**FORECASTING KIJANG EMAS (GOLD) PRICES IN MALAYSIA: A TIME SERIES APPROACH USING ARIMA AND ARFIMA MODELS**

\*\*Double blind review, please do not include authors information in this version \*\*

Received Date: \*date

Accepted Date: \*date

Published Date: \*date

**HIGHLIGHTS**

  - Forecasting is done to better understand about the pattern of Kijang Emas prices in Malaysia.

  - ARIMA and ARFIMA model are used to forecast Kijang Emas prices in Malaysia.

  - The forecast accuracy checking was measured using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Mean Absolute Percentage Error (MAPE)

ABSTRACT

*Gold is known as the most valuable commodity in the world because it is a universal currency recognized by every single bank across the globe.* *Thus, many people were interested in investing in gold since gold market is always steadier compared to other investments. However, the credibility of gold was questionable due to the changes in gold price caused by variety of circumstance. Hence, information on gold price inflation were needed to understand the trend in order to plan future economies in accordance with international gold price standards. The aim of this study was* *to identify the trend of Kijang Emas monthly average prices in Malaysia from 2010 to 2021, to determine the best fit time series model for Kijang Emas prices in Malaysia and using univariate time series models to forecast Kijang Emas prices in Malaysia. The ARIMA and ARFIMA models were used in this study to forecast price of gold in Malaysia that were known as Kijang Emas. Each of the actual monthly Kijang Emas prices for 2021 were found within 95% predicted interval of both ARIMA and ARFIMA models. The performances for each model were checked by considering the values of MAE, RMSE and MAPE. From the findings, all the MAE, RMSE and MAPE values shows that ARFIMA model emerged as the best model in forecasting the Kijang Emas prices in Malaysia compared to the ARIMA model.*

*Keywords: Kijang Emas Price, Time Series Modelling, ARIMA model, ARFIMA model*

# INTRODUCTION 

In the commodity market, gold is a precious metal and also known as stock contract goods. According to World Gold Council (2019), gold has been described as a highly liquid but limited commodity that can be bought as an investment as well as a luxury item like jewelleries. As a result, gold can provide long-term returns. During the global economic crisis of 2008, a wide range of financial goods performed poorly except for gold market (Yang, 2019). During the disease outbreak of Coronavirus Disease 2019 (COVID-19), the increasing number of COVID-19 cases globally has positively impacted gold prices because the uncertainty about upcoming economic situations created due to the pandemic (Yousef and Shehadeh, 2020). Based on data from Bank Negara Malaysia (BNM), the market price of the commodity rose to acquire the highest possible level in the country's history of RM9,188 per ounce on 7 August 2020, when the average level of gold price is between RM6,748 and RM8,483.

When analysing the long-term performance of gold as an investment, the period of time under evaluation of the gold was really essential. According to Eryigit (2017), the price of gold as a safe refuge over the longer terms of period has been proved and it remained important to some investors, but this condition might rely on various factors such as crude oil price, inflation rates, interest rates, and stock indices. According to a research by Guha and Bandyopadhyay (2016), it was said that people, especially economists, need information on gold price inflation in order to plan their future economies in accordance with international gold price standards. Gold investment is one of the most popular types of investment because gold has been regarded as a secure and tempting investment for decades, with the investors purchasing gold not only for the purpose of stockpiling the metal, yet also as well as their risk decrement strategy (Zainab and Neha, 2019). There were two forms of gold property available in Malaysia which were the physical gold and gold accounts (Razimi, Shahril, Romle and Azizan, 2017). Customers can purchase the tangible gold from bank, and each transaction would get a Gold Certicard (certificate). Similarly, a gold account can be bought offline from the counter or via online banking, and all transactions were conducted in Malaysia Ringgit (Johan, 2020).

Volatility of gold price is obvious in every country. Therefore, it will be useful to analyse and forecast the trend of gold prices. In this study, time series models are used to assume advanced predictions. Various time series models had been used to model gold prices both in Malaysia and elsewhere. For example, Autoregressive Integrated Moving Average (ARIMA) is a data analysis model that utilizes time series data to better comprehend the data set or forecast upcoming trends (Hayes, 2021). According to Hayes, an ARIMA model would estimate a company's profitability based on previous periods or predict a stock's future prices based on prior performance. However, in some cases, the time series data also had a long memory correlation especially in econometrics (Granger and Joyeux, 1980). For this situation, the time series analysis method that also can be applied is Autoregressive Fractionally Integrated Moving Average (ARFIMA). According to Bhardwaj and Swason (2006), ARFIMA was said to be better than ARIMA at forecasting accuracy on long memory data and can handle non-stationary data. This was based on the fact that the fractional difference parameter of ARFIMA could describe the causal correlation in data with a short memory, a long memory, or both at the same period. ARFIMA is also commonly applied in the study of macroeconomic and financial data, both of which are covered in econometrics (Bhardwaj, Swason, 2006).

According to Hong (2021), investors bought gold as a protection against economic instability since gold had been proven to hold its value in volatile market conditions. However, gold is not responsible for the large price changes or extreme volatility, but it is acknowledged to always be rising in value as its uses as consumer preferences expand. Furthermore, because gold is a valuable commodity with a constant resource, the marketplaces were worth keeping an eye on, and predicting gold prices within the next ten years could often result in a favorable return within this time frame (Gorn, 2021). According to Hashim, Ramlan, Razali and Nordin (2017) that studied on the macroeconomic indicators that has an effect on the volatility of gold values, it is showed that only crude oil prices and real interest rates were found to have a remarkable positive affiliation with gold prices and the most important factor that influencing the gold prices was the real interest rate. Besides, changes in the US dollar, which serves as the gold price's standard, also have an impact on worldwide gold prices. As a result, lower US dollar is more likely to attract investors to invest in gold. There have recently been demands for gold to be used as a haven asset to protect investors from the volatility of cryptocurrencies (Hong, 2021).

Economic disruption caused by the Movement Control Order during the COVID-19 outbreak has resulted in decreased productivity as a result of decreasing demand for goods and services. Some businesses were negatively affected, and investors were experiencing negative investment returns (Hong, 2021). As a result, some investors had transferred their capital from the stock market and other investments to the gold market, which was seen as a safer shelter for their money. Nevertheless, the news of the revelation of vaccine for COVID-19 has brought comfort to all people on the earth not to mention also including Malaysia which has been the main cause of the recent rapid drop in the price of gold. It is said that the threat of COVID-19 that has caused the gold price to rise prior to the disclosure of the COVID-19 vaccine discovery which had encouraged business analysts and investors to expect a recovery in the global economy, gold appears to have lost its appeal as a haven asset (Bernama, 2020).

In general, the purpose of gold as a safe haven property was questionable (Henriksen, 2018). Therefore, the prediction of gold price is an important study as such situations and trends can be better understood if we could model and future predicts the gold price in Malaysia. Time series models were known as the useful tools for forecasting, and they had been used in variety fields of study including gold. Because of gold has been known as an instrument of investment in econometrics situation and suitable for every period of time, therefore the best forecasting method that can be used to predict the performance of gold price were ARIMA and ARFIMA models. Nevertheless, there has been a shortage of recent articles on forecasting the gold price in Malaysia using time series models involving recent data. The study of forecasting Kijang Emas price in Malaysia conducted by Khamis and Awang (2020) used the data from January 2011 until February 2017 only. Ismail, Razak and Bulot (2020) used the LSTM models based on RNN instead. To the best of our knowledge, there is still a lack of study done using the ARIMA model involving recent data. We also found no study using the ARFIMA model to forecast the Kijang Emas price in Malaysia for the recent year. Therefore, this study is focusing on the prediction of Kijang Emas price in Malaysia for the year 2021 using both the ARIMA and ARFIMA models.

# METHODOLOGY 

**The Dataset**

This study used the data of Malaysia’s gold which was known as Kijang Emas that were obtained from Bank Negara Malaysia’s website. A total of 127 monthly average dataset that were used in this study contains the selling price of one ounce or 31.105 gram Kijang Emas from January 2010 until July 2021. These data were about 11 years that consist of two to five days per week since there was no trace of trade prices for some selected occasions because of holidays or market closing days. The calculation of monthly average data based on daily data. It was done using Microsoft® Excel®. Figure 1 presents a time series plot of the data.

![](6119f0730da8c_media/media/image1.emf)**Figure 1** Plot of the Monthly Kijang Emas Prices from January 2010 to July 2021

**Time Series Modelling Procedure**

For the purpose of time series modelling, a total of 127 observation for monthly average Kijang Emas prices were divided into two parts. The first part were used to fit the ARIMA and ARFIMA models by using data from January 2010 to December 2020. The remaining 7 observations in the second part from January 2021 to July 2021 were applied for post sample forecast accuracy checking. The model fitting for ARIMA and ARFIMA models were done by using the computer software ‘ITSM2000’. The forecast accuracy checking were measured using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Mean Absolute Percentage Error (MAPE) as given in Equations (1), (2) and (3) respectively,

MAE = \(\frac{\sum_{t = 1}^{n}|x_{i} - {\ \widehat{x}}_{i}|}{n}\) (1)

RMSE = \(\sqrt{\frac{\sum_{t = 1}^{n}{(x_{i} - {\widehat{x}}_{i})}^{2}}{n}}\) (2)

MAPE = \(\frac{\sum_{t = 1}^{n}\left| \frac{x_{i} - {\widehat{x}}_{i}}{x_{i}} \right|}{n}\) x 100% (3)

where \(x_{i}\) is the actual observed value,\({\ \widehat{x}}_{i}\) denotes the expected values and *n* denotes the number of expected values.

The Autoregressive Integrated Moving Average (ARIMA) models combined the autoregressive (AR) and moving average (MA) functions. Sato (2013) claimed that the ARIMA model was invented by George Box and Gwilym Jenkins as an attempt to describe changes on the time series using a mathematical approach in the 1970s. According to Lazim (2016), this model was created when the variable’s stationarity assumption was not met. ARIMA (p, d, q) is the general term for this ARIMA model, where p is the order of the AR process, *q* is the order of the MA process, and *d* is the order of the integration or differencing process. The definition of *d* is a non-negative integer, then {*X<sub>t</sub>*} is an ARIMA (*p, d, q*) process if *Y<sub>t</sub> =* (*1-*B)<sup>d</sup> *X<sub>t</sub>* is a causal ARMA (*p, q*) process (Brockwell and Davis, 2002). Brockwell and Davis (2002) stated that {*X<sub>t</sub>*} satisfies a difference equation of the form as given in Equation (4),

𝛷(𝐵) (1 − 𝐵)<sup>d</sup> 𝑋<sub>t</sub> = (𝐵)𝑍<sub>t</sub> (4)

where 𝛷(*z*) and (*z*) are polynomials of degrees *p* and *q*, respectively, where 𝛷(*z*)=0 for *|z|*≤ 1 and *B* is the backward shift operator. The polynomial 𝛷*\**(*z)* has a zero of order *d* at *z=*1. The process {*X<sub>t</sub>*} is stationary if and only if *d*=0*,* in which case it simplifies to an ARMA (*p, q*) process.

Figure 2 displays the time series for Kijang Emas prices from January 2010 to December 2020, which includes consists of 120 observations. Since the plot in Figure 2 was not stationary and shows a linear trend, a Box-Cox transformation with the parameter value (λ=0) was used to stabilize the variability and differencing at lag 1 was apply in the series to eliminate the trend. Figure 3 shows the plot of the transformed series after the mean was deducted. Figure 4 displayed the plot of autocorrelation function (ACF) and partial autocorrelation function (PACF) which were useful in finding the appropriate models for the series.

![](6119f0730da8c_media/media/image2.emf)

Figure 2. The time series plot for Kijang Emas prices in Malaysia from January 2010 to December 2020

![](6119f0730da8c_media/media/image3.emf)

Figure 3. The stationaries series after transformation and differencing

![](6119f0730da8c_media/media/image4.emf)

Figure 4. The ACF and PACF after transformation and differencing

The model fitting for ARIMA models was done by using the computer software ‘ITSM2000’. “Autofit” option was used in finding the maximum likelihood estimators with the lowest AICC value. AICC implies to the bias-corrected version of Akaike Information Criterion (AIC) (Brockwell and Davis, 2002). After specifying the upper bound and lower bound for *p* and *q*, the program will choose the pair (*p, q*) with the smallest AICC statistic value that can be describe through Equation (5),

AICC = − 2ln L (\(\phi_{p},\ \theta_{q},\ \frac{S(\phi_{p},\ \theta_{q})}{n}\) + \(\frac{2\left( p + q + 1 \right)n}{(n - p - q - 2)}\) (5)

where L(\(\phi,\theta,\sigma^{2})\) = \(\frac{1}{\sqrt{\left( 2\pi\sigma^{2} \right)^{n}r_{0}\ldots r_{n - 1}}}e( - \frac{1}{2\sigma^{2}}\sum_{j = 1}^{n}\frac{{(x_{j} - {\widehat{x}}_{j})}^{2}}{r_{j} - 1})\) and S(\(\phi,\theta) = \sum_{j = 1}^{n}{\frac{{(x_{j} - {\widehat{x}}_{j})}^{2}}{r_{j} - 1}}\)

The Autoregressive Fractionally Integrated Moving Average (ARFIMA) model (*p, d, q*) can be extended by including AR and MA and also the fractional difference. ARFIMA is a stationary process with a slower declining autocorrelation function (ACF) that be use in modelling long-run memory process with 0 \< |*d*| \< 0.5. The ARFIMA process {y<sub>t</sub>} is written as in Equation (6),

(1 − 𝐵)*<sup>d</sup>*𝛷(𝐵)𝑋*<sub>t</sub>* = \(\theta\)(𝐵)𝑍*<sub>t</sub>* (6)

where 𝑍<sub>t</sub>\~𝑊𝑁(0, 𝜎<sup>2</sup>). \(\phi\)(𝑧) and \(\theta\)(𝑧) are polynomials of degrees p and q, satisfying \(\phi\)(𝑧) ≠ 0 and \(\theta\)(𝑧) ≠ 0 respectively for all z such that |𝑧| ≤ 1, and *B* is the backward shift operator. (1 − 𝐵)*<sup>d</sup>* is defined by the binomial expansion as in Equation (7),

(1 -\(\ \text{Β)}\)*<sup>d</sup>* = \(\sum_{i = 0}^{\infty}\pi_{i}B^{j}\) with n<sub>0</sub> = 1 and 𝜋<sub>i</sub> = \(\prod_{0 < k \leq i}^{}\frac{k - 1 - d}{k}\) for 𝑖 = 0,1,2, … (7)

where \(\prod\) indicating the gamma function and the parameter *d* can be assumed as any real number (Brockwell and Davis, 2002).

The procedure for fitting the ARFIMA model was similar to that of the ARIMA model. However, ARFIMA model must be specify as fractionally integrated model by setting the non-integer *d* value before running the ‘autofit’.

# FINDINGS AND DISCUSSION

**Modelling of Kijang Emas Price**

Based on the lowest AICC value obtained using the “autofit” option in “ITSM2000”, ARIMA (1, 1, 1) model proved to be the best model in its class. The model was given by Equation (8),

> 𝛸<sub>*t*</sub> = −0.8398X<sub>(*t*−1)</sub> – 𝛧(t) + 1.000 𝛧<sub>(*t*−1),</sub> (8)

where { 𝛧*<sub>t</sub>* } \~ WN (0.001029). The actual values together with their 95% confidence interval for monthly Kijang Emas prices in Malaysia form January 2021 to July 2021 using the ARIMA (1, 1, 1) model and their plots were displayed in Table 1 and Figure 5, respectively. Each of the actual Kijang Emas prices fell inside the 95% estimate intervals when forecast using the ARIMA (1, 1, 1) model.

**Table 1:** The Actual Prices, Forecast Values and 95% Forecast Interval of Kijang Emas Price in Malaysia from January to July 2021 using ARIMA (1, 1, 1) Model.

| **Month** | **Actual Prices (RM)** | **Forecast Values** | **95% Forecast Interval** |
| --------- | ---------------------- | ------------------- | ------------------------- |
| January   | 8010                   | 8109                | (7615, 8635)              |
| February  | 7785                   | 8086                | (7343, 8903)              |
| March     | 7509                   | 8185                | (7289, 9190)              |
| April     | 7679                   | 8181                | (7139, 9375)              |
| May       | 8092                   | 8264                | (7105, 9613)              |
| Jun       | 8063                   | 8274                | (7003, 9777)              |
| July      | 8032                   | 8347                | (6974, 9989)              |

<span class="chart">\[CHART\]</span>

**Figure 5:** The Plot of Actual Price, Forecast Values and 95% Forecast Interval of Kijang Emas Prices from January to July 2021 using ARIMA (1, 1, 1) Model

Based on the lowest AICC value obtained in ‘ITSM2000’, ARFIMA (1, 0.07694, 1) model emerged as the best model in its class based on the lowest AICC value. The model was given by Equation (9),

> (1 − 𝐵)<sup>0.07694</sup>(\(Χ_{t}\) + 0.8447\(Χ_{t - 1}\)) = 𝜃\(\theta\)(𝐵)\(Ζ_{t}\) + 0.9694\(Ζ_{(t - 1)}\) (9)

where {\(Ζ_{t}\)} \~ WN (0.001056). The actual values together with their 95% confidence interval for monthly Kijang Emas prices in Malaysia from January to July 2021 using the ARFIMA (1, 0.07694, 1) model and their graph were shown in Table 2 and Figure 6, respectively. When forecast using the ARFIMA (1, 0.07694, 1) model, all of the actual Kijang Emas prices also fell inside the 95% forecast intervals.

**Table 2:** The Actual Prices, Forecast Values and 95% Forecast Interval of Kijang Emas Price in Malaysia from January to July 2021 using ARIMA (1, 0.07694, 1) Model.

| **Month** | **Actual Prices (RM)** | **Forecast Values** | **95% Forecast Interval** |
| --------- | ---------------------- | ------------------- | ------------------------- |
| January   | 8010                   | 8091                | (7594, 8621)              |
| February  | 7785                   | 8068                | (7306, 8908)              |
| March     | 7509                   | 8162                | (7218, 9230)              |
| April     | 7679                   | 8161                | (7047, 9451)              |
| May       | 8092                   | 8244                | (6985, 9730)              |
| Jun       | 8063                   | 8257                | (6863, 9935)              |
| July      | 8032                   | 8331                | (6811, 10190)             |

<span class="chart">\[CHART\]</span>

**Figure 6:** The Plot of Actual Price, Forecast Values and 95% Forecast Interval of Kijang Emas Prices from January to July 2021 using ARIMA (1, 0.07694, 1) Model

**Comparison between the ARIMA and ARFIMA Models**

The performances of the ARIMA and ARFIMA models in forecasting the Kijang Emas prices in Malaysia will be presented in this section. Table 3 shows the values of MAE, RMSE and MAPE for Kijang Emas prices in Malaysia. It is noticeable that the ARFIMA (1, 0.07694, 1) model performed better in predicting Kijang Emas prices because the values of MAE, RMSE and MAPE were smaller compared to the ARIMA (1, 1, 1) model.

Table 3. The MAE, RMSE and MAPE values of Kijang Emas prices in Malaysia for  
ARIMA (1, 1, 1) and ARFIMA (1, 0.07694, 1) model.

| **Models**             | **MAE**      | **RMSE**     | **MAPE**  |
| ---------------------- | ------------ | ------------ | --------- |
| ARIMA (1, 1, 1)        | **325.1429** | **374.6831** | **4.19%** |
| ARFIMA (1, 0.07694, 1) | **306.2857** | **357.6758** | **3.95%** |

**CONCLUSION AND RECOMMENDATIONS**

It is obvious that forecasting could be done to better understand about the pattern of gold price in Malaysia. In the first part of the study, the overall pattern of Kijang Emas price in Malaysia from the year 2010 to the year 2020 showed a constant fluctuation trend in the monthly average prices. This study purpose is to forecast the Kijang Emas prices in Malaysia using time series models which are the ARIMA and ARFIMA models. The data of Kijang Emas from January 2010 until December 2020 and total of 120 observations were used to forecast the Kijang Emas prices throughout the year 2021. The entire actual monthly Kijang Emas prices for the year 2021 were inside the range of 95% forecast interval of the ARIMA and ARFIMA models. The performances for each model were checked by considering the MAE, RMSE and MAPE values. From the findings, all the MAE, RMSE and MAPE values shows that ARFIMA model emerged to be the best model in forecasting the Kijang Emas prices in Malaysia compared to the ARIMA model. In this study, the best fit model was the ARFIMA model which is a long memory model and not the short memory model, that is the ARIMA model.

From this study, the proposed models were univariate time series models which only considered the data alone. Therefore, for further study, we recommend applying various forecasting models, for example, Vector Autoregression (VAR) or Generalized Autoregressive Conditional Heteroscedasticity (GARCH) to do the forecasting. From prior research by Azzutti (2016), the ARIMA model had previously been proven to be the best, but this is not the case in this study. As a result of this study, it is evident that updating the models when new data becomes available is both recommended and necessary.

**ACKNOWLEDGMENTS**

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