, 1, 2)
/ GARCH(1, 1)

414.4506

ESS
N

ARIMA(3, 1, )

193 .3087

158.8801

155.5007

⌢
1
∑ Yt − Yt
n t =1
⌢
1 n Yt − Yt
× 100%
∑
n t =1 Yt

392.6509

1348.9835

122.8083

126.7645

6.8052

2.8309

2.4178

2.5528

RSME
⌢ 1 n
1
Y 2 + ∑Y 2
∑ n t =1
n t =1

0.0237

0.0192

0.0161

0.0157

n

MAE

MAPE

n

U- Statistics

Table 3: Error Sum of Square (ESS) according to α and β values

α
β

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

19345

81246

46341

31941

24941

21943

121426

31244

451427

0.2

18311

78242

47398

34742

23946

20846

110478

30142

410672

0.3

14567

61965

35943

33943

22647

20143

101495

30245

401421

0.4

13425

56349

26332

22946

21548

20959

100433

30456

400231

0.5

12833

45247

22344

21948

21041

22932

221409

32537

421502

Table 4: Output of the double exponential smoothing model
Parameters

Values

α

0.9800

β

0.1000

Sum of squared residuals

187364

Root mean squared error

234.3465

Mean

4764.2345

Trend

-32.3465

Table 5: Estimation of ARIMA (3, 1, 2)
Variables

Coefficient

Standard error

Z-statistic

p-value

Constant

19.2319

28.2345

0.6934

0.4562

AR(2)

-0.7843

0.0453

-18.4653

0.0001*

AR(3)

0.1287

0.0321

2.8675

0.0023*

MA(2)

0.9128

0.0234

20.3465

0.0002*

MA(3)

0.3465

0.0458

28.9874

0.0134*

*significant at 0.005 level

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Vol.4, No.17, 2013

Table 6: Estimation of GARCH (1, 1)
Mean Equation
Variables

Coefficient

Standard error

Z-statistic

p-value

Constant

2.2319

18.2345

0.1934

0.8502

5452.87

3214.03

2.8642

0.0123*

0.3128

0.1026

2.3465

0.0302*

0.6487

0.0858

8.9814

0.0134*

Conditional Variance Equation
Constant

ε

2
k −1
2
k −1

σ

Table 7: Estimation of ARIMA (3, 1, 2)/GARCH (1, 1)
Variables

Coefficient

Standard error

Z-statistic

p-value

ARIMA(3, 1, 2)
Constant

12.2337

18.2311

0.6341

0.5568

AR(2)

-0.7423

0.0403

-18.4639

0.0021*

AR(3)

0.2384

0.0125

3.4672

0.0053*

MA(2)

0.8023

0.0336

30.5462

0.0042*

MA(3)

0.2662

0.0151

27.3873

0.0104*

4405.3452

2345.0987

2.4986

0.8373

0.3254

0.0975

2.4863

0.0034*

0.7654

0.1203

7.9073

0.0002*

GARCH(1, 1)
Constant

ε

2
k −1
2
k −1

σ


Table 10: Regression Statistics with Q2r on LHS and Q2g2211f, Q2e2211f on RHS
Relationship
1

Regression order
1

K
-0.001

Q2g2211f
0.07

2
3
4
5
6
7

1
1
2
2
2
3

0.007
0.010
0.008
0.003
0.010
0.008

Q2j2211f
0

0
0
0.45
0
0.34
-0.89

0

p- value
0.0264

3.23
0
3.41
-17.6
3.04
-17.7

140

Q2e2211f
0
3.01
0
0
2.06
11.3

0.0262
0.0257
0.0263
0.0251
0.0259
0.0349

R-Sq
0%

R-Sq(adj.)
0%

1.50%
4.90%
1.60%
10.60%
5.0%
11.20%

0%
3.40%
0%
7.70%
1.80%
7.60%


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Vol.4, No.17, 2013

Figure 1: AFC and PACF Plots

Figure 2: Time Series Plots of Residuals

.

141

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