{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T08:39:06Z","timestamp":1772699946428,"version":"3.50.1"},"reference-count":67,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,2]],"date-time":"2022-10-02T00:00:00Z","timestamp":1664668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper aims to empirically examine long memory and bi-directional information flow between estimated volatilities of highly volatile time series datasets of five cryptocurrencies. We propose the employment of Garman and Klass (GK), Parkinson\u2019s, Rogers and Satchell (RS), and Garman and Klass-Yang and Zhang (GK-YZ), and Open-High-Low-Close (OHLC) volatility estimators to estimate cryptocurrencies\u2019 volatilities. The study applies methods such as mutual information, transfer entropy (TE), effective transfer entropy (ETE), and R\u00e9nyi transfer entropy (RTE) to quantify the information flow between estimated volatilities. Additionally, Hurst exponent computations examine the existence of long memory in log returns and OHLC volatilities based on simple R\/S, corrected R\/S, empirical, corrected empirical, and theoretical methods. Our results confirm the long-run dependence and non-linear behavior of all cryptocurrency\u2019s log returns and volatilities. In our analysis, TE and ETE estimates are statistically significant for all OHLC estimates. We report the highest information flow from BTC to LTC volatility (RS). Similarly, BNB and XRP share the most prominent information flow between volatilities estimated by GK, Parkinson\u2019s, and GK-YZ. The study presents the practicable addition of OHLC volatility estimators for quantifying the information flow and provides an additional choice to compare with other volatility estimators, such as stochastic volatility models.<\/jats:p>","DOI":"10.3390\/e24101410","type":"journal-article","created":{"date-parts":[[2022,10,8]],"date-time":"2022-10-08T04:04:56Z","timestamp":1665201896000},"page":"1410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Volatility Dynamics of Non-Linear Volatile Time Series and Analysis of Information Flow: Evidence from Cryptocurrency Data"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9213-0616","authenticated-orcid":false,"given":"Muhammad","family":"Sheraz","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Institute of Business Administration, The School of Mathematics and Computer Science, Karachi 75270, Pakistan"},{"name":"Department of Financial Mathematics, Fraunhofer ITWM, 67663 Kaiserslautern, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7195-0941","authenticated-orcid":false,"given":"Silvia","family":"Dedu","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Bucharest University of Economic Studies, 010734 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0686-3718","authenticated-orcid":false,"given":"Vasile","family":"Preda","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014 Bucharest, Romania"},{"name":"\u201cGheorghe Mihoc-Caius Iacob\u201d Institute of Mathematical Statistics and Applied Mathematics of Romanian Academy, 2. Calea 13 Septembrie, nr. 13, Sect. 5, 050711 Bucharest, Romania"},{"name":"\u201cCostin C. Kiritescu\u201d National Institute of Economic Research of Romanian Academy, 3. Calea 13 Septembrie, nr. 13, Sect. 5, 050711 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"424","DOI":"10.2307\/1912791","article-title":"Investigating causal relations by econometric models and cross-spectral methods","volume":"37","author":"Granger","year":"1969","journal-title":"Econometrica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1915","DOI":"10.1080\/00036846.2015.1109048","article-title":"A nonlinear Granger causality test between stock returns and investor sentiment for Chinese stock market: A wavelet-based approach","volume":"48","author":"Chu","year":"2016","journal-title":"Appl. Econo."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s11071-016-3254-7","article-title":"Mutual-information matrix analysis for nonlinear interactions of multivariate time series","volume":"88","author":"Zhao","year":"2017","journal-title":"Nonlin. Dyna."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s11071-019-04974-y","article-title":"Dynamic relationship analysis between NAFTA stock markets using nonlinear, nonparametric, non-stationary methods","volume":"97","year":"2019","journal-title":"Nonlin. Dyna."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1950140","DOI":"10.1142\/S0218348X19501408","article-title":"A look at short- and long-term nonlinear dynamics in family business stock returns listed on Casablanca stock","volume":"27","author":"Lahmiri","year":"2019","journal-title":"Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"122923","DOI":"10.1016\/j.physa.2019.122923","article-title":"Nonlinear analysis of Casablanca Stock Exchange, Dow Jones and S&P500 industrial sectors with a comparison","volume":"539","author":"Lahmiri","year":"2020","journal-title":"Physica A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1108\/SEF-09-2020-0385","article-title":"COVID-19 pandemic and cryptocurrency markets: An empirical analysis from a linear and nonlinear causal relationship","volume":"38","author":"Sahoo","year":"2021","journal-title":"Stud. Econ. Finan."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s40822-022-00206-8","article-title":"Testing for asymmetric non-linear short- and long-run relationships between crypto-currencies and stock markets","volume":"12","author":"Ghorbel","year":"2022","journal-title":"Eurasi. Econ. Revi."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"102803","DOI":"10.1016\/j.frl.2022.102803","article-title":"Nonlinear dynamics analysis of cryptocurrency price fluctuations based on Bitcoin","volume":"47","author":"Tong","year":"2022","journal-title":"Finan. Res. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Sheraz, M., and Nasir, I. (2021). Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach. Risks, 9.","DOI":"10.3390\/risks9050089"},{"key":"ref_11","first-page":"20","article-title":"Risk-neutral densities in entropy theory of stock options using Lambert function and a new approach","volume":"16","author":"Preda","year":"2015","journal-title":"Proc. Roman. Acad. Ser. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2006.12.004","article-title":"Causality detection based on information-theoretic approaches in time series analysis","volume":"441","author":"Vejmelka","year":"2007","journal-title":"Phys. Repor."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring Information Transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Revi. Lett."},{"key":"ref_14","first-page":"275","article-title":"Analysing the information flow between financial time series","volume":"30","author":"Marschinski","year":"2000","journal-title":"Eur. Phys. J. B Cond. Matt. Comp. Syst."},{"key":"ref_15","unstructured":"Baek, S.K., Jung, W.S., Kwon, O., and Moon, H.T. (2005). Transfer entropy analysis of the stock market. arXiv preprint."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2851","DOI":"10.1016\/j.physa.2008.01.007","article-title":"Information flow between composite stock index and individual stocks","volume":"387","author":"Kwon","year":"2008","journal-title":"Phys A Stat. Mech. App."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"68003","DOI":"10.1209\/0295-5075\/82\/68003","article-title":"Information flow between stock indices","volume":"82","author":"Kwon","year":"2008","journal-title":"Europhys. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.intfin.2014.03.004","article-title":"The impact of the financial crisis on transatlantic information flows: An intraday analysis","volume":"31","author":"Dimpfl","year":"2014","journal-title":"J. Int. Finan. Mark Inst. Mon."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.chaos.2014.08.007","article-title":"Effective transfer entropy approach to information flow between exchange rates and stock markets","volume":"68","author":"Sensoy","year":"2014","journal-title":"Chaos. Solit. Frac."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"110084","DOI":"10.1016\/j.chaos.2020.110084","article-title":"Renyi entropy and mutual information measurement of market expectations and investor fear during the COVID-19 pandemic","volume":"139","author":"Lahmiri","year":"2020","journal-title":"Cha. Solit. Frac."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2971","DOI":"10.1016\/j.physa.2011.12.064","article-title":"R\u00e9nyi\u2019s information transfer between financial time series","volume":"391","author":"Jizba","year":"2012","journal-title":"Physica A Stat. Mech. App."},{"key":"ref_22","first-page":"85","article-title":"Using transfer entropy to measure information flows between financial markets","volume":"17","author":"Dimpfl","year":"2013","journal-title":"Stud. Nonlin. Dyn. Econo."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4443","DOI":"10.3390\/e16084443","article-title":"Structure of a global network of financial companies based on tansfer entropy","volume":"16","author":"Leonidas","year":"2014","journal-title":"Entropy"},{"key":"ref_24","first-page":"265","article-title":"Transferentropy-quantifying information flow between different time series using effective transfer entropy","volume":"10","author":"Behrendt","year":"2019","journal-title":"Software X"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"054101","DOI":"10.1103\/PhysRevLett.107.054101","article-title":"Inner composition alignment for inferring directed networks from short time series","volume":"107","author":"Hempel","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1140\/epjb\/e2013-31111-8","article-title":"Data-driven reconstruction of directed networks","volume":"86","author":"Hempel","year":"2013","journal-title":"Eur. Phys. J. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1821","DOI":"10.1007\/s11071-014-1251-2","article-title":"Segmented inner composition alignment to detect coupling of different subsystems","volume":"76","author":"Wang","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2439","DOI":"10.1007\/s11071-014-1823-1","article-title":"The coupling analysis of stock market indices based on cross-permutation entropy","volume":"79","author":"Shi","year":"2015","journal-title":"Nonlin. Dyna."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2050250","DOI":"10.1142\/S0218127420502508","article-title":"Shortcomings of Transfer Entropy and Partial Transfer Entropy: Extending Them to Escape the Curse of Dimensionality","volume":"30","author":"Papana","year":"2020","journal-title":"Int. J. Bifur. Chaos."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Rozo, A., Morales, J., Moeyersons, J., Joshi, R., Caiani, E.G., Borz\u00e9e, P., Buyse, B., Testelmans, D., Van Huffel, S., and Varon, C. (2021). Benchmarking Transfer Entropy Methods for the Study of Linear and Nonlinear Cardio-Respiratory Interactions. Entropy, 23.","DOI":"10.3390\/e23080939"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Bossomaier, T., Barnett, L., Harre, M., and Lizier, J.T. (2016). An Introduction to Transfer Entropy: Information Flow in Complex Systems, Springer International Publishing.","DOI":"10.1007\/978-3-319-43222-9"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1016\/j.ribaf.2017.05.010","article-title":"Price dynamics and speculative trading in bitcoin","volume":"41","author":"Blau","year":"2017","journal-title":"Rese. Int. Bus. Finan."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chan, S., Chu, J., Nadarajah, S., and Osterrieder, J. (2017). A Statistical Analysis of Cryptocurrencies. J. Risk. Finan. Manag., 10.","DOI":"10.3390\/jrfm10020012"},{"key":"ref_35","first-page":"43","article-title":"Bitcoin Cash: Stochastic Models of Fat-Tail Returns and Risk Modelling","volume":"3","author":"Sheraz","year":"2020","journal-title":"Econ. Comp. Econ. Cyb. Stud. Rese"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Chu, J., Chan, S., Nadarajah, S., and Osterrieder, J. (2017). GARCH modelling of cryptocurrencies. J. Risk. Finan. Manag., 10.","DOI":"10.2139\/ssrn.3047027"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.frl.2015.10.008","article-title":"Bitcoin, Gold and the Dollar\u2014A GARCH Volatility Analysis","volume":"16","author":"Dyhrberg","year":"2016","journal-title":"Finan. Rese. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.irfa.2018.08.012","article-title":"Volatility connectedness in the cryptocurrency market: Is Bitcoin a dominant cryptocurrency?","volume":"60","author":"Yi","year":"2018","journal-title":"Int. Rev. Financ. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.intfin.2017.11.001","article-title":"Virtual relationships: Short-and long-run evidence from Bitcoin and altcoin markets","volume":"52","author":"Ciaian","year":"2018","journal-title":"J. Int. Financ. Mark. Inst. Money"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.frl.2018.10.005","article-title":"Volatility co-movement between Bitcoin and Ether","volume":"30","author":"Katsiampa","year":"2019","journal-title":"Financ. Res. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"126484","DOI":"10.1016\/j.physa.2021.126484","article-title":"Using transfer entropy to measure information flows between cryptocurrencies","volume":"586","author":"Assaf","year":"2022","journal-title":"Physica A Stat. Mech. App."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"102556","DOI":"10.1016\/j.frl.2021.102556","article-title":"Information sharing among cryptocurrencies: Evidence from mutual information and approximate entropy during COVID-19","volume":"47","author":"Assaf","year":"2022","journal-title":"Finan. Rese. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Garc\u00eda-Medina, A., and Gonz\u00e1lez Far\u00edas, G. (2020). Transfer entropy as a variable selection methodology of cryptocurrencies in the framework of a high dimensional predictive model. PLoS ONE, 15.","DOI":"10.1371\/journal.pone.0227269"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1086\/296072","article-title":"On the Estimation of Security Price Volatilities from Historical Data","volume":"53","author":"Garman","year":"1980","journal-title":"J. Bus"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1086\/296071","article-title":"The Extreme Value Method for Estimating the Variance of the Rate of Return","volume":"53","author":"Parkinson","year":"1980","journal-title":"J. Bus"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1214\/aoap\/1177005835","article-title":"Estimating Variance from High, Low and Closing Prices","volume":"1","author":"Rogers","year":"1991","journal-title":"Ann. Appl. Prob."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1086\/209650","article-title":"Drift-Independent Volatility Estimation Based on High, Low, Open, and Close Prices","volume":"73","author":"Yang","year":"2000","journal-title":"J. Bus"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1061\/TACEAT.0006518","article-title":"The long-term storage capacity of reservoirs","volume":"116","author":"Hurst","year":"1951","journal-title":"Trans. Amer. Soc.Civil. Eng."},{"key":"ref_49","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell. Syst. Tech. J."},{"key":"ref_51","first-page":"547","article-title":"On measures of entropy and information","volume":"1","year":"1961","journal-title":"Proc. Fourth. Berkeley. Symp. Math. Stat. Prob."},{"key":"ref_52","unstructured":"Vicente, R., and Wibral, M. (2014). Directed Information Measures in Neuroscience, Understanding Complex Systems, Springer-Verlag."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1016\/j.physa.2017.04.089","article-title":"Comparison of transfer entropy methods for financial time series","volume":"482","author":"He","year":"2017","journal-title":"Phys. A Stat. Mech. App."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"383","DOI":"10.2307\/2325486","article-title":"Efficient capital markets: A review of theory and empirical work","volume":"25","author":"Fama","year":"1970","journal-title":"J. Finan."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1086\/258792","article-title":"New methods in statistical economics","volume":"71","author":"Mandelbrot","year":"1963","journal-title":"J. Polit. Econ."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1750003","DOI":"10.1142\/S2010495217500038","article-title":"A Statistical Risk Assessment of Bitcoin and Its Extreme Tail Behavior","volume":"12","author":"Osterrieder","year":"2017","journal-title":"Ann. Fin. Econ."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"5950","DOI":"10.1080\/00036846.2018.1488076","article-title":"Some Stylized Facts of the Cryptocurrency Market","volume":"50","author":"Zhang","year":"2018","journal-title":"Appl. Econ."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Kaya Soylu, P., Okur, M., \u00c7at\u0131kka\u015f, \u00d6., and Altintig, Z.A. (2020). Long Memory in the Volatility of Selected Cryptocurrencies: Bitcoin, Ethereum and Ripple. J. Risk Financ. Manag., 13.","DOI":"10.3390\/jrfm13060107"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"52985309","DOI":"10.1080\/00036846.2020.1761942","article-title":"Long memory and efficiency of Bitcoin under heavy tails","volume":"52","author":"Wu","year":"2020","journal-title":"Appl. Econ."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.chaos.2017.12.018","article-title":"Long-range memory, distributional variation and randomness of bitcoin volatility","volume":"107","author":"Lahmiri","year":"2018","journal-title":"Chaos Soli. Fract"},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Rambaccussing, D., and Mazibas, M. (2020). True versus Spurious Long Memory in Cryptocurrencies. J. Risk Financ. Manag., 13.","DOI":"10.3390\/jrfm13090186"},{"key":"ref_62","unstructured":"Peters, E.E. (1994). Fractal Market Analysis\u2014Applying Chaos Theory to Investment and Economics, John Wiley & Sons."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"200863","DOI":"10.1098\/rsos.200863","article-title":"Information-theoretic measures for nonlinear causality detection: Application to social media sentiment and cryptocurrency prices","volume":"7","author":"Keskin","year":"2020","journal-title":"R. Soc. Open Sci."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/j.physa.2018.10.048","article-title":"Group transfer entropy with an application to cryptocurrencies","volume":"516","author":"Dimpfl","year":"2019","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"109725","DOI":"10.1016\/j.econlet.2021.109725","article-title":"Are cryptocurrencies becoming more interconnected?","volume":"199","author":"Aslanidisa","year":"2021","journal-title":"Econ. Lett."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"14","DOI":"10.37394\/23207.2020.17.3","article-title":"Bankruptcu Risk Assessment Measures of Polish SMEs","volume":"17","author":"Chlodnicka","year":"2020","journal-title":"Wseas Trans. Busi. Econ."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Moradi, M., Appolloni, A., Zimon, G., Tarighi, H., and Kamali, M. (2021). Macroeconomic Factors and Stock Price Crash Risk: Do Managers Withhold Bad News in the Crisis-Ridden Iran Market?. Sustainability, 13.","DOI":"10.3390\/su13073688"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1410\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:45:35Z","timestamp":1760143535000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,2]]},"references-count":67,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["e24101410"],"URL":"https:\/\/doi.org\/10.3390\/e24101410","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,2]]}}}