Forecasting Gold Price. A Comparative Study

Transcription

1 Course of Financial Econometrics FIRM Forecasting Gold Price A Comparative Study Alessio Azzutti, University of Florence Abstract: This paper seeks to evaluate the appropriateness of a variety of existing forecasting techniques (6 methods) at providing accurate forecasts for gold price. Special consideration is given to the ability of these techniques at providing forecasts, which outperforms the random walk, used as a benchmark in the comparison. Interestingly, the results shows that only the ARIMA model is able to outperform the random walk at every horizons and on average the ARIMA model is seen providing the best forecasts in terms of the lowest root mean squared error over the 36 month forecasting horizons. Moreover, other four precious metals (silver, platinum, palladium and rhodium) data were used to run a multiplicative regression model; but, again, the optimal ARIMA reported finer results. 1 Introduction Gold serves several function in the world economy, and its link with financial and macroeconomic variables are well-established (Pierdzioch et al., 2014). It has a monetary value and it is sought after by central banks to be part of their international reserves, which fulfil many purposes (Gupta et al., 2014). It has industrial uses and it can be transformed into jewellery. In modern finance, it is used as a hedge against inflation and a safe haven during crises. In fact, unlike stocks and bonds, the gold price has been constantly thought as the less risky asset. Gold has also other many distinguished features. Its supply is accumulated over the years and its global physical production can be as small as 2% of total supply, thereby in contrast to other commodities its annual production may not sway its price as other factors do. It is noticeable that given the significance of gold in the modern world, the ability to provide accurate forecasts into the future price of gold will be of primary importance. Moreover, there are benefits from finding the right model that forecasts the gold price more accurately than others do. Out-of-sample forecasting offers informational availability advantage for monetary and policymakers, hedge fund managers and international portfolio managers which can be used in gauging future inflation, estimating demand for jewellery, discerning investment in precious metals and other commodities and assessing the future movement of the dollar exchange rate. The figure below shows the time series for gold price from January 1970 to August In general, it has an increasing trend for the whole period and it seems to portray an exponential growth over the last 15 years. A first look at the figure shows signs of two major shocks post 1980 and 2010, which create structural breaks in the time series.

2 The aim of this paper is to evaluate the use of a variety of forecasting models representing both parametric and nonparametric techniques for obtaining accurate forecasts for the price of gold. Whilst there exists various metrics, which are used for comparing between two different out-ofsample forecasts, the paper relies on the Root Mean Squared Error (RMSE). Although the RMSE criterion is in the process of gaining its popularity, it is necessary to briefly describe this measure at the outset so that the reader has a clear understanding of the results reported in this paper. In the very recent past, the RMSE criterion has been adopted as a popular measure in a range of forecasting studies (e.g., Altavilla and De Grauwe, 2010; Hassani et al.,2009, 2013; Beneki and Silva, 2013). The RMSE can be computed as follows: RMSE = n (ŷ T+h,i y T+h,i ) 2 i=1 n where, ŷ T+h is the h-step ahead forecast obtained by the model used to fit data, y t is the actual values and N is the number of the forecasts. This paper shows the results obtained by the application of several different forecasting techniques over 36 horizons from 1 month ahead until 36 months ahead. This time horizon enables to capture both short and medium run effectiveness of a given forecasting model at accurately predicting the future price of gold. The models evaluated in the study include an Autoregressive integrated moving average (ARIMA), an Autoregressive fractionally integrated moving average (ARFIMA), an Exponential smoothing (ETS), an Exponential smoothing state space model with Box-Cox transformation, ARMA errors, Trend and Seasonal components (TBATS) and a Multiple linear regression (MLR) with other four precious metals monthly prices as explanatory variables. Thereafter each out-of-sample forecasting result will be compared along with a Random Walk (RW) model. The results from the six competitive models, selected based on the average lowest Root Mean Squared Error (RMSE) are reported in this paper. Note that unlike most of the existing literature on forecasting the price of gold, which analyses the role of financial and macroeconomic variable in predicting gold price, this study primarily concentrates on univariate approaches. In fact, the univariate approach relieves the problem of,

3 choosing macroeconomic and financial variables that defines the state of the world economy, given that gold is a globally traded asset. Thereafter, a multiple regression model, including prices of four (silver, platinum, palladium and rhodium) other precious metals, is used and it is compared with univariate approaches. Finally, it is worth noting that using the above described approaches, which can handle non-stationarity of the data and hence the price of gold is forecasted and not the gold returns as in done in the literature (Shafiee and Topal, 2010) 1. The remainder of the work is organized as follows. Section 2 describes the methodology underlying the various forecasting techniques whilst Section 3 is dedicated to an analysis of the data. Section 4 reports the empirical results and the paper concludes in Section 5. 2 Methodology 2.1 Forecasting Models Random Walk The random walk model is used as a benchmark, as it is a widely accepted practice that a forecasting technique which is recommended for a particular forecast should at least be more accurate than a random walk. In brief, today s value for gold is forecasted to be tomorrow s value for gold. And if the series being fitted by a random walk has an average trend that is expected to continue in the future, a so called drift might be taken into account. Autoregressive Integrated Moving Average (ARIMA) ARIMA models are the most general class of models for forecasting a time series, which can be made to be «stationary» by differencing, perhaps in conjunction with logging or deflating if necessary. The random variable is viewed as a combination of signal and noise. An ARIMA model can be thought as a filter that tries to separate the signal from the noise, and the signal is then extrapolated into the future to obtain forecasts. The ARIMA forecasting equation for a stationary time series is a linear equation in which the predictors consist of lags of the dependent variable and/or lags of the forecast errors. A nonseasonal ARIMA model is classified as an ARIMA(p,d,q) model, where: p is the number of autoregressive terms, d is the number of nonseasonal differences, q is the number of lagged forecast errors in the prediction equation. 1 Shalfiee and Topal (2010) address the issue of non-stationary gold prices by proposing a model that has three components: along term trend reversion component, a diffusion component and a jump or dip component.

4 The optimal ARIMA model, which is referred to as automatic-arima, is provided through the forecast package for R. A more detailed description of the algorithm underlying automatic-arima can be found in Hyndman and Khandakar (2008). The general Box-Jenkins model for y is written as: y = Øy t 1 + Øy t Øy t p + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q, where Ø and θ are unknown parameters and the ɛ are i.i.d. normal errors with zero mean. Once the number of differences (d) has been determined through the unit root test, the algorithm that minimises the Akaike Information Criterion (AIC) is used in order to determine the value of p and q. The following is the formula to be minimised: AIC = log(l) + 2(p + q + P + Q + k), where k = 1 if c 0 and 0 otherwise and L represents the maximum likelihood of the fitted model. Autoregressive Fractionalized Integrated Moving Average (ARIMA) The study relies on the ARFIMA modelling process provided through the forecast package in R. Once again, the modelling algorithm automatically estimates and selects the p and q for an ARFIMA (p,d,q) model based on the Hyndman and Khandakar (2008) algorithm whilst the d and parameters are selected based on the Haslett and Raftery (1989) algorithm. Exponential Smoothing (ETS) The ETS technique incorporate the foundations of exponential smoothing and is made available through the forecast package for the R software. ETS overcomes a limitation found in earlier exponential smoothing models which did not provide a method for easy calculation of prediction intervals (Makridakis, Wheelwright and Hyndman, 1998). The ETS model from the forecast package consider the error, trend and seasonal components along with over 30 possible options for choosing the best exponential smoothing model via optimization of initial values and parameters using the MLE and selecting the best model based on the AIC criterion. A detailed description of ETS can be found in Hyndman and Athanasopoulos (2013). Figure 2 summarises in table format the several ETS formula s that are evaluated in the forecast package to select the best model to fit the data. Note that ((Hyndman and Athanasopoulos, 2013): ell t denotes the series level at time t, b t denotes the slope, s t denotes the seasonal component of the series, M denotes the number of season in a year, α, β, γ and ϕ are smoothing parameters, φ h = φ + φ φ h and h m + = [(h 1)modm] + 1.

5 Figure 2: Formulae for recursive calculations and points forecast (Hyndman and Athanasopoulos, 2013). Exponential smoothing state space model with Box-Cox transformation, ARMA errors, Trend and Seasonal components (TBATS) The TBATS model is an exponential smoothing state space model with Box-Cox transformation, ARMA errors, Trends and Seasonal components. The result is a technique, which is aimed at providing accurate forecasts for time series with complex seasonality. A detailed description of TBATS model can be found in De Livera et a. (2011) and is therefore not reproduced here. Even if the idea of non-seasonality of gold prices is definitely common, seasonal models are used to verify this characteristic of gold price time series. Multiple linear regression (MLR) The general form of a multiple linear regression is y i = β 0 + β 1 x 1,i + β 2 x 2,i + + β k x k,i + e i, where y i is the variable to be forecast and x 1,i,, x k,i are the k predictor variables. Each of the predictor variables must be numerical. The coefficients β 1,, β k measure the effect of each predictor after taking account of the effect of all other predictors in the model. Thus, the coefficient measure the marginal effects of the predictor variables. As for simple linear regression, when forecasting we require the following assumption for the errors (e i,, e N ): The errors have mean zero; The errors are uncorrelated with each other;

6 The errors are uncorrelated with each predictor x j,i. As mentioned before, tomorrow s gold prices is forecast through the information available of today s other four precious metals prices (silver, platinum, palladium and rhodium). Thus, the general formula of the multiple regression under analysis will be in the following form: GOLD t = β 0 + β S SILVER t 1 + β PL PLATINUM t 1 + β PA PALLADIUM t 1 + β R RHODIUM t 1 + e t. 3 Data The data used in this study relates to the prices of gold, silver, platinum, palladium and rhodium. The price of gold is determined through trading in the gold and its derivatives markets, however a procedure known as the Gold Fixing in London which sprang from September 1919, provides a daily benchmark price to the industry. The afternoon fixing was introduced only in 1968 to provide a price when US markets are open which is caused by time differences. The other precious metals use the similar pricing model. A monthly adjusted close price of gold from August 1992 to September 2014 (266 observed prices) is used for the analysis. The data are freely available at It was possible to gather data for all the five metals only starting from August 1992, whereas the endpoint of the sample is purely driven by data availability at the time of conducting this study. The analysis evaluates out-sample forecast for horizons of h = 1 step, up to h = 36 steps ahead, and thereby enables capturing and evaluating both short and long run forecasting abilities of the given forecasting models. The table below presents some descriptive statistics of data as a help to understand the structure of the data and the time series. Table 1: Descriptive statistics for metals (Aug 1992 Sep 2014) Series Mean Median SD CV Skewness S-W(p) ADF Gold <0.01* -0.20ꜝ Silver <0.01* -1.36ꜝ Platinum <0.01* -1.40ꜝ Palladium <0.01* -1.09ꜝ Rhodium <0.01* -2.64ꜝ Note: * indicates data is not normally distributed based on a Shapiro-Wilk test at p=0.01. ꜝ indicated a nonstationary time series based on the Augmented Dickey-Fuller test at p=0.01. As the main interest of this paper is the price of gold, hereafter the discussion will focus only describing this data. The first observation is that based on the Shapiro-Wilk Test, gold is not normally distributed with 99% confidence level. Then a first look at Figure 3 suggests that gold time series has unit root problems, and this has been confirmed by the Augmented Dickey-Fuller test. The coefficient of variation (CV) statistic is reported in order to enable comparing the variation in gold during the period with other metals.

7 Figure 3 It is observable an initial flat trend until 2000, then the price of gold starts to grow faster and faster reaching a peak on 2011, and finally gold price dropped of more than $300 dollars/oz during the last 4 years. What is worth to notice by looking at the correlograms of gold price is that there are high ACFs and PACFs. The gold prices autocorrelations are very significant until lag 36 and they show a decreasing behaviour, and for what concerns the partial-autocorrelations, there are quite significant spikes at different lags. Figure 4 By taking logarithm of the data, the variance become more stable but data has still an increasing trend. As a consequence, the first difference of logarithms is taken. The differenced logarithm series (called returns in finance) shown in Figure 5 looks definitely more stationary than the original one. Figure 5

8 Here there are reported the statistics for the gold return series and the histogram of the series plotted against a normal distribution: Table 2 Mean Median SD Skewness S-W(p) ADF * ꜝ Note: * indicates data is not normally distributed based on a Shapiro-Wilk test at p=0.01. ꜝ indicated this time a stationary time series based on the Augmented Dickey-Fuller test at p=0.01. Figure 6 As expected, both the Shapiro-Wilk test and the histogram confirm that data are not normally distributed. In addition, by performing the Augmented Dickey-Fuller unit-root test on the monthly returns, the ADF test statistic is for lag order 1 and a p-value smaller than 0.01 is recorded. With stationary as the alternative hypothesis, the null hypothesis that there is a unit-root in the series is rejected. Thus, the return series is stationary. The graphics below show the ACFs and PACFs for the return series, which are definitely pattern less and statistically not significant. Figure 7 It was a duty the study of the logarithm return series, because of some of the models that will be processed needs to deal with such features of data. For example, it will be the case of the ARIMA model.

9 4 Empirical results 4.1 Fitting the models In order to develop the above-described models for the gold price, the available dataset, which consist of 266 input vectors and their corresponding output vectors from the historical data of gold price, was separated into training and test sets. For achieving the aim, 230 observations (from August 1992 to September 2011) are first applied to formulate the model and the last 36 observations (from October 2011 to September 2014) are used to reflect the performance of the different constructed models. For each model, the procedure consists of computing one-step ahead forecast and then refit the model on a window training set of the same length as the one of the initial fit. A technique called rolling forecast. Random Walk To forecast from a simple Random Walk model the function rwf from the forecast package in R software is used. A non-zero constant term was included into the model, due to the fact that gold price have shown an average upward trend that might continue into the. Figure 8 shows the 36 forecast values compared to the realized gold prices. Figure 8 ARIMA model For what concern the ARIMA model, R software is used for estimating the coefficient 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), the best model to forecast gold prices is ARIMA(0,1,1) with drift since it contains the least AIC ratio and the third lowest BIC ratio.

10 Table 3 ARIMA (p,d,q) AIC BIC ARIMA(0,1,0) with drift (3) (2) ARIMA(1,1,0) with drift (2) (4) ARIMA(0,1,1) with drift (1) (3) ARIMA(1,1,1) with drift (4) (5) ARIMA (0,1,0) (5) (1) The resulting coefficient of the selected ARIMA model fitted to the initial window of the training set are: Table 4 MA1 Drift An ARIMA(0,1,1) corresponds in practice to a MA(1). This is only the initial selection of the best fitting ARIMA model to the data; in fact, at any iteration of the algorithm, R will re-estimate the model before to predict the consequent forecast value, given d = 1 so that the gold monthly returns will be modelled. Figure 9 illustrates the 36 resulting forecast compared to the actual gold prices. Figure 9 ARFIMA model To fit an ARFIMA model to data, it can be used the R function fracdiff from the homonymous package. The coefficient of the ARFIMA model selected by the software and applied to the initial training set are: Table 5 Coefficient d Sigma Log likelihood

11 Figure 10 shows the resulting plot of forecast values against the actuals clearly shows the inaccuracy of this model especially until May It mostly seems to underestimate the price of gold. Figure 10 ETS model In order to fit an ETS model to our data, it needed the application of the ets function from the forecast package in R. Figure 11 indicates that no-seasonal effect was encountered. As expected, gold price has not any seasonal component; this is mainly due to the special features of the metal. The resulting smoothing parameters and sigma of the selected ETS model fitted to the initial window of the training set are: Table 6 Parameters Value Alpha Beta Sigma Figure 11

12 Based on the AIC criterion, R software selected an Exponential Soothing model with multiplicative errors and trend, but no seasonal effect. Next figure shows the 36 predictions based on the rolling forecast: Figure 12 TBATS model A TBATS model is fitted to the data through the R function tbats from the forecast package. Also Figure 13 confirms that there is no-seasonal effect on gold data, in fact the level plot does not show any seasonal adjustment in respect to the observed values. Table 7 Parameters Value Lambda Alpha Beta Damping 1 Sigma Figure 13

13 Figure 14 shows the forecast observations compared to the actual 36 gold prices, once a TBATS model is fitted to the windows of data. Figure 14 Multiple Linear Regression As anticipated in the introduction at this paper, the aim of the usage of this model is to forecast tomorrow s gold price based on today s silver, platinum, palladium and rhodium prices. The general formula of the multiple regression model is as follow: GOLD t = β 0 + β S SILVER t 1 + β PL PLATINUM t 1 + β PA PALLADIUM t 1 + β R RHODIUM t 1 Data for the four explanatory variables from August 1992 to September 2014 were used to fit the model and get predictions. It is worth to remember that there is not a unique estimated linear relationship between gold and the other four metals. In fact, as already done for the previous models, an iterated estimation was conducted. At any iteration, once the coefficients for each predictor are estimated, only the 1-period ahead actual prices of the 4 metals is needed to get the prediction value of gold price. Figure 15 illustrates the resulting forecast values of gold price based on the iterated MLR model.

14 Figure Comparing the results Once all the 36 predictions of golf prices from October 2011 to September 2014 are obtained from the six models of this study, all that remains is to compare these results. Figure 16 gives a first graphical comparison of the forecast values under different techniques. Figure 16 Since it is quite an hard task to compare the 6 different models by only looking at the their forecast plot, Table 8 and Figure 17 report the absolute errors for each forecast period and the mean absolute errors (MAE) that is an average of the formers. The Score item indicates how many time a particular model was able to outperform among others.

15 Table 8 Time RW.Drift ARIMA ETS TBATS ARFIMA MLR ,91 65,99 33,99 35,01 315,11 346, ,96 22,08 1,11 0,28 296,80 219, ,54 224,15 247,60 246,75 56,95 96, ,79 168,65 139,20 140,99 351,81 449, ,34 45,69 27,65 32,11 299,10 212, ,52 111,98 131,80 129,03 160,94 48, ,08 43,97 69,68 68,15 182,43 102, ,73 113,08 134,51 133,08 96,36 73, ,39 9,11 13,16 9,32 178,48 256, ,39 8,41 0,01 3,05 192,25 293, ,16 19,21 5,94 8,77 206,57 273, ,31 116,33 107,88 110,97 318,11 289, ,17 45,15 56,11 52,26 199,48 29, ,87 13,61 29,96 27,33 212,74 146, ,15 90,30 98,54 95,81 136,68 0, ,62 19,39 36,63 32,49 169,23 177, ,80 90,38 103,63 100,29 92,90 24, ,24 16,83 29,89 18,69 136,44 120, ,28 142,36 150,88 146,80 9,79 3, ,05 107,88 117,88 112,17 3,11 102, ,50 227,40 231,98 214,56 155,17 44, ,64 88,88 95,31 99,82 83,28 239, ,23 86,79 107,02 105,76 134,60 234, ,13 62,45 55,24 55,62 30,12 6, ,67 19,38 16,28 9,89 53,44 111, ,01 81,06 76,15 72,39 10,44 1, ,99 67,37 55,68 58,12 21,26 57, ,46 30,67 45,33 38,73 59,53 117, ,02 73,70 93,20 80,82 123,83 184, ,09 30,69 12,84 23,95 54,23 77, ,22 14,85 2,07 6,35 59,88 120, ,71 46,95 34,24 38,85 24,39 94, ,12 51,06 62,80 59,40 107,85 143, ,83 28,81 13,50 21,60 52,39 44, ,25 10,73 1,21 2,45 61,74 62, ,56 28,26 16,00 20,08 42,96 102,70 MAE 66,47 67,32 68,19 66,99 130,29 136,42 Score

16 Figure 17 It is observable how much close are the MAEs for the RW with Drift, ARIMA, ETS, TBATS models, whereas ARFIMA and MLR resulted with a more than double measure. According to the absolute error criterion, ETS surprisingly outperformed other models on 10 over 36 forecast periods. In addition, the simple RW made a good job on fitting and predicting future gold prices, outperforming the others 8 times. Although the MLR reported to be the worst model according to the MAE criterion, it was able to beat its competitive models 9 times. Table 9, instead, reports the RMSE for out-of-sample forecasting results. The analysis continues by comparing between the RMSE values. First and foremost, it is pertinent to point out that no single model is able to provide the best forecast for the gold price at all horizons. However, based on the lowest average RMSE, we can conclude that the auto-selected optimal ARIMA model is best for forecasting the gold price. Table 9 Time RW+Drift ARIMA ETS TBATS ARFIMA MLR ,92 65,99 33,99 35,01 315,11 346, ,84 49,20 24,04 24,76 306,09 290, ,64 135,51 144,29 143,89 252,08 243, ,14 144,51 143,04 143,17 280,36 307, ,47 130,86 128,53 128,86 284,20 291, ,05 127,90 129,08 128,89 267,63 266, ,03 119,58 122,38 122,07 257,19 249, ,25 118,78 123,96 123,50 242,98 235, ,46 112,03 116,95 116,48 236,68 237, ,41 106,32 110,95 110,51 232,62 243, ,24 101,53 105,80 105,40 230,38 246,76

17 ,02 102,85 105,98 105,87 238,92 250, ,49 99,60 103,00 102,75 236,12 240, ,62 96,05 99,58 99,28 234,53 235, ,28 95,68 99,51 99,05 229,31 227, ,07 92,77 96,78 96,25 226,02 224, ,44 92,63 97,20 96,49 220,43 218, ,70 90,10 94,72 93,88 216,62 213, ,55 93,59 98,48 97,38 210,85 208, ,88 94,35 99,54 98,18 205,51 204, ,11 104,60 109,54 106,64 203,40 199, ,66 103,93 108,93 106,34 199,51 201, ,46 103,25 108,85 106,31 197,14 202, ,44 101,88 107,15 104,69 193,08 198, ,37 99,89 105,04 102,60 189,49 195, ,57 99,24 104,08 101,60 185,82 192, ,25 98,24 102,69 100,33 182,39 188, ,76 96,64 101,20 98,79 179,46 186, ,92 95,94 100,94 98,22 177,83 186, ,60 94,50 99,27 96,67 175,12 184, ,06 93,00 97,66 95,11 172,61 182, ,92 91,91 96,31 93,86 169,94 180, ,05 90,94 95,47 93,00 168,40 179, ,91 89,73 94,08 91,70 166,15 176, ,61 88,46 92,73 90,38 164,09 174, ,47 87,35 91,47 89,18 161,95 173,04 Average RMSE 105,46 100,26 102,59 101,31 216,94 221,82 Score Average RRMSE 2 0,95-0,98 0,99 0,46 0,45 Interestingly, at horizon of h = 1, 2, 4 and 5, the ETS model is seen outperforming all other models, despite it is more suitable for time series with a seasonal component. The TBATS model provides the second best forecast in terms of lowest RMSE at horizons of 1, 2, from 3 to 13 and from 15 to 19, and the RW model provides the second best forecast at horizon 3 and at 20 steps ahead. According to RMSE criterion, the ARFIMA and the MLR models cannot be considered good predicting models, because their forecasts are very imprecise especially in the first horizons. However, the iterative optimal ARIMA models gain the top position in terms of providing the best forecasts for a large majority of the forecasting horizons with a score of 32 out of 36, where score indicated the number of times that a model reports the lowest RMSE (in comparison to the others) over the 36 forecasting horizons. 2 It represents the Ratio of the RMSE between Model A and Model B, if the resulting ratio is less than 1, then Model A outperforms Model B by 1 Model A Model B percent. In this analysis, Model A corresponds to the optimal ARIMA.

18 5 Conclusions This study evaluated the use of 6 different parametric and nonparametric time series analysis and forecasting techniques. Using monthly gold price data 36 different forecasting horizons, which covers both the long and the short run, were considered. The text reports and compares the results obtained from the 6 forecasting models. Firstly, it is interesting to observe that none of the 6 models are able to provide the most accurate forecast of gold price in both the short and long run. Secondly, the optimal ARIMA model seemed to outperform RW, ARFIMA, ETS, TBATS and MLR models. Nevertheless, based on the RMSE criterion, it is possible to conclude that should one be interested in relying on a single model that can provide the most accurate forecast for gold price across 36 horizons, then ARIMA tops the list of contenders in comparison to 6 models evaluated in this study owing to its lowest RMSE. Finally, based on the RRMSE criterion the ARIMA model provides out-of-sample forecast for gold which are 5%, 2%, 1%, 54% and 55% better than RW, ETS, TBATS, ARFIMA and MLR model in terms of the RMSE.