Rescaled Range Statistic Research Articles

Uncovering Information Linkages between Bitcoin, Sustainable Finance and the Impact of COVID-19: Fractal and Entropy Analysis

This study aimed to uncover the impact of COVID-19 on the leading cryptocurrency (Bitcoin) and on sustainable finance with specific attention to their potential long memory properties. In this article, the application of the selected methodologies is based on a fractal and entropy analysis of the econometric model in the financial market. To detect the regularity/irregularity property of a time series, approximate entropy is introduced to measure deterministic chaos. Using daily data for Bitcoin and sustainable finance, namely DJSW, Green Bond, Carbon, and Clean Energy, we examine long memory behaviour by employing a rescaled range statistic (R/S) methodology. The results of the research present that the returns of Bitcoin, the Dow Jones Sustainability World Index (DJSW), Green Bond, Carbon, and Clean Energy have a significant long memory. Contrastingly, an interdisciplinary approach, namely wavelet analysis, is also used to obtain complementary results. Wavelet analysis can provide warning information about turmoil phenomena and offer insights into co-movements in the time–frequency space. Our findings reveal that approximate entropy shows crisis (turmoil) conditions in the Bitcoin market, despite the nature of the pandemic’s origin. Crucially, compared to Bitcoin assets, sustainable financial assets may play a better safe haven role during a pandemic turmoil period. The policy implications of this study could improve trading strategies for the sake of portfolio managers and investors during crisis and non-crisis periods.

Investigation of temporal long-range correlation and anti-correlation of the fluctuations in a radiofrequency produced plasma in a diverging magnetic field configuration

Long range temporal correlation of the fluctuations in both the floating potential (ϕ f ) and the ion-saturation current (I s ) have been investigated in a radio frequency produced magnetized plasma, in which the magnetic field diverges from a narrower quartz tube into a bigger stainless steel expansion chamber. Both argon and helium have been used as the filling gases. The Hurst exponents estimated from the rescaled range statistics indicate that the fluctuations in the ion-saturation current in argon plasma are anti-correlated inside a conical surface. The same fluctuations exhibit a transition to long range correlated behaviour over a narrow radial region of the conical surface. For helium plasma, the fluctuations in I s are mostly long range anti-correlated. The log-log plot of the power spectra of the fluctuations in I s and ϕ f have been observed to follow power laws and the same happens in the power spectrum of the particle flux also, indicating that the self-organized-criticality may exist in the present system.

Long Memory in the Volatility of Selected Cryptocurrencies: Bitcoin, Ethereum and Ripple

This paper examines the volatility of cryptocurrencies, with particular attention to their potential long memory properties. Using daily data for the three major cryptocurrencies, namely Ripple, Ethereum, and Bitcoin, we test for the long memory property using, Rescaled Range Statistics (R/S), Gaussian Semi Parametric (GSP) and the Geweke and Porter-Hudak (GPH) Model Method. Our findings show that squared returns of three cryptocurrencies have a significant long memory, supporting the use of fractional Generalized Auto Regressive Conditional Heteroscedasticity (GARCH) extensions as suitable modelling technique. Our findings indicate that the Hyperbolic GARCH (HYGARCH) model appears to be the best fitted model for Bitcoin. On the other hand, the Fractional Integrated GARCH (FIGARCH) model with skewed student distribution produces better estimations for Ethereum. Finally, FIGARCH model with student distribution appears to give a good fit for Ripple return. Based on Kupieck’s tests for Value at Risk (VaR) back-testing and expected shortfalls we can conclude that our models perform correctly in most of the cases for both the negative and positive returns.

The Hurst phenomenon and the rescaled range statistic

In his 1951 study of Nile River data, H.E. Hurst introduced the rescaled range statistic-the R/S statistic. He argued via a small simulation study that if Xi, i=1,…,n, are i.i.d. normal then the R/S statistic should grow in the order of n. However, Hurst found that for the Nile River data, the R/S statistic increased not in the order of n, but in the order nH, where H ranged between 0.75 and 0.80. He discovered that the effect also appeared in other sets of data. This is now called the Hurst phenomenon. We shall establish some unexpected universal asymptotic properties of the R/S statistic, which show conclusively that the Hurst phenomenon can never appear for i.i.d. data.

Testing of Long Memory in Indian Stock Market using ARFIMA model

This paper examines the presence of long memory property and market cycles in the Indian stock market by using daily closing price of Nifty Index from May 2009 to April 2015. The study has used Unit Root Test, Autocorrelation Test, Rescaled Range (R/S) statistics and ARFIMA Models to determine the long memory property in Indian stock market. The result of the study shows that Indian stock market exhibits a high degree of positive long-term persistence. The results of ARFIMA model also indicates that stock price index exhibits strong evidence of long memory and contradicts the evidence against efficient market hypothesis. This implies that future stock price can be predicted. The study concludes that there is an arbitrage opportunities available for international investors and they can make abnormal profits by investing in Indian stock market.

On the Use of Simple Scaling Stochastic (SSS) Framework to the Daily Hydroclimatic Time Series in the Context of Climate Change

Climate Change hypothesis pushed the scientific community to question the characteristics of the classical statistics such as mean, variance, standard deviation, covariance, etc. in the hydroclimatic field. Many studies have revealed that the climate has always changed and that these changes are closely related to the Hurst phenomenon detected in long hydroclimatic time series and in stochastic term which is equivalent to a simple scaling behavior of climate variability on the time scale. A new statistical framework taking into account the climatic variability is now applied. Most studies are at annual scale where variability at finer scales is not taken into account. This paper proposes to verify the validity of the new statistical framework at finer time scale: the daily time scale. Twelve (12) daily time series of flows, rainfalls and temperatures with 18,628 observations, each one, were studied. Four different methods, such as Rescaled range Statistic (R/S) method, R/S modified method, Aggregate Variances method and Aggregated Standard Deviation (ASD) were applied to determine the Hurst exponent (H). All methods lead to the conclusion that the investigated time series have a long-term persistence phenomenon. Contrary to annual time series where variability corresponds to a Simple Scaling Stochastic (SSS) process, the daily time series seem to correspond to a process having both a SSS component and a deterministic component.

The impact of estimation methods and data frequency on the results of long memory assessment

The main goal of this paper is to examine the effects of selected methods of estimation (the Geweke and Porter-Hudak, modified Geweke and Porter-Hudak, Whittle, R/S Rescaled Range Statistic, aggregated variance, aggregated absolute value, and Peng’s variance of residuals methods) and data frequency on properties of Hurst exponents for stock returns, volatility, and trading volumes of 43 companies and eight stock market indices. The calculations have been performed for a time series of log-returns, squared log-returns, and log-volume (based on hourly and daily data) by nine methods. Descriptive statistics and distribution laws of Hurst exponents depend on the method of estimation and, to some extent, on data frequency (daily and hourly). While by and large in log-returns no long memory has been detected, some estimation methods confirm the existence of long memory in squared log-returns. All of the applied estimation methods show long memory in log-volume data.

Precise asymptotics in the law of the iterated logarithm for "Equation missing" statistic

Abstract Let { X , X n , n ≥ 1 } be a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with zero mean and possibly infinite variance, Q ( n ) = R ( n ) / S ( n ) be the rescaled range statistic, where R ( n ) = max 1 ≤ k ≤ n { ∑ j = 1 k ( X j − X ¯ n ) } − min 1 ≤ k ≤ n { ∑ j = 1 k ( X j − X ¯ n ) } , S 2 ( n ) = ∑ j = 1 n ( X j − X ¯ n ) 2 / n and X ¯ n = ∑ j = 1 n X j / n . Then two precise asymptotics related to probability convergence for Q ( n ) statistic are established under some mild conditions in this paper. Moreover, the precise asymptotics related to almost surely convergence for Q ( n ) statistic is also considered under some mild conditions. MSC:60F15, 60G50.

Long Memory Properties in Return and Volatility: An Application of the Impact of Arab Spring in Turkey Financial Market

The Arab Spring which began on 17 December 2010 with the civil rebellions, revolutionary wave of demonstrations and protests in the Tunisia, Egypt, Libya, Yemen, Bahrain and Syria. The Arab Spring not only created a domino effect between Arabic countries but also it reflected a significant influence on the financial markets all over the world. The objective of this study is to analyze the impact of the Arab Spring in Turkey Financial Market in consideration of long memory. Long memory can be defined as the persistence of the unexpected shocks on the underlying has long lasting effects. Modeling long memory in stock returns and volatility has also attracted great deal of attention from finance literature recently. Existence of long memory is determined both for the returns and volatility of the time series by using different methods. Existence of long memory can be tested by Rescaled Range Statistics (R/S), Geweke and Porter-Hudak (GPH) Model and Gaussian Semi Parametric (GSP) Method. In consequence of these tests, if the stock returns have long memory affect then respectively Fractionally Integrated Autoregressive Moving Average Model (ARFIMA) and the Fractionally Integrated Generalized Autoregressive Conditional Heteroscedasticity (FIGARCH) model are used to detect the long memory in respectively return and volatility. In this study, the impact of the Arab Spring is investigated by modeled the long memory in Istanbul Stock Exchange using ISE 30 index prices in between December 17, 2010 and April 02, 2012.

Classification of emotional states from electrocardiogram signals: a non-linear approach based on hurst

BackgroundIdentifying the emotional state is helpful in applications involving patients with autism and other intellectual disabilities; computer-based training, human computer interaction etc. Electrocardiogram (ECG) signals, being an activity of the autonomous nervous system (ANS), reflect the underlying true emotional state of a person. However, the performance of various methods developed so far lacks accuracy, and more robust methods need to be developed to identify the emotional pattern associated with ECG signals.MethodsEmotional ECG data was obtained from sixty participants by inducing the six basic emotional states (happiness, sadness, fear, disgust, surprise and neutral) using audio-visual stimuli. The non-linear feature ‘Hurst’ was computed using Rescaled Range Statistics (RRS) and Finite Variance Scaling (FVS) methods. New Hurst features were proposed by combining the existing RRS and FVS methods with Higher Order Statistics (HOS). The features were then classified using four classifiers – Bayesian Classifier, Regression Tree, K- nearest neighbor and Fuzzy K-nearest neighbor. Seventy percent of the features were used for training and thirty percent for testing the algorithm.ResultsAnalysis of Variance (ANOVA) conveyed that Hurst and the proposed features were statistically significant (p < 0.001). Hurst computed using RRS and FVS methods showed similar classification accuracy. The features obtained by combining FVS and HOS performed better with a maximum accuracy of 92.87% and 76.45% for classifying the six emotional states using random and subject independent validation respectively.ConclusionsThe results indicate that the combination of non-linear analysis and HOS tend to capture the finer emotional changes that can be seen in healthy ECG data. This work can be further fine tuned to develop a real time system.

Precise asymptotics in the law of the logarithm for the rescaled range statistic

For a sequence of i.i.d. zero mean random variables belonging to the domain of attraction of the normal law, two results concerning the rescaled range statistic are investigated in this paper. More specifically, we obtain precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence under some mild conditions.

Seasonality of low flows and dominant processes in the Rhine River

Low flow forecasting is crucial for sustainable cooling water supply and planning of river navigation in the Rhine River. The first step in reliable low flow forecasting is to understand the characteristics of low flow. In this study, several methods are applied to understand the low flow characteristics of Rhine River basin. In 108 catchments of the Rhine River, winter and summer low flow regions are determined with the seasonality ratio (SR) index. To understand whether different numbers of processes are acting in generating different low flow regimes in seven major sub-basins (namely, East Alpine, West Alpine, Middle Rhine, Neckar, Main, Mosel and Lower Rhine) aggregated from the 108 catchments, the dominant variable concept is adopted from chaos theory. The number of dominant processes within the seven major sub-basins is determined with the correlation dimension analysis. Results of the correlation dimension analysis show that the minimum and maximum required number of variables to represent the low flow dynamics of the seven major sub-basins, except the Middle Rhine and Mosel, is 4 and 9, respectively. For the Mosel and Middle Rhine, the required minimum number of variables is 2 and 6, and the maximum number of variables is 5 and 13, respectively. These results show that the low flow processes of the major sub-basins of the Rhine could be considered as non-stochastic or chaotic processes. To confirm this conclusion, the rescaled range analysis is applied to verify persistency (i.e. non-randomness) in the processes. The estimated rescaled range statistics (i.e. Hurst exponents) are all above 0.5, indicating that persistent long-term memory characteristics exist in the runoff processes. Finally, the mean values of SR indices are compared with the nonlinear analyses results to find significant relationships. The results show that the minimum and maximum numbers of required variables (i.e. processes) to model the dynamic characteristics for five out of the seven major sub-basins are the same, but the observed low flow regimes are different (winter low flow regime and summer low flow regime). These results support the conclusion that a few interrelated nonlinear variables could yield completely different behaviour (i.e. dominant low flow regime).

Investigation of long-range temporal correlation in electron cyclotron resonance produced linear magnetized plasma of the MaPLE device

Long range temporal correlation of the low frequency fluctuations is investigated in a linear electron cyclotron resonance produced magnetized plasma at different radial positions and filling gas pressures. These fluctuations turn from chaotic to coherent and again chaotic as one moves radially outwards from the center towards the edge region. The power spectrum of these fluctuations shows three distinct frequency regions characterized by their power exponents. Long range temporal correlation of these fluctuations is investigated by estimating the self similarity parameter (Hurst exponents) using rescaled range (R/S) statistics as well as from power spectrum analysis. Dependence of this long range temporal correlation on filling gas pressure has also been investigated.

On the link between extreme floods and excess monsoon epochs in South Asia

This paper provides a synoptic view of extreme monsoon floods on all the nine large rivers of South Asia and their association with the excess (above-normal) monsoon rainfall periods. Annual maximum flood series for 18 gauging stations spread over four countries (India, Pakistan, Bangladesh and Nepal) and long-term monsoon rainfall data were analyzed to ascertain whether the extreme floods were clustered in time and whether they coincided with multi-decade excess monsoon rainfall epochs at the basin level. Simple techniques, such as the Cramer’s t-test, regression and Mann–Kendall (MK) tests and Hurst method were used to evaluate the trends and patterns of the flood and rainfall series. MK test reveals absence of any long-term tendency in all the series. However, the Cramer’s t test and Hurst-Mandelbrot rescaled range statistic provide evidence that both rainfall and flood time series are persistent. Using the Cramer’s t-test the excess monsoon epochs for each basin were identified. The excess monsoon periods for different basins were found to be highly asynchronous with respect to duration as well as the beginning and end. Three main conclusions readily emerge from the analyses. Extreme floods (>90th percentile) in South Asia show a tendency to cluster in time. About three-fourth of the extreme floods have occurred during the excess monsoon periods between ~1840 and 2000 AD, implying a noteworthy link between the two. The frequency of large floods was higher during the post-1940 period in general and during three decades (1940s, 1950s and 1980s) in particular.

Time series analysis of monthly rainfall in Nigeria with emphasis on self-organized criticality

Monthly rainfall data of twenty-one years (1980 – 2000) were analyzed for the six regions of Nigeria using the rescaled range (R/S) statistic, the standard fluctuation analysis (FA) and the detrended fluctuation analysis (DFA). The results indicated that the distribution of monthly rainfall has a Hurst exponent of 1 in the short term which is an evidence of self-organized criticality, and a Hurst exponent of about 0.5 in the long run which is an evidence of a random walk. The Hurst exponents decreased from around 1 for small window sizes and reached 0.5 after 36 months and then remained fairly constant. The R/S statistic was found to perform better than the FA and the linear DFA used in the analysis.

ROLE OF HURST EXPONENT IN PREDICTION OF MARKET EFFICIENCY IN KSE-100 INDEX

Purpose:- This paper aims to investigate the Efficient Market Hypothesis (EMH) and validity of Random Walk Model (RWM) in KSE-100 index starting from 1992 till 2014 taking monthly averages of index. Methodology:- Main focus of the paper is to evaluate the efficiency in KSE-100 index with respect to application of Hurst Exponent and Rescaled Ranged Statistics. Although many researchers have previously explained the working of EMH in KSE-100 index but rarely anyone has explained it using Hurst Exponent Analysis on over all longest period since the establishment of KSE-100 index Feb, 1992 to Dec, 2014. Annual Rescaled Range Statistics are also calculated to explain the good or bad years according to Estimated Hurst Statistics. All statistical analysis has been performed on Gretl which gives the good grasp over Hurst Exponent Analysis. Results:- The results revealed that overall KSE-100 index is not following the random walk and is not performing efficiently, and yearly break up shows that market was persistent in few years but mostly it was antipersistent (long-run memory prevails). Practical Implication:- Implementing Hurst Exponent Analysis enabled us to get rigorous result about performance of the Pakistan stock market in terms of efficiency that implied chances of arbitrage opportunity prevail significantly.

FRACTAL DETRENDED FLUCTUATION ANALYSIS OF CHINESE ENERGY MARKETS

In this paper, we analyze and compare long-range power-law correlations of returns, absolute returns, squared returns, cubed returns and square waved returns for sixteen individual stocks from the block of energy sources of Chinese stock market and five stock indices (Shanghai Composite Index, Shenzhen Component Index, Dow Jones Industrial Average index, Nasdaq Composite Index, the Standard and Poor's 500 Index) by using a detrended fluctuation analysis approach. The empirical evidence suggests that Shanghai Composite Index is very close to Shenzhen Component Index and Nasdaq, DJIA is very close to S&amp;P 500 in all cases. And the exponent trends of the returns are close to that of square waved returns. Also, five indices deviate from other sixteen individual energy stocks in all cases except square waved returns. Further, there are long-range correlations and persistence in volatility series of absolute returns and squared returns. Moreover, we investigate the long-term memory of these returns by applying Lo's modified rescaled range statistic. We find that the China energy market exhibits fractal and persistence properties.

LONG-TERM MEMORY EFFECT IN STOCK PRICES ANALYSIS

The Hurst exponent is widely applied for time series analysis. The Hurst exponent is a statistical measure used to classify time series. Using the Hurst parameter processes are classified into long range dependence, antipersistence and white noise processes. R/S analysis method is one of the few methods that evaluate the Hurst exponent. This method uses the rescaled range statistic (R/S statistic). The R/S statistic is the range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation. A log-log plot of the R/S statistic versus the number of points of the aggregated series should be a straight line with the slope being an estimation of the Hurst exponent. However, there are many methods of evaluating the Hurst exponent such as ratio variance of residuals, the periodogram method, the Whittle method, the Abri-Veitch method, etc. Investigation object – the Baltic sector indices. The latter represent tendencies of different sector activity indices in the stock market. The work concentrates on calculating the Hurst parameter, evaluated Hurst parameters of the Baltic sector indices are given for different periods of time.

Study on self organized criticality of China power grid blackouts

Based on the complex system theory and the concept of self organized criticality (SOC) theory, the mechanism of China power grid blackout is studied by analyzing the blackout data in the China power system from 1981 to 2002. The probability distribution functions of various measures of blackout size have a power tail. The analysis of scaled window variance and rescaled range statistics of the time series show moderate long time correlations. The blackout data seem consistent with SOC; the results obtained show that SOC dynamics may play an important role in the dynamics of power systems blackouts. It would be possible to propose novel approaches for understanding and controlling power systems blackouts.

Long memory in international equity markets: revisited

This study provides empirical evidence of the long-range behaviour in international equity markets. We test for long memory in the daily returns using the modified rescaled range statistic R/S proposed by Lo (1991) and the rescaled variance V/S statistic developed by Giraitis et al. (2003). Long memory is found to be weak in the return series when using R/S but some evidence of long memory is found in USA and Germany based on V/S analysis. Our results confirm those reported by Lo (1991) using only the rescaled range analysis and should be useful to regulators, practitioners and derivative market participants, whose success depends on the ability to forecast stock price movements.