Forecasting of Gold Prices (Box Jenkins Approach)

Transcription

1 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, 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. Keywords-- ARIMA, Forecasting, Stationary Time Series, Gold Price, Mean Absolute Error, Root Mean Square Error 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). 662 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 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 ( ), = First component or drift ( ) = Second component or the range of random movement = Third component or jump or dip 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 (α 2 t) 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.

2 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 Prais- Winsten 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: 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. ΔYt = Aus Can Jap EU US Future Oil 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: PAUD t = α + β +PAUD t 1 + U t t = 2, T. Where U t is a white noise and T = Sample size Testing for Simple Random Walk (SRW) and found that the preferred model is: PAUD = PAUD t 1 + U t 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: 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. 663

3 Data Preparation Identification Difference data to obtain stationary series Model Selection Examine data, ACF, PACF to identify potential (choosing tentative p, q, d) Estimation Estimate parameters in potential model and testing. Select best model using suitable criterion Diagnostic Figure Gold Price TABLE 4.1 Correlogram of GP 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). 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. We therefore decided to transfer the gold price series to changed to 1 st difference data and tested again. Figure 4.2 shows the 1 st 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 1 st difference data, the ACFs and PACFs are pattern less and statistically not significant. Figure st Differential 664

4 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: H 0 : ρ= 1 (Non stationary) H 1 : ρ 0 (Stationary) Table 4.3 depicts Augmented Dickey Fuller Test; t- statistic value for direct values is so we do not reject H 0 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 , so we reject H 0 and it shows that series is stationary at first difference I (1). 665

5 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: *Significant at 1% Table 4.5 Unit Root Test Stationery Test (KPPS) ADF PP Level 1 st Diff Level 1 ST Diff Level 1 st Diff With Intercept * * * Trend and * * * Intercept No Intercept and Trend (None) * 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: ARIMA (p,d,q) TABLE 4.6 AIC (1,1,0) (2,1,0) (3,1,0) (4,1,0) (5,1,0) (0,1,1) * * (0,1,2) (0,1,3) (0,1,4) (1,1,1) (2,1,2) (3,1,3) (4,1,4) *Lowest Value of AIC and SIC SIC 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: ŷ= + AR (n) + β MA (n) + We are using MA(1) and AR(1) models based on selected criteria s B y t = MA (1) The ARIMA(1,1,0) model is: D (GP) = AR (1) 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 ARIMA(0,1,1) ARIMA(1,1,0) RMSE MAE MAPE 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 1 st Feburary,2012 to 1 st 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 Forecast within 2 St. Error 2/06/2012 2/13/2012 2/20/2012 2/27/2012 GPF2 Figure 4.3 Forecast: GPF2 Actual: GP Forecast sample: 2/01/2012 3/01/2012 Included observations: 22 Root Mean Squared Error Mean Absolute Error Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion

6 Date Real Data Table 4.7 ARIMA(0,1,1) Resulted Data Error 1-Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Mar Date Real Data Table 4.8 ARIMA(1,1,0) Resulted Data Error 1-Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Mar Forecast Within 2 St. Error(110) 2/06/2012 2/13/2012 2/20/2012 2/27/2012 GPF Figure 4.4 Forecast: GPF Actual: GP Forecast sample: 2/01/2012 3/01/2012 Included observations: 22 Root Mean Squared Error Mean Absolute Error Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion 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 $ However, in the last; (22 nd value) the error is $ 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 ( Figure 4.5 shows the comparison for real and forecasted values for ARIMA(0, 1, 1) 667

7 V. CONCLUSION In order to develop a univariate Time Series Model, we used London Fix gold prices from January 02, 2003 till March 01, 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. After the test of stationary, we conclude that the data is 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. Acknowledgement I would like to acknowledge and thank Mr. Muhammad Haseeb (Faculty of IoBM) for his help and guidance during this research. I am also thankful to Mr. Muhammad Farhan Ali Khan, who helped me during the preparation of this paper. REFERENCE [1] Cooray T.M. J.A, (2008), 'Applied Time Series, Analysis and Forecasting' Narosa Publishing House India. [2] DeLurgio, Stephen A. (1993), 'Forecasting Principles and Application', University of Missori USA [3] Frank Ahrans, September 24, 2009 "Economy Watch- The Daily Washington Post" [4] Gaynor, Patricia E and Kirkpatrick Rickey C, (1994) 'Introduction to Time Series Modeling and Forecasting in Business and Economics' Mc Graw Hill International USA [5] Govelt, M.H., Govelt GJS (1982), 'Gold Demand and Supply', Resources Policy [6] Greene, William H, (2003) Econometrics Analysis Fifth Edition Pearson Education Inc USA [7] Gujrati, Damodar N, (2004) Basic Econometrics Fourth Edition [8] Ismail, Z., Yahya A. And Shabri A.. (2009) ' Forecasting Gold Prices Using Multiple Linear Regression Method'. American Journal of Applied Sciences 6: [9] Kanfmann, T.D. and Winters, R.a., (1989), 'The Price of Gold; A Simple Model'A Resources Policy 15, [10] Kennedy Peter, (2008) ' A Guide to Econometrics' 6th Edition, Blackwell Publishing USA [11] Lineesh, M.C, Minu, K.K. and Jessy John, e. (2010), "Analysis of Nonstationary Nonlinear Economic Time Series of Gold Price A Comparative Study." International Mathematical Forum, 5, [12] Newaz, M.K. (2008), 'Comparing the Performance of Time Series Models for Forecasting Exchange Rate' BRAC University Journal, Vol V, No [13] Said Nasser and Scacciavillant, Fabio, (2010), 'The Case for Gold as Reserve Asset: Gold, the GCC and Khaliji", Dubai International Financial Center [14] Selvanathan, E.A, (1991) 'A Note on the Accuracy of Business Economists' Gold Price Forecast'. Australian Journal of Managements. 16 [15] Selvanathans S. Selvanathan, E.A. (1999). 'The Effect of the Price of Gold on its Production; A Time Series Analysis', Resource Policy 25, [16] Shafiee Shahriar and Topal Erkan, (2010), "An Overview of Global Gold Market and Gold Price Forecasting' Resources Policy, Journal 35, [17] WGC, 2012, World Gold Council Publications Achieves, : World Gold Council [18] Lin, Jing, Empirical Study of Gold Price Based on ARIMA and GARCH Models [19] London Fix Historical gold- result [20] Khaemasunun Pravit, 'Forecasting Thai Gold Prices', Australian Journal of Management, Vol. 16, No.1 [21] of forecasting [22] 668

8 Appendix 01 Correlogram of 2 nd Difference of Gold price Appendix 01 ADF for 2 nd Difference gold price series 669

9 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 Appendix 02 ARIMA(0,1,1) Variable Coefficient Std. Error t-statistic Prob. C MA(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted MA Roots.09 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 Variable Coefficient Std. Error t-statistic Prob. C AR(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots