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Gold Price Forecasting Using ARIMA Model
Banhi Guha and Gautam Bandyopadhyay
Department of Management Studies, National Institute of Technology, Durgapur, India
Email: banhi.guha@gmail.com, math_gb@yahoo.co.in
Abstract This study gives an inside view of the application
of ARIMA time series model to forecast the future Gold
price in Indian browser based on past data from November
2003 to January 2014 to mitigate the risk in purchases of
gold. Hence, to give guideline for the investor when to buy
or sell the yellow metal. This financial instrument has
gained a lot of momentum in recent past as Indian economy
is curbed with factors like changing political scenario, global
clues & high inflation etc, so researcher, investors and
speculators are in search of different financial instrument to
minimize their risk by portfolio diversification. Gold earlier
was only purchased at the time of marriage or other rituals
in India but now it has gained importance in the eyes of
investors also, so it has become necessary to predict the
price of Gold with suitable method.
Index Terms gold price, ARIMA, forecasting
I. INTRODUCTION
All that glitters is not gold is a proverb use to depict
that not everything that looks attractive is not so, in
reality. This yellow metal has grab lot of attention for
every class of people as investment purpose.
People investing in gold have mainly two primary
objectives, one being it is a hedge against inflation as
over a period of time, the return on gold investment is in
line with the rate of inflation, next to mix your investment
basket and hence diversify the risk and will help you
reduce the overall volatility of your portfolio.
Investing in gold have evolved over a period of time
for traditional ways by buying jewelries or by modern
way as purchasing gold coins and bars (which is available
in scheduled banks nowadays) or by investing in Gold
Exchange traded fund (Gold ETF). Gold ETF is in
financial instrument of mutual fund in nature which in
turn invests in gold and these are listed in a stock index.
Gold Fund of Funds and Equity based Gold Funds are
other instrument where investor can choose to invest in
Gold having some variation like Gold Fund of Funds is
an investment made on behalf of the investor without
holding a Demat account and Equity based Gold Funds
are investment which are not made directly in Gold, but
investing in the companies, which are related to the
mining, extracting and marketing of the Gold.
Importance of the yellow metal has changed over a
period of time. In India few thousands of years ago,
countless Kings & Emperor, the then rulers of land in
different parts of the country having different monetary
Manuscript received July 6, 2014; revised October 14 , 2014.
system, but only Gold was treated as common exchange
commodity.
India is known to demand of gold mainly for jewelry
fabrication where it makes in the top list of imports of
gold as the production of gold in India for mining
activities is very limited, but in our country the demand
for the yellow metal is seasonal and are high during
wedding season, Post harvest season and festival season
and demand are down during monsoon season. As of
today the exchange (National Stock Exchange & Bombay
Stock Exchange) has introduced different instrument
linked with Gold investment to simplify to purchase of
gold from exchange without forgoing with different
charges associated with purchase of jewelries which
ultimately reduces the return of investors while
investment in yellow metal, so it is important for
investors to be well informed about fluctuation in the
price so that he can take wise decision in investing in
Gold. Hence this paper gives an insight of forecasting of
Gold price through time-series ARIMA model.
II. LITERATURE REVIEW
Ref. [1] Banerjee, D., (2014) in her paper Forecasting
of Indian Stock Market using Time-series ARIMA
Model has applied ARIMA model based on which she
predict the future stock indices which have a strong
influence on the performance of the Indian economy. In
her paper she first determined the ARIMA model then
she forecasted the sensex through model validation and at
the end the recurrence validation was done.
Ref. [2] Abdullah Lazim (2012) in his paper has
addressed the forecasting of gold bullion coin prices
through ARIMA model and had concluded by suggesting
that the gold bullion coin selling prices are in upward
trends and could be considered as a worthy investment.
Wouter Theloosen in his research paper A review on the
determinants of the price of gold has cited the different
factors associated with the gold price fluctuation.
Ref. [3] Baber. P., Baber. R, Thomas. G in their study
Factors affecting Gold prices: a case study of India list
different factors affecting the gold price in India and
gives special emphasis on rise in gold price in the decade
from 2002 to 2012.
Ref. [4] Mohamad As’ad (2012) has modeled the peak
daily electricity demand using half hourly demand date.
He coined for ARIMA Models based past three, six, nine
and twelve months of data and suggested that the ARIMA
model build based on past three months data is the best
model in terms of forecasting two to seven days ahead
117
2016 Engineering and Technology Publishing doi: 10.12720/joams.4.2.117-
and ARIMA model based on the past six months data is
the best model to forecast one day ahead.
Ref. [5] Aidan Meyler, Geo Kenny and Terry Quinn
(1998) had forecasted Irish inflation though ARIMA
model. This research paper gives an insight of ARIMA
Modeling by step by step approach for forecasting using
ARIMA Model.
Deepika M G, Gautam Nambiar & Rajkumar M (2012)
has tried to study the forecasting of gold price through
ARIMA model & Regression but their finding suggests
that suitable model was not identified to forecast Gold
price through ARIMA Model hence Regression analysis
was carried out in the later part of their study.
III. OBJECTIVE OF STUDY
To forecast the price of Gold using time-series
ARIMA Model.
IV. DATA & METHODOLOGY
Our study is based on secondary monthly data for Gold
price which is collected from Multi Commodity
Exchange of India Ltd (MCX) ranging from November
2003 to January 2014. MCX is a commodity future
exchange based in India which started its operations from
November 2003.
MCX has different commodities product list for
trading in varied commodity futures contracts across
segments which includes bullion, ferrous and non-ferrous
metals, energy, Agri-based and agricultural commodities.
Gold is segmented as bullions in MCX, its other future
contracts in this category are Gold Guinea, Gold M, Gold
Petal, Gold Petal(New Delhi), Platinum, Silver, Silver M,
Silver Micro, Silver 1000.
Data of total Quantity in terms of 1000 grams and
Value in Rs. Lakhs was taken for a particular month from
MCX website and the price of Gold per 10 grams was
found out tabulated in Table VI inserted in Appendix A.
SPSS 15 evaluation version was used for computation
& graphical plotting of data.
After collecting data it was tested for its suitability for
time series analysis. For this purpose Durbin-Watson Test
was carried out to understand the nature of data.
According to James Durbin and Geoffrey Watson test
statistics was developed to detect the presence of
autocorrelation for its suitability for regression analysis.
Autocorrelation is interrelated between the values with
suitable time lag.
Durbin-Watson (DW) 2[1-(1)], where (1) is the 1st
order auto-correlation.
Ref. [1] If DW value lies between 0 to 1.5 or between
2.5 to 4 then the data is longitudinal i.e. dependent on
time, so time-series analysis can be done, but if DW
value is between 1.5 to 2.5 then it is cross-sectional data
i.e. independent of time hence regression analysis should
be carried out on the collected data.
As test statistics indicate the suitability of time-series
analysis, we can move further in our finding to fulfill our
objective of forecasting using ARIMA Model. The first
steps involved in finding out autocorrelation and partial
auto-correlation between the values of the data.
A. Autocorrelation
It is defined by ACF = corr(Xt , Xt+k) i.e. relationship
between each other. Here Xt is the current observation
and Xt+k is observation after k period. It ranges from -1 to
+1.
B. Partial Auto-Correlation
Yet another important characteristic is a partial
autocorrelation function (PACF) which is conditional
correlation of Xt+k with Xt. PACF is defined for positive
lag only, their value also lies between -1 and +1. Both the
characteristic, ACF & PACF are equally important, but
ACF is relatively easier to calculate than PACF.
####### TABLE I. PROPERTIES OF ACF & PACF FOR AR, MA & ARMA
Properties AR (p) MA (q) ARMA (p, q)
ACF Decay Cuts after q lag Decay
PACF Cuts after p lag Decay Decay
C. Auto Regressive Integrated Moving Averages
Ref. [2] ARIMA as it is better known as is a time
series forecasting technique for short run, which is widely
used in todays world since the evolution of sophisticated
statistical software package.
ARIMA has four major steps in model building-
Identification, Estimation, Diagnostics & Forecast. With
these four steps first tentative model parameters are
identified through graphs ACF and PACF then coefficient
are determined and find out the likely model, next steps
involves is to validate the model and at the end use
simple statistics and confidence intervals to determine the
validity of the forecast and track model performance.
ARIMA model uses the historic data and decomposes
it into Autoregressive (AR) Indicates weighted moving
average over past observations, Integrated (I) Indicates
linear trends or polynomial trend and Moving Average
(MA) Indicates weighted moving average over past errors.
Therefore, it has three model parameters AR(p), I(d) and
MA(q) all combined to form ARIMA(p, d, q) model
where
p = order of autocorrelation
d = order of integration (differencing)
q = order of moving averages
Ref. [1] A non-seasonal stationary time-series can be
modeled as a combination of the past values and the
errors which can be denoted as ARIMA (p, d, q) or can
be expressed as
Xt = 0 + 1 Xt-1+ 2 Xt-2+ +p Xt-p+ et – 1 et-1 – 2 et-2 – qet-q
(1)
Table I gives an idea to assume the initial values of
ARIMA (p, d, q) for further computation.
Ref. [5] After estimating the three parameters p, d and
q, evaluating the model with fit statistics (goodness of Fit)
is required to measure the performance of forecast as
shown in Table II with its acceptable limits.
Some of the statistical measures are as follows
2016 Engineering and Technology Publishing
####### TABLE II. FIT STATISTICS
Fit Statistics Expression Remark
Root-mean-
square error
(RMSE)
Relatively Low
Mean absolute
percentage error
(MAPE) (^) Minimum Mean absolute error (MAE) Minimum Bayesian information criterion (BIC) Where rss = residual sum of squares. k = no. of coefficients estimated = 1+p+q+P+Q n= no. of observations. Lungs Box Q statistics n=the number of residuals h = number of time lags includes in the test ^2 k = the residual autocorrelation at lag k
For the model to be adequate the significance level for
q statistic should be significant
V. RESULT & DISCUSSION
The value of Durbin-Watson (DW) was 0.091 for the
sample data of the Gold price from November 2003 to
January 2014 which indicates that the data is suitable for
time-series analysis. As DW 2[1-(1)] hence (1) =
0.945 which indicates that gold price shows high 1st order
auto-correlation.
The ACF and PACF correlogram was plotted to
identify the model of ARIMA
Lag Number
987654321 3231302928272625242322212019181716151413121110
ACF
1.
0.
0.
-0.
-1.
Gold_Price
Lower Confidence Limit
Upper Confidence Limit
Coefficient
Figure 1. ACF-Gold price correlogram
Ref. [1]-[2], [5]-[6] From the above correlogram
shown in Fig. 1 and Fig. 2, initial assumption can be
taken as p = 1 of ARIMA(p, d, q) since PACF cuts after
lag 1 but about the value of q & d it was undeceive so we
had to test for all possible values of d = 0 & 1 & q =1,2,
to get the model parameters according to model based on
parsimony principle of optimality. Comparing six
different combinations of p, d, q of ARIMA model given
in Table V inserted in Appendix where value of p =1. Fig.
1 shows that ACF correlogram decays hence we are sure
about the parameter p of ARIMA which can be assumed
as 1 as tabulated in Table I.
Lag Number
987654321 3231302928272625242322212019181716151413121110
Pa
rti
al ACF
1.
0.
0.
-0.
-1.
Gold_Price
Lower Confidence Limit
Upper Confidence Limit
Coefficient
Figure 2. Partial ACF-gold price correlogram
Ref. [1], [5] After comparing the fit statistics as
tabulated in Table II, only ARIMA (1, 1, 1) satisfy all the
criteria, hence we obtained the model ARIMA (1, 1, 1)
which is used to forecast the future gold price.
The generalized ARIMA (1, 1, 1) model is in the form
Xt = +Xt-1 + (Xt-1 Xt-2) et-1 (2)
where = Constant (1- )
= AR Coefficient
= MA Coefficient
The Table III below shows the parameter estimate of
ARIMA (1, 1, 1) with respective significance level.
####### TABLE III. ESTIMATE TABLE-ARIMA(1,1,1)
Estimate SE t Sig.
Constant 190.708 70.176 2.718.
AR Lag 1 - .734 .170 - 4.326.
Difference 1
MA Lag 1 - .869 .124 - 7.011.
Thus the model using the above table is
Xt = 190.708(1+0.734) + Xt-1 -0.734 (Xt-1 Xt-2) +
0.869 et-1 (3)
Here all the significant values are less than 0.05 so
ARIMA (1, 1, 1) was taken into consideration.
####### TABLE IV.^ FORECASTED MONTHLY GOLD PRICE^
Month^
Observed
values^
Model
validation^
Recurrent
validation^
Feb- 14 29482.91^ 29386.40^ 29352.^
Mar- 14 29670.43^ 29614.93^ 29500.^
Apr- 14 28514.64^ 29850.13^ 29913.^
May- 14 27812.81^ 30009.72^ 28477.^
June- 14 26813.15^ 30224.71^ 28067.^
July- 14 27867.11^ 30399.11^ 26753.^
2016 Engineering and Technology Publishing
Now coming back to the objective to predict the future
gold price with the model from the equation 3 is shown in
Table IV below with the relevant graph.
Date
1 121111101918171615141312111
Num
ber
35 ,00 0
30 ,
25,
20,
15,
10,
5,
Gold_Price-Model_
Observed
Figure 3. Graph showing gold price-observed value in y-asis vs time
in x-asis
Fig. 3 shows the trend of gold price over the period of
ten years till January 2014.
Date
1 121111101918171615141312111
Num
ber
35 ,
30 ,
25,
20,
15,
10,
5,
Gold_Price-Model_
FitObserved
Figure 4. Graph showing gold price-observed value and fit value in y-
asis vs time in x-asis
Fig. 4 shows the plotting of gold price and price
calculate from model as shown in equation (3) which also
tabulated in Table IV. The blue worm in the figure is the
model data crawls with the red worm is gold price. As the
gap between the blue and red worm is minimum so the
model equation can be taken into consideration for
forecasting of gold price beyond January 2014.
Fig. 5 shows the plotting of forcasted value with the
help of the model values for six months.
Date
1 121111101918171615141312111
Num
ber
35 ,00 0
30 ,
25,
20,
15,
10,
5,
Gold_Price-Model_
Forecast
Fit
Observed
Figure 5. Graph showing gold price-observed value and fit value &
forecast value in y-asis vs time in x-asis
VI. CONCLUSION
Analysis of performance of the gold price from
preceding 10 years traded value in MCX gives us
ARIMA (1, 1, 1) model which helps us in predicting the
future values of Gold. ARIMA (1, 1, 1) was chosen from
six different model parameters as it provides the best
model which satisfies all the criteria of fit statistics while
other five failed the fit statistics.
VII. LIMITATION
There are certain limitations in forecasting a data with
ARIMA modeling. This technique is used for short run
only, to detect small variation in the data. In case of
sudden change, in the data set (when the variation is large)
in case of change in government policies or economic
instability (structural break) etc. it becomes difficult to
capture the exact change, hence this model becomes
ineffective to forecast in this scenario more over the
forecasting with this method is based on assumption of
linear historic data but there is no evidence that the gold
price is linear in nature.
VIII. FUTURE SCOPE OF STUDY
Forecasting of gold price with ARIMA model was
done with the basic assumption that it follows a perfectly
linear pattern, hence implementing non-linear forecasting
techniques using soft computing techniques can be
considered with less white noise term.
APPENDIX^ A TABLES^
####### TABLE V. ARIMA MODEL SUMMARY
ARIMA (p, d, q) R-squared RMSE MAPE MAE
Normalized
BIC
Lungs Box Q
(18) statistics
(sig)
p value of
Constant AR (1)
####### MA
lag 1 lag 2 lag 3
ARIMA(1,0,1) 0.978 1300. 41 4.846 595.112 14.458 1 0.366 0 0.266 - -
ARIMA(1,0,2) 0.978 1295.7 0 4.829 594.48 0 14.49 0 1 0.34 0 0 0.304 0.854 -
ARIMA(1,0,3) 0.979 1274.9 0 4.707 570.170 14.497 1 0.231 0 0.05 0.888 0.
ARIMA(1,1,1) 0.993 719.18 3.245 47 7.33 0 13.274 0.646 0.008 0 0 - -
ARIMA(1,1,2) 0.993 720.74 3.261 474.84 0 13.318 0.63 0.005 0.004 0.002 0.575 -
ARIMA(1,1,3) 0.993 716.35 3.135 463.15 0 13.345 0.841 0.019 0.106 0.047 0.738 0.
2016 Engineering and Technology Publishing
####### TABLE VI. MONTHLY GOLD PRICE IN RS/10GRAMS FROM NOVEMBER 2003 TO JANUARY 2014
Month Gold Price Month Gold Price Month Gold Price Month Gold Price Month Gold Price
Nov- 03 5884.183 Dec- 05 7624.102 Jan- 08 11263.45 Feb- 10 16516.16 Mar- 12 27961.
Dec- 03 6132.507 Jan- 06 7941.226 Feb- 08 11774.79 Mar- 10 1 6602.01 Apr- 12 28622.
Jan- 04 6223.322 Feb- 06 8018.163 Mar- 08 12554.32 Apr- 10 16748.32 May- 12 28823.
Feb- 04 6081.285 Mar- 06 8112.945 Apr- 08 11789.38 May- 10 17984.35 Jun- 12 29912.
Mar- 04 6032.368 Apr- 06 9055.945 May- 08 12116.98 Jun- 10 18722.85 Jul- 12 29554.
Apr- 04 5902.734 May- 06 10048.49 Jun- 08 12350.31 Jul- 10 18265.4 Aug- 12 30366.
May- 04 5732.307 Jun- 06 8993.265 Jul- 08 13037.99 Aug- 10 18509.24 Sep- 12 31706.
Jun- 04 5855.261 Jul- 06 9611.196 Aug- 08 11805.17 Sep- 10 19103.68 Oct- 12 31208.
Jul- 04 6001.504 Aug- 06 9683.527 Sep- 08 12273.34 Oct- 10 19583.55 Nov- 12 31592.
Aug- 04 6086.221 Sep- 06 9046.995 Oct- 08 12736.15 Nov- 10 20162.13 Dec- 12 31130.
Sep- 04 6132.289 Oct- 06 8809.648 Nov- 08 12084.21 Dec- 10 20628.66 Jan- 13 30754.
Oct- 04 6272.087 Nov- 06 9150.099 Dec- 08 12837.27 Jan- 11 20253.88 Feb- 13 30205.
Nov- 04 6463.56 Dec- 06 9256.871 Jan- 09 13474.71 Feb- 11 20495.05 Mar- 13 29599.
Dec- 04 6423.59 Jan- 07 9090.71 Feb- 09 14912.04 Mar- 11 20919.54 Apr- 13 27449.
Jan- 05 6133.366 Feb- 07 9589.743 Mar- 09 15256.39 Apr- 11 21635.71 May- 13 26551.
Feb- 05 6099.13 Mar- 07 9403.5 Apr- 09 14450.24 May- 11 22189.03 Jun- 13 27066.
Mar- 05 6237.06 Apr- 07 9442.006 May- 09 14543.05 Jun- 11 22465.34 Jul- 13 26893.
Apr- 05 6202.608 May- 07 8923.453 Jun- 09 14631.97 Jul- 11 22824.97 Aug- 13 30386.
May- 05 6056.98 Jun- 07 8796.646 Jul- 09 14682.79 Aug- 11 26268.63 Sep- 13 30798.
Jun- 05 6181.291 Jul- 07 8763.533 Aug- 09 14922 Sep- 11 27471.55 Oct- 13 29590.
Jul- 05 6087.641 Aug- 07 8867.74 Sep- 09 15690.97 Oct- 11 26742.87 Nov- 13 30000.
Aug- 05 5738.967 Sep- 07 9340.204 Oct- 09 15855.3 Nov- 11 28560.84 Dec- 13 28891.
Sep- 05 6583.637 Oct- 07 9699.008 Nov- 09 17101.92 Dec- 11 28188.29 Jan- 14 29095.
Oct- 05 6821.654 Nov- 07 10273.67 Dec- 09 17222.07 Jan- 12 27692.
Nov- 05 7107.171 Dec- 07 10297.75 Jan- 10 16696.9 Feb- 12 28357.
2016 Engineering and Technology Publishing
REFERENCES
[1] D.Banerjee, Forecasting of Indian stock market using time-series ARIMA model,inProc.Conference Paper, ICBIM- 14 , 2014. [2] L. Abdullah,ARIMA model for gold bullion coinselling prices forecasting, International Journal of Advances in Applied Sciences, vol. 1, no. 4 , pp. 153- 158 , 2012. [3] P. Baber, R.Baber, and G.Thomas, Factors affecting gold prices:
A case study of India, in Proc. Evolving Management
Paradigms in Manufacturing and Service Sectors,March 13. [4] M.As’ad,Finding the best ARIMA model to forecast daily peak electricity demand, in Proc. the Fifth Annual ASEARC Conference, 2012. [5] AMeyler, G.Kenny, and T.Quinn.(1998).Forecasting Irish inflation using ARIMA models.[Online].MPRA PaperNo. 11359, posted 3. November 2008 14:34 UTC. Available: http://mpra.ub.uni-muenchen.de/11359/ [6] C.Chatfield, The analysis of time series an introduction.
BanhiGuhais Research Scholar, National Institute of Technology,
Durgapur, India and has obtained MBA for same Institute.
GautamBandyopadhyayhas obtained his PhD
form JadavpurUniversity. He is also fellow
member of the Institute of Cost & Works
Accountants. He is presently guiding good
number of PhD students and has already
produced PhD too. He is presently serving NIT
Durgapur as Associate Professor.