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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 3, March 2013)
Forecasting of Gold Prices (Box Jenkins Approach)
Dr. M. Massarrat Ali Khan
College of Computer Science and Information System, Institute of Business Management, Korangi Creek, Karachi
Abstract— In recent years, the global gold price trend has
attracted a lot of attention and the price of gold has
frightening spike compared to historical trend.
In times of uncertainty investors consider gold as a hedge
against unforeseen disasters so the forecasted price of gold has
been a subject of highest amongst all.
In this paper an attempt has been made to develop a
forecasting model for gold price. The sample data of gold
price (in US$ per ounce) were taken from January 02, 2003 to
March 1, 2012. Data up till January 02, 2012 were used to
build the model while remaining were used to forecast the
gold price and to check the accuracy of the model.
We have used Box-Jenkins, Auto Regressive Integrated
Moving Average (ARIMA) methodology for building
forecasting model. Results suggest that ARIMA(0,1,1) is the
most suitable model to be used for predicting the gold price.
For testing the forecasting accuracy Root Mean Square Error,
Mean Absolute Error, and Mean Absolute Percentage Error
are calculated.
II. LITERATURE REVIEW
In recent past most of the literature on gold is based on
analysis and review of gold markets, its supply and demand
situations and forecasting of gold prices by various
methods. Some of recent works on gold price forecasting
are discussed below:
Shahriar and Erkan (2010) carried out a study on global
gold market and gold price forecasting. They analyzed the
world gold market from January 1968 to December 2008.
They applied a modified econometric version of the longterm trend reverting jump and dip diffusion model for
forecasting natural resource commodity prices. Estimates
of dynamic jump and dip as parameters were obtained
using the model
(
)
,
Where
Keywords-- ARIMA, Forecasting, Stationary Time Series,
Gold Price, Mean Absolute Error, Root Mean Square Error
= First component or drift
(
) = Second component or the range of random
movement
= Third component or jump or dip
I. INTRODUCTION
Gold is one of the most important commodities in the
world. It is the only commodity that retains its value
during all the periods of crises – economies, financial or
political. As long as the world economy remains uncertain
and investors feel inflation and sovereign default, gold will
keep its allure (The Economist – July8, 2010). The price of
gold has been rising day by day, the fear of the world
economy have caused the price of gold to roar. When
foreign nations that hold billions of dollars in US debt
starts buying gold because they fear the value of the dollars
will go down, the rising price of gold becomes more than a
novelty (Frank Ahrans – September 24, 2009 – Economy
Watch – The Daily Washington Post). Price of gold cannot
be controlled; however it can be measured and forecasted
for future decisions. Forecasting models based on time
series data (univariate methods) and relationship between a
series of other indicators (causal or multivariate methods)
are very popular, since they are more effective and have
less residual errors and forecast errors.
The purpose this study is to develop a forecasting model
for the price of gold.
The study is based on the time series data of London
gold price per ounce, in US Dollars from Jan 02, 2003 to
March 1, 2012 (London Gold Price Fixing).
Where;
, if gold prices have jump and
if the gold
prices do not have any jump,
, if gold prices have
dip and
if the gold prices do not have any dip and
is the historical volatility of gold prices.
To evaluate the jump or dip, the model reviews the
historical price trend of jump and dip and then estimates
the same trend for future. The unit root test for nominal
gold price has also been conducted before applying the
model. The first component (α2t) in the model is the longterm trend component; this component shows that gold
price should be reverting to the historical long-term trend.
Shahriar and Erkan (2010) discussed the results of three
above mentioned components for gold price from Jan 1968
to Dec 2008 and found that gold price on average increased
by $1.12 per ounce per month, jump and dip volatility is
25% of the current gold price per month whereas the gold
price goes down at $18 per ounce decreasing a dip period
and goes up at $20 per ounce during the jump period.
Shahriar and Erkan used the model for forecasting next
10 years gold prices. Results indicated that, assuming the
current price jump started in 2007 behaves in the same
manner as that experienced in 1978, the gold price would
remain high up to the end of 2014.
662
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Ismail, Yahya and Shabri (2009) developed a forecasting
model for predicting gold price using Multiple Linear
Regression (MLR). They obtained four different models
based on several economic factors. In this study PraisWinsten procedure was employed to estimate the
regression coefficients and they found this procedure
successfully solving the problem of correlated error terms.
Ismail, et al. (2009) considered different commodities of
factors such as Commodity Research Bureau future index
(CBR) , Euro/USD (Euro/USD foreign exchange rate),
Inflation Rate (INF) Money Supply (M1), New York Stock
Exchange (NYSE), Standard and Poor 500 (SPX), Treasure
Bill (T-Bill) and US Dollar Index (USDX) to estimate the
gold price. They found that CRB, Euro/USD foreign
exchange rate, and M1 as significant factors in forecasting
the gold price communicated with low P-values.
Khaemasunun (2006) forecasted Thai gold price, using
Multiple Regression and Auto-Regression Integrated
Moving Average (ARIMA) model. While fitting the model
Khaemasunun (2006) considered the effect of nine
currencies (United States, Australia, Canada, Peru, Hong
Kong, Japan, German and Italy, Signapore and Colombia),
Oil Prices, Set Index, Interest Rate, Gold Derivation on
Thai gold price. Khaemasunun (2006) also used Chinese
New Year Gift as dummy variable.
The model to be tested is as follows:
Using various lags of time and observing the ACF (Auto
correlation Function) and PACF (Partial Auto-correlation
Function), Khaemasunun (2006) described to use
ARIMA(1,1,1), as it contained the least Mean Absolute
Percentage Error (MAPE), to forecast Thai Gold Prices.
Selvanatha (1991) used London daily gold price in
Australian dollars (PAUD) for the period 3 August, 1987 to
20 July, 1988 to construct the gold price forecasted model.
Using Box–Jenkins technique, Selvanatha (1991) found
that a suitable model could be:
PAUDt = α + β +PAUDt – 1 + Ut
t = 2, ……T.
Where Ut is a white noise and T = Sample size
Testing for Simple Random Walk (SRW) and found that
the preferred model is:
PAUD = PAUDt – 1 + Ut
Selvanathan has analyzed the accuracy of the gold price
forecasts gathered from a panel of gold experts and
concluded that forecasts from a simple random-walk model
are superior to the ERC panel forecasts and simple randomwalk model forecasts are cheap as compared to the efforts
of the panel of experts.
III. FORECASTING MODEL
The Box-Jenkins ARIMA is one of the most
sophisticated techniques of time series forecasting. It is so
common is econometrics that the terminology ―time series
analysis‖ referred to the ‗Box Jenkins approach to
modeling time series. (Kennedy, 2008)
The general Box-Jenkins (ARIMA) model for y is
written as:
Yt(Gold Price) = β1US + β2Aus + β3Can + β4Per +
β5HK + β6Jap + β7EU + β8SIN + β9Col + β10Oil
+ β11Set + β12Int + β13Future + β14Gift
The first stage results showed that only five currencies
were significantly affecting Thai gold price namely AUS,
US, Jap, EU and CAN.
At second stage Khaemasunun (2006), used five
currencies and five other variable namely OIL (Oil Price),
SET index (Portfolio Theory), INT (Interest Rate),
FUTURE (Gold Derivation) and GIFT (Chinese New Year
– Chinese Thai people have belief that gold represent
prosperity, giving gold as a gift alike giving the prosperity
to other).
At final stage simple regression model analysis based on
the five currencies, FUTURE and OIL was identified and
following significant factors were identified.
Where and θ are unknown parameters and the are
independent and identically distributed normal errors with
zero mean, p is the number of lagged value of y*, it
represents the order of auto regressive (AR) dimensions, d
is the number of times y is differed, and q is the number of
lagged values of the error terms representing the order of
moving average (MA) dimension of the model. The term
―integrated‖ means that to obtain a forecast for y from this
model it is necessary to integrate the forecast y*. ARIMA
methodology may be represented by the following diagram.
ΔYt = 0.022039321137 + 0.6624022662Aus + 0.41996153
Can + 0.436812817Jap + 0.1504451898EU +
0.8546501878US
+
0.121974604Future
+
0.04365593688Oil
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Data Preparation
Identification
Difference data to obtain stationary series
Model Selection
Examine data, ACF, PACF to identify potential
(choosing tentative p, q, d)
Figure 4.1 - Gold Price
Estimation
TABLE 4.1
Correlogram of GP
Estimate parameters in potential model and testing.
Select best model using suitable criterion Diagnostic
Diagnostic
Check ACF/PACF of residuals (are the estimated
residuals are white noise?)
Forecasting
Use model to forecast
The principle of parsimony is adapted to determine p, d,
and q. The Box-Jenkins approach is valid for the variables
which are stationary having constant mean and variance
over time. Many research studies have shown that most of
the macro economy data are non-stationary, as it carries
few characteristics of a random walk (highly correlated in
adjacent observations).
We therefore decided to transfer the gold price series to
changed to 1st difference data and tested again. Figure 4.2
shows the 1st difference gold price series; this series may
have a mean of zero and are distributed as white noise.
Table 4.2 shows the ACFs and PACFs for 1st difference
data, the ACFs and PACFs are pattern less and statistically
not significant.
IV. DATA COLLECTION AND ANALYSIS
The data of gold price (in US per ounce) were collected
from January 02, 2003 to March 1, 2012 (London Gold
Price Fixing).
A. Stationary Test:
The line diagram for gold price data from Jan 02, 2003
to March 03, 2012 (London Gold Price Fixing) is shown
Figure 4.1. The correlogram of gold price is shown in
Table 4.1. Both Figure 4.1 and Table 4.1 show random
walk behavior. Furthermore the price is showing
fluctuation but overall trend is upward. Table 4.1 shows
there is high ACFs and PACFs.
Figure 4.2 - 1st Differential
664
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TABLE 4.2
Correlogram of D (GP)
TABLE 4.3
TABLE 4.4
The Unit Root Test:
After stationary test, the unit root test is done by
―Augmented Dickey Fuller‖ and ―Philips Perron‖ unit root
test by setting hypothesis as:
H0 : ρ= 1 (Non stationary)
H1 : ρ ≠ 0 (Stationary)
Table 4.3 depicts Augmented Dickey Fuller Test; tstatistic value for direct values is -0.686623 so we do not
reject H0 at 5% level of significance. The gold price series
(level) is non stationary. However Table 4.4 shows ADF
unit root test statistic for first differential gold price data
has significant value of test statistic, which is -53.0557, so
we reject H0 and it shows that series is stationary at first
difference I(1).
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Consolidated Unit Root Test (ADF and PP) and KPSS
stationary test ‗with intercept‘, ‗trend and intercept‘ and ‗no
intercept and trend‘ is shown below in Table 4.5:
ARIMA(0,1,1) and ARIMA(1,1,0) models statistics are
shown in Appendices.
Once the model is identified after evaluating basic
assumptions we fitted the following model:
Table 4.5
Unit Root Test
With Intercept
Trend and
Intercept
No Intercept and
Trend (None)
ADF
Level
1st Diff
Level
0.6866 -53.055* 0.808
-2.1406 -53.085* -2.071
PP
1ST Diff
-53.3329*
-53.410*
2.00
-53.119
-52.951* 2.604
ŷ= +
Stationery Test
(KPPS)
AR (n) + β MA (n) +
We are using MA(1) and AR(1) models based on
selected criteria‘s
Level 1st Diff
5.506 0.2316*
1.045 0.0161*
yt = 0.590786 – 0.090876MA (1)
The ARIMA(1,1,0) model is:
D (GP) = 0.590848 – 0.087114 AR (1)
*Significant at 1%
B
Forecasting Accuracy:
There are several methods of measuring accuracy and
comparing one forecasting method to another, we have
selected Root Mean Square Error (RMSE). Mean Absolute
Error (MAE) and Mean Absolute Percentage Error
(MAPE). The RMSE, MAE and MAPE are as follows
Model identification and Coefficient Estimates:
After the test of stationary, we conclude that the data is
stationary at first difference. The repressor that would be
chosen from the model is selected from various iteration for
AR(p) and MA(q), the selection is based on observing the
ACFs and PACFs.
We used E-views for estimating the coefficients and
testing the goodness of fit of the model.
The search algorithm tried number of different
coefficient values, after several iterations, and based on
comparing Akaike Information Criteria (AIC), and
Schwarz Information criteria (SIC), the best model to
forecast gold prices is ARIMA (0,1,1) since it contains the
least AIC and SIC ratios.
Table 4.6 shows the AIC and SIC value for various
ARIMA (p,d,q) iterations:
AIC
ARIMA(1,1,0)
22.44866
18.85051
1.086433
The above table shows that the Root Mean Squared
Error and Mean Absolute Error are less in ARIMA(0,1,1)
as compared to ARIMA(1,1,0).
C
Forecast Result Analysis:
The gold prices from 1st Feburary,2012 to 1st
March,2012 were used to predict the gold prices by using
both MA(1) and AR(1) model. The results of forecasted
actual values are shown in Figure 4.3, Table 4.7 and Figure
4.4, Table 4.8.
TABLE 4.6
ARIMA
ARIMA(0,1,1)
22.38152
18.78972
1.082841
RMSE
MAE
MAPE
SIC
(p,d,q)
(1,1,0)
8.000928
8.005803
(2,1,0)
8.001443
8.008758
(3,1,0)
8.002672
8.012428
(4,1,0)
8.003700
8.015900
(5,1,0)
8.004865
8.015900
(0,1,1)
8.000171*
8.005044*
(0,1,2)
8.000700
8.008040
1720
(0,1,3)
8.001538
8.011284
1680
(0,1,4)
8.002094
8.014277
1640
(1,1,1)
8.001164
8.008477
(2,1,2)
8.002196
8.009510
(3,1,3)
8.002877
8.010195
(4,1,4)
8.009990
8.017310
Forecast within 2 St. Error
1880
Forecast: GPF2
Actual: GP
Forecast sample: 2/01/2012 3/01/2012
Included observations: 22
1840
1800
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
1760
2/06/2012
2/13/2012
2/20/2012
2/27/2012
GPF2
Figure 4.3
*Lowest Value of AIC and SIC
666
22.38152
18.78972
1.082841
0.006411
0.138334
0.678907
0.182759
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Table 4.7
Table 4.8
ARIMA(0,1,1)
Real Data Resulted Data
Date
1-Feb-12
2-Feb-12
3-Feb-12
6-Feb-12
7-Feb-12
8-Feb-12
9-Feb-12
10-Feb-12
13-Feb-12
14-Feb-12
15-Feb-12
16-Feb-12
17-Feb-12
20-Feb-12
21-Feb-12
22-Feb-12
23-Feb-12
24-Feb-12
27-Feb-12
28-Feb-12
29-Feb-12
1-Mar-12
1740
1751
1734
1719
1724
1746
1748
1711.5
1720
1722
1733
1713
1723
1733
1748
1752
1777
1777.5
1772
1781
1770
1714
1743.518
1744.22
1744.801
1745.393
1745.984
1746.574
1747.165
1747.756
1748.347
1748.938
1749.529
1750.12
1750.71
1751.301
1751.892
1752.483
1753.074
1753.665
1754.255
1754.846
1755.437
1756.028
Date
Error
1-Feb-12
2-Feb-12
3-Feb-12
6-Feb-12
7-Feb-12
8-Feb-12
9-Feb-12
10-Feb-12
13-Feb-12
14-Feb-12
15-Feb-12
16-Feb-12
17-Feb-12
20-Feb-12
21-Feb-12
22-Feb-12
23-Feb-12
24-Feb-12
27-Feb-12
28-Feb-12
29-Feb-12
1-Mar-12
3.518
-6.78
10.801
26.393
21.984
0.574
-0.835
36.256
28.347
26.938
16.529
37.12
27.71
18.301
3.892
0.483
-23.926
-23.835
-17.745
-26.154
-14.563
42.028
1900
Forecast: GPF
Actual: GP
Forecast sample: 2/01/2012 3/01/2012
Included observations: 22
1800
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
1750
1700
1650
1740
1751
1734
1719
1724
1746
1748
1711.5
1720
1722
1733
1713
1723
1733
1748
1752
1777
1777.5
1772
1781
1770
1714
ARIMA(1,1,0)
Resulted Data
1743.439
1744.03
1744.621
1745.212
1745.802
1746.393
1746.984
1747.575
1748.166
1748.756
1749.347
1749.938
1750.529
1751.12
1751.71
1752.301
1752.892
1753.483
1754.073
1754.664
1755.255
1755.846
Error
3.439
-6.97
10.621
26.212
21.802
0.393
-1.016
36.075
28.166
26.756
16.347
36.938
27.529
18.12
3.71
0.301
-24.108
-24.017
-17.927
-26.336
-14.745
41.846
Figure 4.3 and Figure 4.4 shows the forecast evaluation
statistics for gold price with the two standard error bands;
and we observe that the forecasted for ARIMA(0,1,1) is
better as compared to ARIMA(1,1,0).
The results in Table 4.7, ARIMA(0,1,1) show that there
is less error in forecasting the first 21 values and the
constant are in the range of $26.15 to $37.12. However, in
the last; (22nd value) the error is $42.028 which was caused
due to change in stance of U.S Federal Reserve. Federal
Reserve Chairman stopped the monetary easing
(purchasing of gold and other precious metals-which
people have been hoping for), which had impacted on the
demand of gold and causing the decline in the price of gold
by nearly 4% for its biggest one day drop
(http://www.assettrend.com).
Figure 4.5 shows the comparison for real and forecasted
values for ARIMA(0, 1, 1)
Forecast Within 2 St. Error(110)
1850
Real Data
22.44866
18.85051
1.086433
0.006430
0.143425
0.674298
0.182277
1600
2/06/2012 2/13/2012 2/20/2012 2/27/2012
GPF
Figure 4.4
667
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Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 3, March 2013)
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V. CONCLUSION
[11]
In order to develop a univariate Time Series Model, we
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March 01, 2012.
In this paper, we have developed a systematic and
iterative methodology of Box-Jenkin ARIMA forecasting
for gold price.
A unit root test was applied to the long term daily gold
prices. This concludes that the gold price series is nonstationary.
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stationary at first difference, E-views software is used for
fitting the coefficient of the model, using graphs, statistics,
ACFs and PACFs of residuals and after several iterations,
the model selected is ARIMA(0,1,1).
There are several ways of measuring forecasting
accuracy; we have used Mean Absolute Error, Root Mean
Square Error and Mean Absolute Percentage Error.
We may use this model for forecasting the gold prices
for future.
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Acknowledgement
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I would like to acknowledge and thank Mr. Muhammad
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 3, March 2013)
Appendix 01
Correlogram of 2nd Difference of Gold price
Appendix 01
ADF for 2nd Difference gold price series
669
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Appendix 02
ARIMA(0,1,1)
Appendix 02
ARIMA(1,1,0)
Dependent Variable: D(GP)
Method: Least Squares
Date: 06/20/12 Time: 18:24
Sample (adjusted): 1/06/2003 1/31/2012
Included observations: 2367 after adjustments
Convergence achieved after 3 iterations
Dependent Variable: D(GP)
Method: Least Squares
Date: 06/20/12 Time: 16:16
Sample (adjusted): 1/03/2003 1/31/2012
Included observations: 2368 after adjustments
Convergence achieved after 5 iterations
Backcast: 1/02/2003
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.590786
0.246744
2.394331
0.0167
MA(1)
-0.090876
0.020479
-4.437470
0.0000
R-squared
0.007917
Mean dependent var
0.591385
Adjusted R-squared
0.007497
S.D. dependent var
13.25649
S.E. of regression
13.20671
Akaike info criterion
8.000171
Sum squared resid
412670.8
Schwarz criterion
8.005044
F-statistic
18.88034
Prob(F-statistic)
0.000015
Log likelihood
Durbin-Watson stat
Inverted MA Roots
-9470.202
1.996447
.09
670
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
AR(1)
0.590848
-0.087114
0.249796
0.020490
2.365327
-4.251519
0.0181
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.007585
0.007165
13.21171
412808.8
-9467.098
2.003790
Inverted AR Roots
-.09
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.591339
13.25929
8.000928
8.005803
18.07541
0.000022