International Journal of Economics, Commerce and Research (IJECR) ISSN(P): 2250-0006; ISSN(E): 2319-4472 Vol. 4, Issue 1, Feb 2014, 7-14 © TJPRC Pvt. Ltd.
GOLD PRICE DATA ANALYSIS USING RESCALED RANGE (R/S) ANALYSIS
SHAIKH YUSUF H1, KAPSAE S. K.2, KHAN A R3 & G. RABBANI4 1Shivaji Arts, Commerce and Science College, Kannad, India 2S.B. Science College, Aurangabad, Maharashtra, India 3,4Maulana Azad College Dr. Rafiq Zakaria Campus, Dr. Rafiq Zakaria Marg, Aurangabad, Maharashtra, India
ABSTRACT
Gold has been at centre of attraction from various angles, particularly for value and financial importance and served as a reference. There has been a continuous rise in the international gold prices since for ages. Gold is a major part of savings of investors, this has caused concern as to whether any correction in gold prices will have destabilizing implications on the financial markets. With this backdrop, we make an attempt to analyze the gold prices over a period of more than last 32 years to explore the variability in the gold price in India. The gold market has recently attracted a lot of attention and the global price of gold is relatively higher than its historical changing pattern and trend. There has been a drastic increase in international gold prices in the last few years, though there was one large correction in 2008. The gold price data for about 32 years is divided into four sets for the purpose of implementation of Rescaled Range (R/S) Analysis. R/S analysis is an effective technique for analyzing random data in the form of time series in order to bring out trends and any memory effects present in the behavior of the changing data. Details of implementation of the R/S analysis technique and estimation of the Hurst exponent H and the fractal dimensions are presented and findings and results discussed, it is shown that the international gold prices exhibit a consistent trend of persistence and long term memory effect with fractal dimension close to unity.
KEYWORDS: Gold Market, Gold Prices, Time Series, R/S Analysis, Fractal Dimension, Persistence, Memory Effect
INTRODUCTION
Like share prices the international gold prices exhibit a random variation and the parameters governing the gold prices can not exactly be enlisted. Randomly varying prices pose difficulties in predicting trends and future patterns which is of interest to investors and financial agencies. Such situations are at time handled using methods based on the concept of fractals and fractal dimensions and the approach is found to be effective for non-stationary signals such as stock market data [1 to 5]. These apparently irregularly changing signals are considered to be driven by external non stationary factors. Standard methods such as Fourier analysis assume that the signals are stationary in temporal windows. Such an assumption does not strictly apply for gold prices and stock market data as it changes constantly with an element of unpredictability and the factors governing the prices keep on changing from time to time. Such irregular phenomena are at times handled using the concept of fractals and fractal dimensions [6 to 8], fractals possess characteristics of self similarity and scale invariance. Fractal based methods such as rescaled range analysis does not impose the above assumption and therefore are better suited for the analyzing the data and has better predictability. In this paper, we use rescaled range analysis [9 to 15] to analyze gold price (in IRS) data for more than last 32 years which gives statistical insight into the trends and tendencies of fluctuations. The data spread over 32 years is divided into four part of approximately 8 years each (Set – I to IV). The four sets of data is independently subjected to R/S analysis and the findings are presented, it is shown that rescale
8 Shaikh Yusuf H, Kapsae S. K, Khan A R & G. Rabbani range analysis can be used for assessing any trends of persistence or anti persistence that can be used for predicting future trends.
GOLD PRICE DATA
Gold and Share market [16] form the fulcrum of economic trends in the market for ages and it has its own importance in establishing strength or weakness of the prevailing economic trends. It is known that domestic gold prices and international gold prices are closely interlinked. Variations in the international gold prices find almost similar echo in the domestic gold prices. For a long time, macro fundamental such as international commodity prices, US exchange rate and equity prices used to be the most dominating factors affecting the international gold prices and their impact used to be identical both in the long-run and short-run[17]. However this linkage started showing weakness and the mutual dependence was found to be more complex and the relative proportion of the two main deciding factors was found to appreciably change.
Lot of standard software are available out there and for analyzing such data that in fact is a time series to evaluate the volatility and assess the variation patterns, however these try to forecast future prices based on statistical theories and historical weightages and at times are liked with news and technicals. Most of the times, statistically based parameters fail drastically if the prices drastically change due to economic events (also known as news) or issues like change of rule or Government, at times natural calamities and wars also shake the market drastically.
RESCALE RANGE ANALYSIS OF TIME SERIES
For the study natural phenomena possessing random character such as the flow of the Nile River in relation to long term correlation Hurst introduced approach. This approach was based on the statistical assessment of many observations of the natural phenomena. After studying 800 years of records, he showed that the flow of the Nile River was not random, but had a pattern. He introduced a constant, K, which measures the bias of the fractional Brownian motion. In 1968 Mandelbrot defined such pattern as fractal. There are many algorithms to calculate fractal characteristics such as fractal dimensions, which is a number that quantitatively describes how an object fills its space. He renamed the constant K to H in honor of Hurst. Rescaled range (R/S) analysis is a statistical method to analyse the records of natural phenomena in the form of a time series. The theory of the rescaled range analysis was first given by Hurst. Mandelbrot and Wallis further refined the method. Feder (1988) gives an excellent review of the analysis of data using time series [18, 19], history, theory and applications, and adds some more statistical experiments to establish the effectiveness of this approach. The parameter H, the Hurst exponent provides insight into the trends and patterns shown by the time series in respect of as to whether the time series is random or not. It is also related to the fractal dimension, the Rescaled range analysis (R/S) approach of estimating H is helpful in distinguishing a completely random time series from a correlated time series and. The value of H reveals persistence of trends in a given time series.
METHODOLOGY
The approach for estimation of the Hurst exponent ‘H’ follows simple statistical rules, for a time series like data, start with the whole observed data set Z (t) that covers a total duration and calculate its mean over the whole of the available data ‹Z›.
1 Z Z(t) t1 (1)
Gold Price Data Analysis Using Rescaled Range (R/S) Analysis 9
Then Sum the differences from the mean to get the cumulative total of the deviation from mean at each time point,
X ( t, ), from the beginning of the period up to any time t using
X (t, ) Z t Z t1 (2)
Then find the X max the maximum of X (t, ), X min the minimum of X(t, ) and calculate the self adjusted range R() defined as the difference between maximum and minimum accumulated influx X
R ( ) = max X (t, ) - min X (t , ) (3)
The standard deviation S, of the values, Z (t) of the observation over the period,
1 1 2 2 S( ) Z t Z t1 (4)
Hurst used a dimension less ratio R/S where S( ) is the standard deviation as a function of
(R/S) = ( / 2) H (5)
Hurst exponent H can be found by plotting log (R/S) against log (/2), the slope of the resulting straight line is the Hurst exponent. The Hurst exponent gives a measure of the smoothness of a fractal object where H varies between 0 and 1. Low ‘H’ values indicate high levels of roughness or variability. High values of H indicate low value of roughness or high levels of smoothness. The fractal dimension D is related to H as H = 2 – D and higher the value of D, larger is the variability or complexity associated with the time series. We use H and D to analyze the gold price data as it has a broad applicability to signal processing due to its robustness. These parameters can be calculated by the rescaled range analysis method. It is useful to distinguish between random and nonrandom data points/time series. If H equals 0.5, then the data is determined to be random. If the H value is less than 0.5, it represents anti-persistence, meaning, if the signal is up/down in the last period then more likely it will go down/up in the next period (i.e. if H is between 0 and 0.5 then an increasing trend in the past implies a decreasing trend in the future and decreasing trend in past implies increasing trend in the future). If the H value varies between 0.5 and 1, this represents persistence which indicates long memory effects, where, if the signal is up/down in the last period, then most likely in the next period the signal will continue going up/down. This also means that the increasing trend in the past implies increasing trend in the future also or decreasing trend in the past implies decreasing trend in the future. It is important to note that persistent stochastic processes have little noise whereas anti-persistent processes exhibit presence of high frequency noise.
The relationship between fractal dimensions D f and Hurst exponent H can be expressed as
D f = 2 – H (6)
From the Hurst exponent H of a time series, the fractal dimension can be found from this equation. When D f = 1.5, there is normal scaling. When D f is between 1.5 and 2, time series is anti-persistent and when D f is between 1 and 1.5 the time series is persistent. For D f =1, time series is a smooth curve and purely deterministic in nature and for D f = 1.5 time series is purely random.
10 Shaikh Yusuf H, Kapsae S. K, Khan A R & G. Rabbani
R/S ANALYSIS OF GOLD PRICE TIME SERIES
The international gold price data (in IRS) spread over about 32 years is split into four parts with 2048 data points in each set (Set – I to IV) For the purpose of implementation of R/S analysis. Set – I is for the years 2005 to 2013, Set – II for 1997 to 2005, Set – III is for 1989 to 1997 and set – IV is for 1981 to 1989. This covers a period from 19/11/1981 to 12/04/2013; daily record of international gold price [20] in Indian Rupees is used. Figure 1 is the complete plot showing the changing gold prices over 32 years, the period under study. The initial part of the plot looks like more or less flat compared to the later part giving a feel that there was no appreciable rising trend in the initial part. To explore the reality, instead of price versus time a plot of log of price versus time is shown in Figure 2, it is seen from this plot that rising trend persist over the entire period and is superimposed with fluctuations. For a different comparison, the four sets of data are plotted on a graph shown in Figure 3, for ease of comparison all the prices in each set are normalized to unity at the end of each series. This figure clearly indicates that there has been a continuously rising trend in all the four sets however with varying trends and the Set – I representing the latest gold prices show relatively faster rising trend compared to the rest of the three sets. The R/S analysis was implemented individually on each set taking one set at a time.
Figure 1: Shows Gold Prices per 10g from 02/01/1979 to 12/04/2013
Figure 2: Shows Log of Gold Prices per 10g from 02/01/1979 to 12/04/2013
Gold Price Data Analysis Using Rescaled Range (R/S) Analysis 11
Figure 3: Represent Time Series of 4 Sets of 2048 Data Points
The results of R/S analysis implemented individually on each sets (Set – I to IV) taking one set at a time is presented in Table 1 First column is the log(τ/2) value and the remaining four columns show the corresponding values of log(R/S) for each set. The log log plot of R/S against τ/2 is shown in Figure 4, it is interesting to note that all the four plots for the four sets lie close to each other showing identical trend and almost superimpose. The slope of the straight line best fitting the data point lie between 0.5 and 1.0, in fact more than 0.9, indicating persistent trend implying that the rising prices continue to rise and the falling prices continue to fall that is in agreement with what is seen from the plots, of course there is always a variability and there are turning points where the prices show the trend reversal and that is one point or two and rest of the points follow certain trend.
Table 1: Mean R/S and /2 for Gold Price Data (from 19/11/1981 to 12/04/2013) S Log(R/S) Log(T/2) No Set – I Set – II Set – III Set – IV 1 0.30103 0.0015 0.0769 0.0640 0.0178 2 0.6021 0.2805 0.2804 0.2716 0.2491 3 0.9031 0.5635 0.5326 0.5234 0.5005 4 1.2041 0.8533 0.8118 0.8129 0.7804 5 1.5051 1.1482 1.0871 1.1020 1.0711 6 1.8062 1.4460 1.3759 1.3845 1.3801 7 2.1072 1.7454 1.6787 1.6953 1.6824 8 2.4082 2.0456 1.9801 1.9367 1.9731 9 2.7093 2.3462 2.2705 2.3094 2.3216 10 3.0103 2.6470 2.5822 2.6444 2.5785
Figure 4: Gives Rescale Range Analysis of 4 Sets of Gold Price Analysis of 2048 Data Points
12 Shaikh Yusuf H, Kapsae S. K, Khan A R & G. Rabbani
The value of the Hurst exponent estimated from the slope of the graph and the fractal dimension calculated from the Hurst exponent are tabulated in Table 2 the values of R2 are shown in the last column which indicated the least square fit line best describes the data as the points lie well along this straight line and the value of R2 is close to unity. The value of Hurst exponent H ranges approximately from 0.94 to 0.98 confirming the persistence in trends, meaning that a rising trend is precursor of further rise. The fractal dimension D f for Set – I and IV are on the lower side indicating relatively less fluctuations and complexity of trend as compared to the remaining two sets.
Table 2 S. Hurst Fractal Set R2 No. Exponent H Dimension Df 1 I 0.9796 1.0204 0.9998 2 II 0.9383 1.0617 0.9980 3 III 0.9560 1.0440 0.9970 4 IV 0.9650 1.0350 0.9983
RESULTS AND DISCUSSIONS
Random data like gold prices and share market data that possesses a high degree of variability and unpredictability can be analysed using R/S analysis and Hurst exponent H. Hurst exponent is an indicative of the presence of noise or fluctuations in the data which in turn gives information on the volatility of the prices and turbulence in the price. The results of R/S analysis as applied to international gold prices (in IRS) show that there is persistence in the trends exhibited by the gold prices over the entire period studied as the value of H likes between 0.5 and 1.0. This also confirms long term memory effect that the trends tend to continue. As the value of H range from about 0.94 to 0.98, it indicates that H being on the higher side tending to unity, imply that the time series lean towards deterministic nature and show limited roughness or variability. A closer examination of the four sets of data and their respective H and D values shows that Set – I has the highest value of H and Set – 2 has the lowest showing that the variability for Set - I is less as compared to that for Set – II which is clearly visible from Figure 2 as there are more ups and down in Set – II as compared to Set – I. The remaining two sets exhibit intermediate characteristics as compared to the two. However all the four sets clearly confirm persistence of the trend and existence of long term memory effect with limited degree of local fluctuations.
REFERENCES
1. Lo A. W., ‘Long Term memory in stock market prices’ Econometrica 59, 1279 – 1313.
2. Cheung, Yin-Wong, ‘A search for long memory in international Stock Market returns’ Journal of International Money and Finance 14, 597 – 615(1993).
3. Lo Andrew W. A Craig Mac Kinlay, Long – Term memory in stock market Prices Chapter 6 in A Non-Random Walk Down Wall Street (Princeton Unversity Press. Princeton NJ).
4. Sidra Malik, Shahid Hussain and Shakil Ahmed, “Impact of Political Event on Trading volume and Stock Returns: The Case of KSE ‘International Review of Business Research Papers Vol. 5 No. 4 Pp. 354-364 (June 2009)
5. Mazharul H. Kazi, ‘Stock Market Price Movements and Macroeconomic Variables ’International Review of Business research Papers Vol. 4 No.3 June Pp.114-126(2008).
6. B. B. Mandelbrot, ‘The Fractal geometry of Nature Freeman’, (San Francisco). (1982).
Gold Price Data Analysis Using Rescaled Range (R/S) Analysis 13
7. B.B. Mandelbrot, Fractals and Scaling in Finance, Springer Verlag, New York (1997).
8. B. B. Mandelbrot, Scaling in Financial Prices: IV Multifractal Concenration uantitative Finance 1(6), 641- 649.
9. Razdan ‘Scaling in the Bombay stock exchange index’ Pramana 58, 3,537(2002).
10. H.E Hurst., ‘Long-term storage capacity of reservoirs’ Trans. Am. Soc. Civil Eng’116, 770 (1951).
11. Indian Stock Market Research: 5 years Fact sheets and Analysis http://www.equitymaster.com/research-it/
12. X. Zhou, N. Persaud, H. Wang. ‘Scale invariance of daily runoff time series in agricultural watersheds ’Hydrology and Earth System Sciences 10, 79(2006).
13. A.R Rao., D. Bhattacharya, ‘Comparison of Hurst Exponent Estimates in Hydrometeorological Time Series’ J. of Hydrologic Eng, 4, 3, 225,(1999).
14. Granger, C. W. J, ‘Some properties of Time Series Data and Their Use in Econometric Specification’. Journal of Econometrics; 16: 121-130(1981).
15. Dr. S. Kaliyamoorthy, ‘Relationship of Gold Market and Stock Market: An Analysis’ International Journal of Business and Management Tomorrow Vol. 2 No. 6 (2012).
16. Capie, F., Terence, C.M., Wood, G., ‘Gold as a hedge against the dollar’ ‘International Financial Markets, Institution and Money 15, 343-352(2005).
17. T. Di Matteo, T. Aste., Michel M. Dacorogna., ‘Long term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development’ Journal of Banking and Finance 29, 827- 851(2005).
18. J. W. Kantelhardt, B. E Koscielny., H. H Rego A, S Havlin, Bunde A, Physica A 295, 441(2001).
19. T.C. Mills, ‘Statistical analysis of daily gold price data’ Physica A 338 (3–4), 559–566(2004).