A Comparative Study Between Holt's Double Exponential Smoothing and Fuzzy Time Series Markov Chain in Gold Price Forecasting

Gold price is important to a country’s economy as it can be used as a hedge against inflation especially during financial turmoil. Besides, the gold price also has an impact on the stock market price. As an investor, to make a good investment plan, information regarding the fluctuation price of gold is necessary to minimize the risk. Therefore, this study proposes to compare two of the forecasting models, namely Holt's Double Exponential Smoothing and Fuzzy Time Series Markov Chain to forecast the price of gold. Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) are used to determine a better forecasting model with smaller error. Initially, the data price of gold is analysed by using Durbin Watson Test to check the suitability of the data for time series analysis. The finding of this study shows that Fuzzy Time Series Markov Chain is more accurate in predicting gold price as compared to Holt’s Double Exponential Smoothing because it produces smaller values of RMSE and MAPE.


INTRODUCTION
Gold has been used as money for exchange purposes. Therefore, each piece of gold has its value in the monetary system. However, the value of gold itself has dominant power over the economy as a tool in the monetary system. In the nineteenth and twentieth centuries, gold acted as a cash guarantee for issue for banknotes (Uzun & Kirai, 2017). Later, in Malaysia, many people buy gold either for savings or long-term investment instruments because the price of gold itself will either increase or decrease over time. Thus, buying gold is not just for accessory but also as a medium to gain some income. Most people who know how good gold can be in terms of investment will buy a gold bar because the purity level of the gold can affect the selling value of the gold. The purity of the gold is measured by karat, which is pure gold with 24 karats.
Two main factors contribute to determine the price of gold: which are demand and supply. These factors control the fluctuation of gold price. Other factors that influence the change in the price of gold are inflation rates and currency variation. The fear and uncertainty in the global economy will affect the swing of gold price, turning it to be the most attractive asset for all investors. In other words, the price of gold is the mirror of the world economic situation (Ghalayini & Farhat, 2020).
A good forecasting model will produce better forecasting output with minimal error. It may assist the investors to make better decision making. For example, the forecasting of return on gold price for a shorter period of time can help provide valuable information to an investor about the movement of gold price for short and long-term buying and selling strategies. The volatility of gold prices can be predicted more precisely, and it is beneficial for commodity markets and the global economy (Uzun & Kiral, 2017).
Gold price plays an important role in country's economies as it is used as hedging tool against inflation. It is a type of asset that is negatively correlated with another asset or portfolio. Adding a certain percentage of gold in investment portfolio may assist in decreasing the level of risk during financial turmoil since it is not affected by Consumer Price Index (Shakil et al., 2018). In a study of S&P500 stock market index (GSPC), the gold price has the highest impact on the stock market price in long-run and short-run, compared to other variables such as oil price. As an implication, investors should react against changes in the gold price (Gokmenoglu & Fazlollahi, 2015). Due to that reason, the prediction of gold price accurately is important for investors, portfolio managers and policy makers. The prospective investors should consider gold in their portfolios as a store of value and a diversification tool and cautious of the price fluctuation (Chaku et al., 2022).
Many studies have been conducted to forecast gold price and various forecasting models have been used to find the best model. In a study done by Taufik (2020) (Chukwudike et al., 2020). This method has been successfully fitted to the data series chosen. Among those seventeen ANN structures that have been suggested, ANN (2-6-1) was the best structure since it has the least error in MSE and MAE.
This study is only designed to compare between Holt's Double Exponential Smoothing model and Fuzzy Time Series Markov Chain Model in predicting price of gold. To select a better forecast model with least error, MAPE and RMSE are used to analyse the forecast outputs. Initially, the historical data of gold price is analysed using Durbin Watson Test in order to check the suitability of the data for time series analysis. Although this study uses the same method as done by Taufik et al. (2020) but the data used are different in terms of time setting and location. They use daily price of gold in Indonesia from May to July 2020 while this study used monthly price of gold in Malaysia from January 2016 to December 2020. The result cannot be assumed the same. Therefore, this study aims to determine suitable method for gold price prediction in Malaysia.

METHODOLOGY
The models used in this study are Fuzzy Time Series Markov Chain and Holt's Double Exponential Smoothing. The model formulation and calculation for both models' solutions will be discussed throughout this study. Both results will be evaluated using Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) to determine a better model. In this study, monthly data of gold prices from January 2016 until December 2020 are collected from Index Mundi website page.

Method of Data Analysis
Before doing further analysis, the collected data must be checked whether there is any missing data. After that, the data should be tested to see the compatibility of the time series analysis applied to the price of gold monthly data. For that reason, the Durbin-Watson Test must be applied using the formula below: where ( ) 1 first-order autocorrelation of the data p = . If the value of the Durbin Watson is between 0 and 1.5, it shows that the data is considered dependent on time and suitable for time series analysis. The data will then be used to create a model using the Fuzzy Time Series Markov Chain and Holt's Double Exponential Smoothing.

Fuzzy Time Series Markov Chain
Markov Chain method is different from the classic Fuzzy Time Series, and this method has a transition metrics statistic concept in the forecasting calculation. The formula used in this method is shown in equation (2)  D and 2 D are any two positive integers to enable the intervals of data are divided appropriately and evenly.
Then, the universe of discourse U will be divided into equal intervals and specify its length as illustrate in equation (3) below.
where = l the length of an interval = n the number of intervals Next step, for each interval, the midpoint will be calculated and labelled as 1 2 3 ,, u u u until n u .
The fuzzy sets 1 A until n A are defined as represented in equation (4) Then, each data of gold price, t x will be fuzzified into a related fuzzy set.

Next, Fuzzy logic relationship (FLR) is defined as
are the relationship between two consecutive observations. From FLR, Fuzzy Logical Relationship Group (FLRG) is formed. Then, Markov Transition Probability Matrix with the size of nn  is obtained by using equation (5).  If 0 ij P  , the is accessible from state . Hence, we will defuzzify the forecast result from the probability matrix. Finally, the tendency of the forecasting result needs to be adjusted and the calculation of the final output can be performed.

Holt's Double Exponential Smoothing
The equations for Holt's Double Exponential Smoothing are: Equation (6) is the exponentially smoothed series t S in which  and  are the parameters to be determined with values from 0 to 1. It is used to calculate the successive exponential smoothed value used in the trend estimate equation. Next, equation (7) is the trend estimate equation which will be calculated by taking the difference between two successive exponential smoothed value, 1 tt SS − − . The calculation result produces an estimate for a trend in the data.
Furthermore, both equation (6) and (7) will be used to forecast using equation (8). The trend estimate will be multiplied by the number of periods to be forecast, m and the results will then be added to the exponentially smoothed series, t S to eliminate the randomness from the data. Finally, the forecasting results are generated for the model evaluation to compare both methods.

Model Evaluation Methods
The evaluation of the result from each method is done using RMSE and MAPE, given by equation (9) and (10) below: The smaller value of RMSE indicates that the model is better. As for MAPE, if the result is less than 10 per cent, it is of excellent significance level. If the result is 10 percent until 20 percent, it is of good significance level. Next, if the result is 20 percent until 50 percent, it is moderate significance level. While for low significance level, the result is greater than 50 percent.

FINDINGS AND DISCUSSIONS
The data are analysed using Holt's Double Exponential Smoothing and Fuzzy Time Series Markov Chain. Those two models were tested using Durbin Watson Test in equation (1)  Then, the fuzzy set of seven intervals are generated as shown in Table 1.  Table 2 shows that each historical data of gold price is fuzzified and fuzzy logical relation.
Using Equation (5), the Markov Transition Probability Matrix with the size of 7 7 is obtained, shown in Table 5. Finally, the forecast value is calculated and the result is represented in Table 6.  (6). Then, trend estimate equation is used by taking the difference between two successive exponential smoothed value and the result produces an estimate for a trend in the data. We used Microsoft Excel with Solver for this method to find the alpha and beta values with the minimum RMSE and MAPE.
As a result, the value of  is 1, and the value of  is 0.05235. The forecasting values in Table 7 below shows the result based on those α and β.   Smoothing as well as other non-statistical model such as ANN. In order to choose the most accurate method, many accuracy tests can be applied other than RMSE and MAPE, such as MAE and BIC. Besides, this study could be extended for broader topic such as analysis of the factors that affect the price of gold so that different perspectives and ideas can be discussed through those topics.