Gold sales forecasting: The box-Jenkins methodology

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The study employs the Box-Jenkins Methodology to forecast South African gold sales. For a resource economy like South Africa where metals and minerals account for a high proportion of GDP and export earnings, the decline in gold sales is very disturbing. Box-Jenkins time series technique was used to perform time series analysis of monthly gold sales for the period January 2000 to June 2013 with the following steps: model identification, model estimation, diagnostic checking and forecasting. Furthermore, the prediction accuracy is tested using mean absolute percentage error (MAPE). From the analysis, a seasonal ARIMA(4,1,4)×(0,1,1) 12 was found to be the “best fit model” with an MAPE value of 11% indicating that the model is fit to be used to predict or forecast future gold sales for South Africa. In addition, the forecast values show that there will be a decrease in the overall gold sales for the first six months of 2014. It is hoped that the study will help the public and private sectors to understand the gold sales or output scenario and later plan the gold mining activities in South Africa. Furthermore, it is hoped that this research paper has demonstrated the significance of Box-Jenkins technique for this area of research and that they will be applied in the future.

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12
Risk governance & control: financial markets & institutions / Volume 7, Issue 1, Winter 2017
GOLD SALES FORECASTING:
THE BOX-JENKINS METHODOLOGY
Johannes Tshepiso Tsoku*, Nonofo Phukuntsi, Daniel Metsileng
* North West University, South Africa
Abstract
The study employs the Box-Jenkins Methodology to forecast South African gold sales. For a
resource economy like South Africa where metals and minerals account for a high proportion of
GDP and export earnings, the decline in gold sales is very disturbing. Box-Jenkins time series
technique was used to perform time series analysis of monthly gold sales for the period January
2000 to June 2013 with the following steps: model identification, model estimation, diagnostic
checking and forecasting. Furthermore, the prediction accuracy is tested using mean absolute
percentage error (MAPE). From the analysis, a seasonal ARIMA(4,1,4)×(0,1,1) was found to be the
“best fit model” with an MAPE value of 11% indicating that the model is fit to be used to predict
or forecast future gold sales for South Africa. In addition, the forecast values show that there will
be a decrease in the overall gold sales for the first six months of 2014. It is hoped that the study
will help the public and private sectors to understand the gold sales or output scenario and later
plan the gold mining activities in South Africa.
Furthermore, it is hoped that this research paper
has demonstrated the significance of Box-Jenkins technique for this area of research and that
they will be applied in the future.
Keywords: Gold Sales, ARIMA, Box-Jenkins, GDP, MAPE
JEL Classification: C38, L52
DOI: 10.22495/rgcv7i1art7
1. INTRODUCTION
year. However, recent figures published by Statistics
South Africa show that South Africa has been
The Box-Jenkins methodology has gained more
popularity since their book publication in 1970. The
Box-Jenkins method which does not require
establishing assumptions on the interdependence of
variables could be used to test applicability on data
series undergoing dynamic fluctuation. More
significantly, this technique does not introduce too
much personal bias into the process of forecasting.
The Box-Jenkins technique is considered as a suitable
forecasting tool when the components describing the
time series are fluctuating quite rapidly over time
(Bowerman and O’Connell, 1993; Wong et al., 2005).
At the same time, the Box-Jenkins method is a
reliable and convenient tool among numerous
common time series skills. Therefore, this research
paper adopted the Box-Jenkins methodology to
construct a forecasting model for South Africa gold
sales. The technique is based on the idea that a time
series in which successive values are highly
dependent can be regarded as being generated from
a series of independent shocks. Analysing such series
leads to the class of Autoregressive Integrated
Moving Average (ARIMA) models. An autoregressive
(AR) process is fundamentally a regression equation
slipping down the table from the world’s top
producer less than a decade ago to sixth position.
According to the Mineweb (2013), China is
currently leading, followed by Australia, the USA,
Russia and Peru in that order. In 1970, South Africa
produced almost 80% of global gold production now
it manages only around 6%, which is a very big fall
(Mineweb, 2013). There has been a noticeable decline
in gold production (extraction) and mining
contribution to South Africa’s Gross Domestic
Product (GDP). In terms of employment, the mining
industry reported an annual decrease of over 6%
from December 2008 to December 2009 (StatsSA,
2013). South Africa’s mineral industry is export-
oriented, due to the small domestic market for most
commodities.
In another report by Statistics South Africa
(2013), annual mineral sales was estimated to have
decreased by 8.5% in May 2013 and the largest
negative growth rates were recorded for gold with a
value of -42.6%, i.e. from May 2012 to May 2013, gold
sales decreased by 42.6% as illustrated in Table 1.
This was the largest negative growth rate observed
for gold sales in 2013.
where a variable is related to its own previous values
instead of to a set of independent variables
(Chatfield, 2000).
The gold mining sector played a substantial role
as a basis industry in the evolution of South African
industry. The gold mining industry has been the
dominant foreign exchange earner for the country
over the past century. More recent statistics indicate
that gold export earnings in 1980 accounted for over
50% of South Africa’s merchandise exports in that
Table 1. Year-on-year percentage change in gold
sales at current prices
Date % change
December 2012 -16.1
January 2013 12.4
February 2013 -9.1
March 2013 3.3
April 2013 -13.6
May 2013 -42.6
54
12
Risk governance & control: financial markets & institutions / Volume 7, Issue 1, Winter 2017
There is a general decline in the gold sales over
Khan
(2013)
applied
Box-Jenkins’
ARIMA
the years and this could have long negative results in
approach for building forecasting model using the
the future. According to an article in the Mineweb
gold price sample (in US$ per Ounce). The results
(2013),
showed that ARIMA(0,1,1) is the most appropriate
“More recent statistics indicate that gold export
model to be used for forecasting the gold price. Patel
earnings in 1980 accounted for over 50% of South
(2013) investigated the role of gold as a strategic
Africa’s merchandise exports in that year. However,
prophecy against inflation and exchange rate and
recent figures published
by Statistics South Africa
found that gold can act as a hedge against inflation
show that South Africa has been slipping down the
and exchange rate in two different ways. Firstly, gold
table from the world’s top producer less than a decade
price act as an internal hedge against inflation of the
ago to sixth position”.
country. This means, that if inflation increases, gold
The following could be listed as contributing
price would also increase. Secondly, gold price also
factors: sudden changes in gold demand levels, price-
acts as an external hedge. This means if exchange
cutting manoeuvres of the competition, strikes, large
rate decrease, price of gold will increase. Ranson
swings of the economy, interest rates, inflation rates
(2005) examined the role of gold and oil as predictor
and seasonality. A decline in gold sales has affected
of inflation. He found that gold price is more reliable
many
related
sectors
and
contributed
negatively
barometer of the inflation than oil price because the
towards South Africa's annual GDP. The forecasting
effect
on
official
inflation
statistics
is
reliably
of gold sales is essential to help in calculating the
indicated by how far policy actions have allowed the
volume of mining production which affects GDP and
price of gold to rise.
its components. The main objective of this study is
to apply a Box-Jenkins’ ARIMA model approach to
model South Africa’s monthly gold sales and to use
3. METHODOLOGY
the identified ARIMA model to forecast future South
African gold sales and the other objective is to
compare the year-on-year percentage change findings
with the Statistics South Africa gold findings.
The paper is set out as follows. Section 2
discusses some of literature relating to our study.
Section 3 briefly outlines the methodological
framework. Section 4 presents the results and
discussions. Concluding remarks is given in section
5.
The study present uses a monthly general gold
production data for the period January 2000 to June
2013. The Box-Jenkins methodology employed in
this study is based on the analysis of pattern
changes in the past history of the observations
and it uses a four-phase approach (Box, Jenkins and
Reinsel, 1994). Namely: tentative model
identification, model estimation, diagnostic checking
and forecasting.
2. LITERATURE REVIEW
3.1. Tentative Model Identification
Ping, Miswan and Ahmad (2013) carried out a study
on forecasting the prices of Kijang Emas, the official
Malaysian gold bullion. Their study employed two
methods, which are Box-Jenkins ARIMA and
Generalized Autoregressive Conditional
Heteroskedasticity (GARCH). Using Akaike’s
information criterion (AIC) as the goodness of fit
measure and mean absolute percentage error (MAPE)
as the forecasting performance measure, they found
that the gold prices data can be characterized by
GARCH (1,1) model. Their conclusion was based on
the fact that GARCH (1,1) had both a lower SIC and
MAPE value than ARIMA (1,1,1) in forecasting its
future values.
Mahipan, Chutiman and Kumphon (2013)
applied both Box-Jenkins and Artificial Neural
Network methods to prediction the rate of
unemployment in Thailand. In their paper, they
determined the stationary and seasonality of the data
and the Augmented Dickey-Fuller test.
(ADF) and autocorrelation (ACF) were used
A plot of the original data should be run as the initial
point in determining the most appropriate model.
Stationarity tests can be performed to determine if
differencing is necessary. Besides looking at the
graphical presentation of the time series values over
time to determine stationary or non-stationary,
the sample ACF also gives visibility to the data. Non-
stationary data displaying trend behaviour can be
transformed through regular differencing. In this
study more focus is based on first and second
regular differencing.
The initial work on stationarity testing came
from Dickey and Fuller (1979), who conceptualised
the technique as “testing for a unit root”. This is a
formal test employed in this study to check for
stationarity in the time series data. Within the
framework of the Box-Jenkins methodology, there is
an overall model which can be decomposed into
three basic models. The ARIMA can be decomposed
into an AR, Moving Average (MA) and Autoregressive
Moving Average (ARMA) model.
respectively. The ADF test for stationary showed that
the series had a unit root implying that the original
3.2. Model Estimation
series was non-stationary. The data became
stationary after first order difference. Examination of
the correlogram indicated that Seasonal ARIMA
model was appropriate. The Box-Jenkins
methodology proved more efficient to estimate the
This phase involves estimation of the parameters of
the models identified (specified) in the first phase.
The least squares approach is employed in model
estimation.
rate of unemployment in Thailand. MAPE was used to
show that SARIMA, SARIMA(0,1,1) provided
satisfactory representation of the unemployment rate
data.
3.3. Diagnostic Checking
Diagnostic testing in the Box-Jenkins methodology
essentially involves the statistical properties of the
55
0 1
𝑛 𝑦
k
ˆ
l
n l
2
Risk governance & control: financial markets & institutions / Volume 7, Issue 1, Winter 2017
error terms (normality assumption, weak white noise
Holt’s forecast model and a combination
of Box-
assumption) as well as common testing procedures
Jenkins and Holt’s in regression, by providing lowest
on the estimates. As mentioned earlier, εt is expected
to follow a white noise process. Graphical procedure
mean MAPE (Warant, 2006). There are many simple
measures of prediction accuracy, for instance the
and formal testing procedure can be used to test
mean squared error (MSE), mean absolute error (MAE)
adequacy of the model. In the graphical procedure a
and
mean
squared
deviation
(MSD).
However
the
plot
of
the
residuals
is
examined
to
check
for
most appropriate simple error measure for this study
outliers. To check the overall acceptability of the
is the MAPE given by the following equation:
overall model, the Ljung-Box (1978) test can be used
as follows:
H : Model is adequate versus H : Model is
inadequate
𝑀𝐴𝑃𝐸 = 1 𝑛 |𝑦𝑡 − 𝑓| (2)
𝑖=1 𝑡
This test statistic can be used to compare the
Test statistic:
accuracy of forecasts based on two entirely different
series (Hanke and Wichern, 2005). According to Lewis
Q* = n' (n' + 2) '1 r2 (a)
l=1
(1)
(1982), the level of accuracy for the MAPE test is
divided into four stages as shown in Table 2. Each
level of accuracy gives the percentage of the accuracy
of predicted value compared to the original time
where
𝑛1 = 𝑛 − 𝑑,
n
is
the
number
of
series value (Muda and Hoon, 2012).
observations and d is the degree of non-seasonal
differencing used to transform the original time
Table 2. Level of accuracy for MAPE test
series values into stationary. The 𝑟(�̂�) is the square
of the autocorrelation of the residuals at lag l
(Bowerman, O’Connell and Koehler, 2005). If the p-
value is greater than significant level α or
MAPE value
MAPE ≤ 10%
10% < MAPE ≤ 20%
Level of accuracy
Very accurate
Accurate
equivalently 𝑄is less than chi-square distribution,
20% < MAPE ≤ 50%
Medium
the null hypothesis cannot be rejected concluding
50% ≤ MAPE
Less accurate
that the model is adequate. According to Verbeek
(2004), if a model is rejected at this stage; the model-
building cycle has to be repeated.
4.
RESULTS AND DISCUSSION
3.4. Forecasting
This section of the study carries out the four Box-
Jenkins technique’s steps to analyse the gold sales
The
final
and
most
important
stage
of
the
Box-
data.
Jenkins process is forecasting. There are two broad
types of forecasts: one step ahead forecasts are
4.1. Step 1: Model Identification
generated for the next observation only whereas
multi-step ahead forecasts are generated for
1,2,3,…..,s steps ahead. Many researchers suggest
that Box-Jenkins’ ARIMA is the most accurate
forecasting model. ARIMA wins over other models;
The first step in developing a Box-Jenkins model is to
determine if the series is stationary and if there is
any observed pattern. The data is plotted as shown in
Figure 1 below.
Figure 1. Original Plot of Gold Sales
(a) Gold series plot
(b) ACF plot of the original data
56

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