Bitcoin’s multifractal influence: deciphering the relationship with conventional and renewable energy markets

Malik, Ayesha Rasool; Aslam, Faheem; Ferreira, Paulo Article Bitcoin’s multifractal influence: deciphering the relationship with conventional and renewable energy markets Cogent Economics & Finance Provided in Cooperation with: Taylor & Francis Group Suggested Citation: Malik, Ayesha Rasool; Aslam, Faheem; Ferreira, Paulo (2024) : Bitcoin’s multifractal influence: deciphering the relationship with conventional and renewable energy markets, Cogent Economics & Finance, ISSN 2332-2039, Taylor & Francis, Abingdon, Vol. 12, Iss. 1, pp. 1-25, https://doi.org/10.1080/23322039.2024.2395413 This Version is available at: https://hdl.handle.net/10419/321579 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Cogent Economics & Finance ISSN: 2332-2039 (Online) Journal homepage: www.tandfonline.com/journals/oaef20 Bitcoin’s multifractal influence: deciphering the relationship with conventional and renewable energy markets Ayesha Rasool Malik, Faheem Aslam & Paulo Ferreira To cite this article: Ayesha Rasool Malik, Faheem Aslam & Paulo Ferreira (2024) Bitcoin’s multifractal influence: deciphering the relationship with conventional and renewable energy markets, Cogent Economics & Finance, 12:1, 2395413, DOI: 10.1080/23322039.2024.2395413 To link to this article: https://doi.org/10.1080/23322039.2024.2395413 © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group Published online: 23 Aug 2024. Submit your article to this journal Article views: 707 View related articles View Crossmark data Citing articles: 1 View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=oaef20 FINANCIAL ECONOMICS | RESEARCH ARTICLE Bitcoin’s multifractal influence: deciphering the relationship with conventional and renewable energy markets Ayesha Rasool Malik a , Faheem Aslam a,b,c and Paulo Ferreira c,d,e a Department of Management Sciences, COMSATS University, Islamabad, Pakistan; b School of Business Administration, Al Akhawayan University, Ifrane, Morocco; c VALORIZA - Research Center for Endogenous Resource Valorization, Portalegre, Portugal; d Instituto Polit ecnico de Portalegre, Portalegre, Portugal; e CEFAGE-UE, IIFA, Universidade de  Evora,  Evora, Portugal ABSTRACT The annual electricity consumption of cryptocurrency mining has witnessed significant growth in recent years, fueled by an increase in market participation and the escalating complexity of the mining process. This has led to carbon emissions that exceed those generated by several developed nations. The growing impact of global warming and rising environmental concerns has brought increased scrutiny to Bitcoin’s energy consumption, particularly its potential to influence prices in unforeseen ways. This study investigates multifractal behavior in the cross-correlation of the Cambridge Bitcoin Electricity Consumption Index (CBECI) with both conventional and renewable energy prices using the Multifractal Detrended Cross-Correlation Analysis (MFDCCA) method. For renewable energy, we considered WilderHill Clean Energy, S&P Global Eco, S&P Global Clean Energy, OMX Solar Energy, and OMX Renewable Energy Index. For conventional energy, we considered the daily prices of WTI crude oil, Brent oil, heating oil, Newcastle coal, and natural gas. The daily price data range from 2 April 2013, to 29 August 2023, encompassing 1709 observations. Additionally, we employed a rolling window analysis to uncover the time-varying dynamics in the cross-correlations and persistence levels between Bitcoin electricity consumption and energy prices. The findings reveal the existence of a cross-correlation between the CBECI and energy markets. Overall, the CBECI exhibits a persistent cross-correlation with both energy markets; however, it is more persistent in the fossil fuel market, specifically in the coal market. These findings suggest the incorporation of dynamic changes in the CBECI in portfolio management for effective risk management strategies. IMPACT STATEMENT The results of this study, which analyses multifractal cross-correlation of Bitcoin Electricity Consumption Index (CBECI) with both conventional and renewable energy prices, reveal the existence of a cross-correlation between variables under analysis. Results are relevant, suggesting the possibility to use CBECI in portfolio management, but also gives information for policymakers relevant, for example, to issues like global and environmental concerns. ARTICLE HISTORY Received 3 May 2024 Revised 29 July 2024 Accepted 15 August 2024 KEYWORDS Bitcoin; cryptocurrency mining; energy consumption; renewable energy; conventional energy; multifractal analysis; MFDCCA SUBJECTS Finance; Environmental Economics; Economics 1. Introduction Energy consumption associated with Bitcoin mining has become a key concern. Miners compete to validate transactions using a computationally intensive process known as proof of work, which requires significant amounts of electricity. According to the Cambridge Center for Alternative Finance (CCAF), a single Bitcoin transaction consumes an estimated 1200 kWh, roughly equivalent to the energy used in 100,000 VISA transactions. This translates to Bitcoin mining, which accounts for approximately 0.52% of the global energy consumption. To put this into perspective, Bitcoin mining utilizes the same amount of electricity annually as the entire state of Washington. This exceeds the annual electricity consumption CONTACT Paulo Ferreira [email protected] VALORIZA - Research Center for Endogenous Resource Valorization, Portalegre, Portugal Instituto Polit ecnico de Portalegre, Portalegre, Portugal CEFAGE-UE, IIFA, Universidade de  Evora,  Evora, Portugal ß2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. COGENT ECONOMICS & FINANCE 2024, VOL. 12, NO. 1, 2395413 https://doi.org/10.1080/23322039.2024.2395413 of countries such as the United Arab Emirates, Philippines, Finland, and Belgium. Further highlighting the scale, the annual energy consumption of Bitcoin could power all water-boiling kettles in the UK for 26 years (CCAF, 2023). Consequently, 65 megatons of carbon dioxide enters the atmosphere, which is comparable to the emissions of Greece. Nevertheless, the mining industry generates $13 billion globally, with projections indicating a further increase in the foreseeable future. Currently, the estimated revenue is roughly $1.6 billion. Consequently, Bitcoin mining has an impact on the energy market owing to heightened energy demand and a possible increase in energy prices. It has an impact on the fossil fuel market in several ways, including greater demand for these fuels, competition for energy sources, and the interplay of market dynamics (supply and demand). It may also affect the renewable energy market by increasing demand, increasing investment incentives for renewable energy, reducing curtailment, and enhancing efficiency. Cryptocurrency is a digital currency that was first launched in 2008 by Satoshi Nakamoto. Since its introduction, cryptocurrency has become a common transaction method and has attracted considerable interest (Cui & Maghyereh, 2022). The cryptocurrency market has rapidly evolved into an essential component of the global financial industry and an emerging class of assets (Corbet et al., 2018a;2018b). Moreover, cryptocurrency is used by 1.2 billion hyperinflation victims worldwide. In Kenya, Vietnam, Venezuela, and Brazil, the expense and complexity of legacy banking systems, unstable monetary governance, and currency devaluation have forced many individuals to use cryptocurrencies to save, send, and receive remittances; buy basic items; and conduct everyday business. Furthermore, it has been welcomed more in developed countries than in developing ones (Sharma et al., 2021). According to Capgemini (2021), the volume of non-cash transactions has increased to 700 billion; by 2025, non-cash transactions will account for 25% of all transactions worldwide. With 10.4 million bitcoin users, Brazil leads Latin America (Cheikosman, 2022). Comparatively, India overtook the South American region and now accounts for the highest cryptocurrency use for the year 2023 (Triple-A, 2023). Initial studies focused primarily on the technical aspects of bitcoin (Holub & Johnson, 2018). Other scholarly investigations have focused on comprehending the essence of bitcoin (Bariviera et al., 2017;Ji et al., 2019). According to Nakamoto (2008), Bitcoin emerged in response to the global financial crisis in 2008 as a decentralized substitute for conventional (fiat) currency systems, which were scrutinized by central banks. Recently, financial transactions have been replaced with cashless transactions as the preferred method of payment. Compared to other cryptocurrencies, Bitcoin is the least risky one (Gkillas & Katsiampa, 2018). It is a significant element of the Fourth Industrial Revolution in the domain of finance and blockchain technology, which is used to carry out cryptocurrency transactions. (Su et al., 2020). Blockchain is a secure digital ledger that facilitates the storage of data and information (Bondarev, 2020). Multiple studies have investigated the unique dimensions of Bitcoin, including its volatility (Mokni, 2021; Takaishi, 2020), predictability (Adcock & Gradojevic, 2019), and others (Vukovic et al., 2021). In a recent study, Hern andez S anchez et al. (2024) highlighted that the regulation of cryptocurrencies in Spain was confusing and difficult to understand, and tax agencies should provide more information and resources. Bitcoin is not only used as a currency but is also regarded as a speculative asset (Corbet et al., 2018a, 2018b; Yermack, 2015) and is intended to establish an electronic peer-to-peer payment system (Zhang & Balogun, 2018). Bitcoin leads all other cryptocurrencies by value and market capitalization ($444 billion (coinmarketcap, 2023)). It maintains its position as the market leader, and the prices of other cryptocurrencies are reliant on Bitcoin price fluctuations (Corbet, Lucey, et al., 2018). The Commodity Exchange Act (CEA) has recognized Bitcoin as a commodity since 2015, and because it is viewed as a commodity, it is affected by other market commodities and macroeconomic factors (Jalal et al., 2020). Furthermore, cryptocurrency functions in a decentralized manner, unlike fiat money, which is governed by regulatory authorities. As a result, it is highly volatile relative to the fiat currency. El Salvador became the first recognized nation for Bitcoin mining (N a~ nez Alonso et al., 2021). The discussion of Bitcoin and its energy use in mining operations is in the developing stage. Recently, several studies have highlighted the escalating energy crisis associated with Bitcoin mining (Chari et al., 2019; Das & Dutta, 2020; de Vries, 2018; Huynh et al., 2022;K € ufeoglu & € Ozkuran, 2019), which has led to intense debate on its long-term sustainability (De Vries & Stoll, 2021). Several studies, including Badea and Mungiu-Pupazan (2021) and Vranken (2017), point out that Bitcoin mining expenses 2 A. RASOOL MALIK, F. ASLAM, AND P. FERREIRA are a major element in determining whether a Bitcoin miner will be profitable. According to Das and Dutta (2020), mining profits are inversely proportional to bitcoin’s energy usage. As Bitcoin prices have an immediate impact on mining and, consequently, energy consumption, it is difficult to predict the amount of energy that will be used in Bitcoin mining in the future (K€ ufeoglu & € Ozkuran, 2019). Regarding asset prices, Huynh et al. (2022) documented a relationship between Bitcoin energy consumption and its returns and a higher directional impact from Bitcoin trading volumes to its energy consumption. Similarly, cryptocurrency energy-usage showed a sustained and significant impact on the performance of companies listed in the energy sector (Corbet et al., 2021). Proponents of cryptocurrencies believe that they will eventually replace fiat currency, while opponents dismiss such hopes because of their volatility and the negative anthropogenic impacts of the mining process. The massive demand for electricity was initially met by using fossil fuels as the primary source of energy. As of 2021, 70% of all crypto-mining has taken place in China (Jiang et al., 2021), due of the country’s access to inexpensive energy sources (coal, etc.). However, China has outlawed cryptocurrency mining because of concerns about its impact on the environment. Consequently, miners are trying to move to countries such as Kazakhstan and the U.S., which are more dependent on fossil fuels for electricity generation. According to Gallersd€ orfer et al. (2020), non-Bitcoin cryptocurrencies account for approximately 33% of the total power use in the cryptocurrency industry. Furthermore, a single mining transaction consumes energy equal to the weekly electricity use of a typical home. Extremely high levels of carbon dioxide are released into the environment when substantial amounts of electricity are generated to support the mining process, using fossil fuels as an energy source. This contributes not only to existing pollution but also to climate change and shows the seriousness of the situation. Therefore, concerns about the environment and carbon emissions have grown in response to the soaring demand for power (Sarkodie et al., 2022). As a consequence of anthropogenic activities and the production of greenhouse gases, the world is experiencing catastrophic effects in the form of climate change and global warming. Global temperature has already risen by one degree Celsius. These devastating effects include floods, tsunamis, melting of glaciers, water shortages, crop failure, and pollution. Countries are unable to stop the rising global temperature and its terrible repercussions despite adopting strict laws and regulations and accepting agreements such as the Kyoto Protocol, which requires participants to cut their greenhouse gas emissions. The Paris Climate Change Agreement served as an inspiration for the Crypto Climate Accord, which seeks to achieve zero net carbon emissions from electricity use across all crypto-related activities by 2030. However, the Bitcoin mining process worsens an already disastrous situation. Bitcoin mining generates almost 35.95 million metric tons of carbon dioxide annually, or about as much as New Zealand consumes electricity (Kumar, 2021). The mining process involves the use of both conventional and nonconventional energy sources. Conventional energy, such as fossil fuels, adds fuel to the fire by emitting massive amounts of carbon dioxide into the environment during mining operations. In January 2022, conventional energy sources accounted for 62 percent of the total energy combination for Bitcoin mining, whereas renewable energy sources made up only 38 percent, according to the Cambridge Centre for Alternative Finance (CCAF, 2022). The Cambridge Bitcoin Electricity Consumption Index (CBECI), created and maintained by the Cambridge Centre for Alternative Finance (CCAF), monitors the electricity consumption of Bitcoin mining facilities. The Bitcoin Mining Council Report for 2022 reports that miners are gradually switching from conventional to renewable energy sources to achieve an optimal energy balance, with renewable energy accounting for 59.4% of all energy used in Bitcoin mining (BMC, 2022). According to recent research by Neumueller (2022), the use of renewable energy in the crypto-industry, particularly Bitcoin, has shown minimal progress during the year. Miners are aware of the increasingly gloomy forecasts of the huge amounts of carbon emissions that destroy the environment. However, they are reluctant to shift from conventional to sustainable energy sources, mainly because fossil fuels are cheaper than renewable energy sources. As electricity consumption for the mining process grows over time, increasing carbon emissions pose a great threat to the environment and the sustainability of cryptocurrency use. However, as in El Salvador, Bitcoin miners can use geothermal energy as a substitute for conventional energy. Geothermal energy may emerge as the next most important energy source for Bitcoin mining. It may also assist in lowering the carbon impact of Bitcoin mining, and because it originates from hot springs or volcanoes, it may be used continuously all year round (Mnif et al., 2021). The International Energy Agency COGENT ECONOMICS & FINANCE 3 (IEA) documented that a decrease in energy demand results in reduced energy prices. Hence, more mining, leading to a higher demand for energy, will place an upward pressure on energy prices. However, efficient energy sources can lower the energy prices. The price of coal and electricity usage for the mining process have a strong time-varying association with each other (Sibande et al., 2022). The multifractal characteristics of the cross-correlation between the CBECI and its energy sources are ambiguous. Therefore, this study adds to the literature with four main contributions that distinguish it from previous studies. First, we investigate the multifractal cross-correlation between the Cambridge Bitcoin Electricity Consumption Index (CBECI) and the energy sources used in the mining process under the fractal market hypothesis. Second, for a detailed comparison, the prices of five fossil fuel energy sources and renewable energy sources are used: the WTI Crude Oil Index (CL), Brent Crude Index (BRN), Heating Oil Index (HO), Newcastle Coal Index (NEWC), and Natural gas Index (NG), as well as five renewable energy indices: WilderHill Clean Energy Index (ECOTR), S&P Global Eco Index (SPGTECOL), S&P Global Clean Energy Index (SPGTCED), NASDAQ OMX Solar Energy Index (GRNSOLAR), and NASDAQ OMX Wind Energy Index (GRNWIND). Third, to reveal the inner dynamics, a robust technique of Multifractal Detrended Cross-Correlation Analysis (MFDCCA), a combination of DCCA and MFDFA, was employed. Finally, the dynamic changes in the cross-correlations between CBECI and the two energy sources were quantified using a rolling window MFDCCA analysis. Furthermore, daily changes in the persistence level of cross-correlations were documented. The findings of this study have important academic and managerial implications for investors (i.e., investment strategies), policy makers (i.e., mining policies), and academia (i.e., nonlinear modeling). In summary, the hypothesis under study is to assess the existence of multifractal cross-correlation among CBECI and the different assets under analysis. The main findings reveal the existence of a multifractal cross-correlation between the CBECI and energy markets. Although the CBECI exhibits a persistent cross-correlation with both energy markets, it is more persistent in the fossil fuel market, specifically in the coal market. The rest of the paper comprises four sections. Section 2 discusses the related literature on the efficient market hypothesis (EMH), Fractal Market Hypothesis (FMH) and the linkage between cryptocurrency and energy markets including the price behavior, followed by the data and empirical methodology in Section 3.Section 4 presents and discusses the empirical findings, and the concluding remarks appear in Section 5. 2. Literature review The foundation of the financial markets is based on the Efficient Market Hypothesis proposed by Fama (1970). However, financial markets do not always remain efficient, and several studies have shown that they have some flaws, including volatility clustering (Xiao and Wang (2021)), fat tails (Telli and Chen (2020)), multifractality (Aslam et al. (2022)), chaos (Li et al. (2020)) and long-term association (Kononovicius & Ruseckas, 2019). Therefore, fractal models are used to counter these discrepancies because they better reflect realistic market behavior. Consequently, the Fractal Market Hypothesis (FMH) was developed by Peters in 1994, based on fractals, and was developed by Peters (1994). The FMH is provided as an alternative to the traditional EMH according to a study by Ayg€ oren and Umut (2023) and was used to explain the behavior of financial markets in terms of market efficiency (Milos¸ et al., 2020). Several researchers have studied the relationship between cryptocurrencies and the factors necessary for the mining process. According to Stoll et al. (2019), cryptocurrency mining consumes an increasing proportion of the world’s power, which is increasing significantly over time. The incentive for cryptocurrency miners to increase production in response to increased cryptocurrency prices has increased power usage. This increase in energy demand can be associated with ownership verification and transactions, as reported by Gallersd€ orfer et al. (2020). According to Huynh et al. (2022), the Bitcoin trading volume may increase long-term energy usage. This indicates that the electricity consumption for mining may exceed current projections, which is also supported by de Vries (2018). Furthermore, Masanet et al. (2019) evaluated the rise in power growth due to Bitcoin mining and predicted that Bitcoin’s popularity would result in inevitable changes in the global temperature. However, higher costs of energy resources, such as surging oil prices, may impede miners’ability to achieve the breakeven point, which is detrimental to the growth of the Bitcoin market and therefore affects Bitcoin price (K€ ufeoglu & € Ozkuran, 2019). 4 A. RASOOL MALIK, F. ASLAM, AND P. FERREIRA There is no consensus on the relationship between Bitcoin prices and energy prices. For instance, Bastian-Pinto et al. (2021) find no association between electricity costs and crypto prices. Similarly, Bitcoin and the cryptocurrency market are becoming more associated with stock markets, whereas the cryptocurrency market’s association with energy, oil, and electricity becomes significant after Bitcoin mining is impacted by a future worldwide power crisis (Huynh et al., 2022). Cryptocurrencies seem to have varying correlations with energy commodities, such as natural gas, crude oil, and heating oil (Ji et al., 2019; Maiti, 2022). However, Bitcoin is linked to the energy required for mining, although this link is chaotic and nonlinear in nature. Mining activities are significantly influenced by fluctuations in the Bitcoin value. As a result, power consumption reacts to fluctuations in the price of bitcoin (K€ ufeoglu & € Ozkuran, 2019). Moreover, there is long-term confirmation of the volatility-generating impact of Bitcoin on fossil fuels and renewable energy equities (Symitsi & Chalvatzis, 2018). Rehman and Kang (2021) established that lead and lag correlations exist among bitcoin, crude oil, and natural gas. Consequently, Bitcoin miners identified a relationship between the value of Bitcoin and the value of energy, indicating that as Bitcoin prices rise, energy prices will (Meiryani et al., 2022). China’s ban on the highly energy-exhausting sector of crypto-mining is a significant improvement in the global environment. However, profit-oriented miners may opt to shift towards regions with less environmentally friendly energy structures, thus countering the efforts of environmental measures. China’s renewable resources, such as hydropower, added approximately 15% to Bitcoin power production. A thorough study of the coal market by Lin and Li (2015) documented that the mining sector is experiencing a smooth shift, with the use of sustainable energy sources continuing to restrict the market for coal and other fossil fuels owing to technical difficulties and relatively high prices. Furthermore, Neumueller (2022) showed that Bitcoin struggled to increase its use of renewable energy in 2021–2022, making only modest growth in its energy mix. The type of energy employed in crypto-mining operations has a notable impact on the environment, and renewable energy facilitates the shift to a sustainable, reliable, and economically feasible energy alternative according to the International Renewable Energy Agency (IRENA, 2019). Current work on the relationship between cryptocurrency and sustainable energy has mostly concentrated on calculating the energy requirements necessary to maintain cryptocurrency marketplaces (Chari et al., 2019; Krause & Tolaymat, 2018; Stoll et al., 2019). According to the prevailing literature, Bitcoin marketplaces mostly depend on non-renewable energy sources that threaten the environment (Shojaei et al., 2021; Stoll et al., 2019). In contrast, several studies have claimed a correlation between cryptocurrency marketplaces and the renewable energy sector (Corbet et al., 2021; Polemis & Tsionas, 2021). According to Suazo (2021), the emphasis should be on using clean energy in the mining of Bitcoin instead of concentrating on the amount of energy Bitcoin consumes. The carbon footprint and anthropogenic effects of cryptocurrencies may be reduced by transitioning to clean energy. In a recent study, Aslam et al. (2023) applied multifractal detrended cross-correlation analysis (MFDCCA) and documented a cross-correlation of the carbon market with Brent crude oil, Richards Bay coal (RBC), UK Natural gas, and the FTSE350 Electricity index. Furthermore, the adoption of renewable power and environmentally friendly mining technology can help reduce Bitcoin’s carbon footprint. There is a lack of connectivity between clean energy and cryptocurrencies, implying that clean energy might be used as a hedging and diversification strategy for digital currencies in the coming years (Ren & Lucey, 2022). 3. Data and methodology 3.1. Data description This study assesses multifractal behavior in the cross-correlation between the Cambridge Bitcoin Electricity Consumption Index (CBECI) and five fossil fuel and renewable energy indices using the robust technique of MFDCCA analysis. Changes in the global environment and fluctuating energy prices have forced cryptocurrency miners to optimize the energy mix to meet the ever-increasing demand for electricity. According to the Cambridge Centre for Alternative Finance CCAF (2022), the percentage of fossil fuels used in the mining process has dropped slightly from 65% in 2021 to 62.4% in 2022. Coal usage declined from 47% to 37% and mining became more reliant on gas. Furthermore, the proportion of renewable sources, classified as hydro, solar, wind, and nuclear, in the energy mix has increased slightly from 35% to 38% in 2022 compared COGENT ECONOMICS & FINANCE 5 to 2021. However, the share of hydropower dropped from 20% to 15%, mainly because of the ban on mining in China, which was conducted either through hydropower or coal. This study uses daily data from the Cambridge Bitcoin Electricity Consumption Index (CBECI). This index, maintained by the University of Cambridge, provides valuable insights into the daily estimates of power consumption and energy mix usage associated with Bitcoin mining worldwide. The CBECI provides the daily Bitcoin network power demand maintained by the Cambridge Center for Alternative Finance (CCAF). As electricity consumption cannot be estimated exactly, the index provides a hypothetical range of energy consumption within which lies the best estimate of real consumption. 1 The CBECI data were taken from the Cambridge Centre of Alternative Finance (https://ccaf.io/cbnsi/cbeci), while daily energy prices were collected from LSEG (https://www.lseg.com/en/data-analytics) from 2 April 2013, to 29 August 2023. The sample data of fossil fuels include the WTI Crude Oil Index (CL), Brent Crude Index (BRN), Heating Oil Index (HO), Newcastle Coal Index (NEWC), and Natural Gas Index (NG); renewable energy includes the Wilder Hill Clean Energy Index (ECOTR), S&P Global Eco Index (SPGTECOL), S&P Global Clean Energy Index (SPGTCED), NASDAQ OMX Solar Energy Index (GRNSOLAR), and NASDAQ OMX Wind Energy Index (GRNWIND). The fossil fuels and renewable energy indices employed are the energy sources most commonly used to generate electricity for cryptocurrency mining. A list of energy markets, their symbols, and descriptions are provided in Table 1. For MFDCCA, the dates of the energy market indices are matched with the dates of the CBECI. 3.2. Multifractal detrended cross-correlation analysis (MFDCCA) Since the development of multifractal detrended cross-correlation analysis (MF-DCCA, or MF-DXA) by Zhou (2008) to reveal the multifractal features of two cross-correlated signals, DCCA and MF-DCCA have been widely discussed and used (Aslam et al., 2022; Jafari et al., 2007; Z.-Q. Jiang & Zhou, 2011; Kristoufek, 2011; Zou & Zhang, 2020). Recent studies attempted to reveal the inner dynamics of such cross-correlations which exist in many simultaneously recorded time series (Aslam et al., 2022; Podobnik et al., 2009; Shi et al., 2020; Wa¸torek et al., 2019; Xiong et al., 2018; Zhao & Cui, 2021). In a recent study, Dhifaoui (2022) proved that the detrended cross-correlation methods remained robust in the presence of any outliers and can be applied to any financial time series. The summarized algorithm of the MFDCCA by Zhou (2008) is explained as follows: First, two time series ðxiÞ  and ðyiÞ  of the same length are considered, where Nis the total number of observations of both time series, then the MF-DCCA method can be summarized as follow Step 1: Construct the profile We begin by constructing the signal profiles of Xi ðÞ and Yi ðÞ as follows: Xi ðÞ ¼X j i¼1 xi−x ðÞ ,i¼1, 2, 3::::::,N, (1) Table 1. List of energy markets. S.no CBECI & Energy Markets Symbol Description 1 Cambridge Bitcoin Electricity Consumption Index (CBECI) Daily updates of Bitcoin power demand Fossil fuels 2 WTI Crude Oil Index (CL) WTI Crude Oil Spot 3 Brent Crude Oil Index (BRN) Brent Crude Spot 4 Heating Oil Index (HO) Heating Oil Spot 5 Newcastle Coal Index (NEWC) Coal Spot 6 Natural Gas Index (NG) Natural Gas Spot Renewable energy 7 Nasdaq OMX Solar Energy Index (GRNSOLAR) Solar power generation companies traded on NASDAQ 8 Nasdaq OMX Wind Energy Index (GRNWIND) Wind power generation companies traded on NASDAQ 9 WilderHill Clean Energy Index (ECOTR) Clean energy market leaders traded on NYSE 10 S&P Global Clean Energy Index (SPGTCED) Top 100 companies in global clean energy industry from developed and emerging markets 11 S&P Gloal Eco Index (SPGTECOL) Forty largest companies from ecology related industries Data range: 2 April 2013–29 August 2023; Number of observations: 1709. 6 A. RASOOL MALIK, F. ASLAM, AND P. FERREIRA Yi ðÞ ¼X j i¼1 yt−y ðÞ ,i¼1, 2, 3::::::,N, (2) where xand yare the average values of ðxiÞ  and ðyiÞ  : Step 2: the constructed signal profiles were divided into Xi ðÞ and Yi ðÞ into Ns¼int N sboxes of the same length s:Considering the possibility of Nbeing a non-multiple of sfrom the end of the sample, as proposed by Kantelhardt (2011), resulting in 2Nssegments obtained altogether. Step 3: the local trends XvðiÞand YvðiÞof each element is computed, and the variance for each v¼ 1, 2, ...,2Nsis calculated as F2s,v ðÞ ¼1 sX s i¼1 X:v−1 ðÞ sþi ½ −XvðiÞ jj :Y:v−1 ðÞ sþi ½ −Yv:ðiÞ jj (3) for each segment v¼1, 2, ...,Nsand F2s, v ðÞ ¼1 sX s j¼1 X:N−v−Ns ðÞ :sþi ½ −Xvi ðÞ : Y:N−v:−Ns ðÞ sþi ½ −Yv:ðiÞj (4) for v¼Ns,...,2Ns: Step 4: By averaging over all segments the q-order fluctuation function is obtained through the equation below. Fqs ðÞ¼1 2NsX 2Ns v¼1 F2:s,v ðÞ  q=2 () 1=q (5) This equation is considered when q 6¼ 0, and when q¼0 the equation is given below: F0s ðÞ¼exp:1 4NsX 2Ns v¼1 ln F2:s,v ðÞ hi () (6) Now, we obtain the standard DCCA at q¼2 with Fqs ðÞ as an increasing function of s. Step 5: Finally, the multi-scaling behavior of fluctuation is detected through the examination of log-log plots of Fqs ðÞagainst sfor each q: Fqs ðÞsHxy:q ðÞ (7) Here, the power law association between the two nonlinear time series is shown by the scaling exponent Hxy q ðÞ , which expresses the extent of Fqs ðÞagainst the increase in the s scale. Both the IF time series ðxiÞ  and ðyiÞ  are identical, and MFDCCA indicates a special case of MFDFA. As suggested by O swie¸cimka, et al. (2014), the scales are selected according to the series length Nwhile the maximum scale is taken as Smax <N5: In the case of a stationary time series, the Generalized Hurst Exponent Hxy 2 ðÞ , is identical to the classic Hurst Exponent h(Kristoufek, 2011). Moreover, a Hxy 2 ðÞ ¼0:5 shows there is no cross-correlation between the two-time series. However, when Hxy 2 ðÞwas greater than 0.5, cross-correlation persisted between the two-time series, indicating a positive correlation between them. Furthermore, Hxy 2 ðÞless than 0.5 shows anti-persistence and negative cross-correlation. According to Yuan et al. (2009), the multifractality degree DHis defined as DH¼Hmax:q ðÞ −Hmin:ðqÞ(8) The multifractality degree represents the strength of the multifractality. The greater the number of DHvalues, the stronger is the degree of multifractality. Furthermore, a particular value of Hxy q ðÞmay reflect the degree of multifractality along with the succeeding cross-correlations. The following can be used to determine the degree of multifractality using the Legendre transform. COGENT ECONOMICS & FINANCE 7 have a normal distribution. The test results indicate statistical significance at the 1% level, leading to the rejection of the null hypothesis for the goodness-of-fit test, which states that the data have a normal distribution. Lastly, the Augmented Dickey Fuller test is used to assess the stationarity of the dataset and to detect the presence of unit roots within the data. The results confirm that CBECI and energy market index data are stationary at a significance level of 1%. 4.4. Multifractal detrended cross-correlation analysis To examine the existence of cross-correlation between CBECI and the energy markets, a robust technique of Multifractal Detrended Cross-Correlation Analysis was employed. For this purpose, log-log plots Figure 5. Daily percentage returns of Renewable Energy Market Indices. 14 A. RASOOL MALIK, F. ASLAM, AND P. FERREIRA of the fluctuation function were examined with an increasing order of qfrom −5toþ5. Figure 6 illustrates plots of the fluctuation function of LogðFxyqðsÞÞ against s(time length) for each q:The left panel displays the plots for fossil fuel indices and the right panel displays the plots for renewable energy indices. The rising linear trend confirms the power-law association between CBECI and both energy markets, that is, fossil fuels and renewable energy. Moreover, the scaling exponent Hxy q ðÞindicates the power law association, which is also called the cross-correlation exponent. The scaling exponent is the slope of the fluctuation function plot. Figure 7 shows the hq plotted against the increasing order of q, and the results are shown in Table 3. The Hurst exponent shows a diminishing pattern with increasing qorders in all markets. Furthermore, the values decreased with increasing q, as shown in Table 3. This diminishing pattern confirms the multifractality between CBECI and energy markets. For example, the value of NEWC is 0.8291 at q¼–5 which dropped to 0.5569 at q¼þ5, and the value of GRNSOLAR is 0.7663, dropping to 0.4953 when the value of qincreases. A similar declining trend is evident for all the variables. The lowest Hurst exponent value was 0.3955 for SPGTECOL. The values of Hxy q ðÞwhen q¼2 indicate the persistence level between the CBECI and energy markets. Accordingly, all the markets, that is, fossil fuels and renewable energy, have qvalues greater than 0.5, exhibiting persistent cross-correlation with CBECI, except SPGTECOL, which has a score less than 0.5, thereby exhibiting anti-persistent cross-correlation with CBECI. This shows the highest persistence level between coal (NEWC) and CBECI in the case of fossil fuel markets, and the highest persistence level between solar energy (GRNSOLAR) and CBECI in the case of renewable markets. The values of Hxy q ðÞchange with the increasing order of q, illustrating that the cross-correlation is multifractal. Furthermore, the values of Hxy q ðÞare greater when q<0 than q>0 indicating that modest fluctuations in the CBECI and energy markets are persistently cross-correlated. According to Kristoufek (2011), Hxy 2 ðÞ >0:5 represent that the series are cross-persistent and a change (positive/negative) of Dxtythas a higher probability of another positive (negative) value of Dxtþ1ytþ1: Likewise, long-range cross correlation means that both time series exhibit long memory in their own series’lag and a change in one variable has a higher probability of being followed by a significant change in another variable (Podobnik & Stanley, 2008; Yuan et al., 2012). In this context, Higher CBECI likely ties to some energy price movements. This could be due to higher energy demand for mining activities, production costs rising with energy prices, or trade policies affecting energy sources. For further confirmation, Table 4 summarizes the multifractal indices, where the Hurst average values range between 0.5–0.7; the DHindicates the strength of multifractality, the greater DH, the larger the multifractality and the market is more inefficient (Figure 8). The values were greater than zero, indicating that the cross-correlations exhibited multifractal patterns. In addition, the highest multifractality was found between CBECI-NEWC, with a value of 0.2722 in the fossil fuel market. The highest multifractality was found in CBECI-SPGTECOL, with a score of 0.3396 in the renewable energy market. To verify the value of DH,Da was obtained, which illustrates the spectrum width and is used to approximate the multifractal strength, as shown in Figure 4. The broader the spectrum, the stronger the multifractality. Hence, SPGTECOL has a broader spectrum width than the other indices in the renewable energy market, Table 2. Summary statistics of CBECI and energy market indices. CBECI & Energy Markets Mean Maximum Minimum SD Skewness Kurtosis Jarque-Bera test ADF CBECI 0.2698 33.9538 −49.4235 3.8781 −2.7529 35.0160 8123.0005 −10.0507 CL −0.1044 37.6623 −305.9661 6.9460 −33.5922 1425.8274 1038.0009 −12.0069 BRN 0.0204 21.0186 −24.4036 2.4401 −0.3534 12.9519 1274.0003 −11.0797 HO 0.0264 15.0127 −21.9277 2.3076 −0.4078 9.5328 9658.0005 −12.0052 NEWC 0.0449 40.5751 −35.1085 2.2699 0.4563 85.2832 7836.0001 −11.0492 NG 0.0434 21.8943 −16.5282 3.4291 0.2149 3.2915 446.0092 −12.0763 GRNSOLAR 0.1091 12.8075 −17.5787 2.0837 −0.2529 5.4041 275.0054 −11.0991 GRNWIND 0.0717 9.5860 −12.4383 1.6767 −0.1665 4.6450 1560.0008 −12.0055 ECOTR 0.0468 33.7928 −21.2982 2.4162 0.7682 20.0306 1379.0005 −11.0245 SPGTCED 0.0463 11.6647 −11.7477 1.4642 −0.2015 7.8130 1025.0009 −10.0625 SPGTECOL 0.0037 8.1570 −10.1472 1.0540 −0.4346 11.4648 950.0009 −11.0776 denotes 1% level of significance. COGENT ECONOMICS & FINANCE 15 illustrating stronger multifractality with CBECI. Moreover, NEWC has a broader spectrum width when the fossil fuel market is considered. A stronger multifractal behavior indicates more inefficiencies in the market, resulting in EMH failure. The AI represents the asymmetric position of the energy market indices. The scores show that CL, NEWC, NG, SPGTCED, and SPGTECOL are left-skewed when the AI is less than 1. In contrast, BRN, HO, Figure 6. Log-Log plots of Fluctuation function for CBECI & Energy Market Indices.Note: x-axis denotes s(days); y-axis denotes Log(Fxyq (S)). 16 A. RASOOL MALIK, F. ASLAM, AND P. FERREIRA GRNSOLAR, GRNWIND, and ECOTR are right-skewed when AI is greater than 1. In addition, Crepresents the extent of truncation. CL, NEWC, NG, SPGTCED, and SPGTECOL have left-side truncation because Cis greater than 1; however, BRN, HO, GRNSOLAR, GRNWIND, and ECOTR have right-side truncation because Cis less than 1. The results for Cwere similar to those of AI:The indices with left (right) truncation Figure 7. Generalized Hurst exponent for CBECI and Energy Market Indices.Note: x-axis denotes q; y-axis denotes hq. COGENT ECONOMICS & FINANCE 17 suggest the presence of stronger (weaker) singularities, and the cross-correlation shows a multifractal structure that remains unaffected by local fluctuations of small (large) magnitudes (Ihlen, 2012). 4.5. Rolling window analysis To investigate the time-varying changes in the cross-correlation between the CBECI and energy markets, rolling window MFDCCA was employed. Figure 9 represents the rolling window analysis of 1000 trading days in energy markets. In the case of fossil fuel market indices (Panel A), NEWC remained above all other indices except in 2018 and then reverted back to a high position with an increasing trend in the first quarter of 2022, while other indices declined. For Renewable Energy Indices (Panel B), the SPGTCED is clearly above all indices during the sample period. The index returns of both markets are above 0.5 throughout the period, indicating a persistent cross-correlation with CBECI. In accordance with the Delta H values, NEWC had the highest multifractality with CBECI in the fossil fuel market. Coal is the cheapest energy source available to miners, resulting in reduced mining costs and improved mining profits. Consequently, coal accounts for a significant portion of the energy source used in Bitcoin mining, and an increase in Bitcoin power usage will also result in increased coal use in the future. SPGTCED had the highest persistent multifractality with CBECI. The SPGTCED comprises the highest number of global clean energy-related firms in developed and emerging markets. This can be attributed to investors’increased interest in clean energy investments over time, environmental consciousness, and regulatory changes. Overall, the behavior of the fossil fuel and renewable energy markets is the same as that of CBECI. All markets are persistent with CBECI, although the level of persistence changes throughout the sample period. This shows that an increase in Bitcoin power consumption will result in an increase in the fossil fuels and renewable energy market in the future. For instance, market persistence has declined with CBECI in 2021. This could be Table 3. Hurst exponent for CBECI and Energy Markets ranging over qe(-5 to 5). Q CL BRN HO NEWC NG GRNSOLAR GRNWIND ECOTR SPGTCED SPGTECOL −5 0.7490 0.7234 0.7255 0.8291 0.6740 0.7663 0.7249 0.7265 0.7260 0.7351 −4.5 0.7392 0.7136 0.7148 0.8209 0.6660 0.7536 0.7142 0.7166 0.7173 0.7237 −4 0.7286 0.7030 0.7032 0.8118 0.6570 0.7395 0.7027 0.7057 0.7076 0.7110 −3.5 0.7172 0.6918 0.6909 0.8017 0.6480 0.7239 0.6901 0.6934 0.6969 0.6970 −3 0.7049 0.6798 0.6777 0.7906 0.6380 0.7068 0.6767 0.6798 0.6850 0.6816 −2.5 0.6918 0.6672 0.6639 0.7784 0.6280 0.6885 0.6624 0.6648 0.6718 0.6647 −2 0.6780 0.6539 0.6494 0.7651 0.6170 0.6694 0.6473 0.6485 0.6574 0.6465 −1.5 0.6637 0.6403 0.6344 0.7509 0.6060 0.6500 0.6318 0.6312 0.6417 0.6272 −1 0.6488 0.6262 0.6190 0.7358 0.5950 0.6311 0.6161 0.6132 0.6250 0.6070 −0.5 0.6337 0.6120 0.6034 0.7202 0.5840 0.6133 0.6004 0.5948 0.6073 0.5865 0 0.6181 0.5974 0.5875 0.7033 0.5720 0.5970 0.5849 0.5759 0.5885 0.5653 0.5 0.6032 0.5834 0.5724 0.6878 0.5610 0.5818 0.5702 0.5583 0.5707 0.5452 1 0.5881 0.5694 0.5573 0.6716 0.5490 0.5683 0.5561 0.5407 0.5523 0.5251 1.5 0.5734 0.5556 0.5429 0.6554 0.5370 0.5562 0.5428 0.5238 0.5342 0.5056 2 0.5592 0.5424 0.5290 0.6394 0.5240 0.5452 0.5304 0.5078 0.5169 0.4869 2.5 0.5455 0.5296 0.5159 0.6239 0.5120 0.5351 0.5188 0.4928 0.5004 0.4690 3 0.5324 0.5175 0.5036 0.6089 0.4990 0.5259 0.5079 0.4789 0.4848 0.4521 3.5 0.5200 0.5060 0.4921 0.5945 0.4870 0.5174 0.4978 0.4661 0.4704 0.4363 4 0.5082 0.4951 0.4813 0.5811 0.4760 0.5095 0.4883 0.4544 0.4570 0.4216 4.5 0.4971 0.4849 0.4713 0.5685 0.4650 0.5021 0.4796 0.4438 0.4447 0.4080 5 0.4867 0.4754 0.4620 0.5569 0.4540 0.4953 0.4715 0.4341 0.4334 0.3955 Table 4. Summary of multifractal indices. Pair Hurst Average Delta H Delta Alpha AI C CBECI-CL 0.6184 0.2623 0.4441 0.9686 1.0612 CBECI-BRN 0.5985 0.2480 0.4217 1.0315 0.9694 CBECI-HO 0.5904 0.2635 0.4435 1.1252 0.8692 CBECI-NEWC 0.6998 0.2722 0.4504 0.7765 1.4146 CBECI-NG 0.5690 0.2198 0.3890 0.8276 1.2381 CBECI-GRNSOLAR 0.6132 0.2710 0.4465 1.8097 0.5354 CBECI-GRNWIND 0.5912 0.2534 0.4226 1.2867 0.7570 CBECI-ECOTR 0.5786 0.2924 0.4688 1.0440 0.9798 CBECI-SPGTCED 0.5852 0.2926 0.4726 0.8243 1.2989 CBECI-SPGTECOL 0.5662 0.3396 0.5547 0.9580 1.0965 18 A. RASOOL MALIK, F. ASLAM, AND P. FERREIRA linked to the ban on cryptocurrency mining activities imposed by China in 2021 (Charlie, 2021). Furthermore, the Chinese prohibition has led to an increase in the use of fossil fuels for mining. Miners relocate to countries such as the United States and Kazakhstan, resulting in increased reliance Figure 8. Multifractal Spectrum Width for CBECI and Energy Market Indices.Note: x-axis denotes a; y-axis denotes f(a). COGENT ECONOMICS & FINANCE 19 on fossil fuel resources from the United States rather than hydropower resources formerly provided by China (Digiconomist, 2022). Moreover, there was a stronger correlation between bitcoin mining and renewable energy use before the emergence of COVID-19. However, this correlation has diminished over time, with a growing association between fossil fuels and bitcoin mining (Kumari et al., 2023). 5. Concluding remarks The purpose of this study is to examine the multifractal behavior of cross-correlation between the Cambridge Bitcoin Electricity Consumption Index (CBECI) and energy markets, that is, fossil fuel and renewable energy markets. The results of the MFDCCA confirm the existence of cross-correlation between the CBECI and energy markets. In addition, a power law association exists between the series. The Generalized Hurst Exponent Hxy 2 ðÞ is used to explore the degree of persistence, indicating that all indices of fossil fuel and renewable energy markets are persistent with the CBECI, except SPGTECOL, which has an anti-persistent association with the CBECI. In the case of the fossil fuel market, NEWC has the highest degree of multifractality with CBECI, as indicated by the DHvalue, while SPGTECOL has the highest degree of multifractality when the renewable energy market is considered. Likewise, Da the spectrum width, shows that NEWC has the largest spectrum width, indicating greater multifractality, whereas SPGTECOL has the largest spectrum width. These results reaffirm existing studies that conclude a causal association between bitcoin mining and conventional and nonconventional energy sources. Bitcoin electricity usage has a significant influence on the energy sector (Corbet et al., 2021). Additionally, the Cscore showed that CL, NEWC, NG, SPGTCED, and SPGTECOL had left-sided truncation and BRN, HO, GRNSOLAR, GRNWIND, and ECOTR had right-sided truncation. These results are consistent with the results of AI:The indices with left (right) truncation suggest the presence of stronger (weaker) singularities, and the cross-correlation exhibits a multifractal structure that remains unaffected by local fluctuations of small (large) magnitudes. The presence of long-range cross-correlations suggests that previous adjustments to the CBECI values may enhance energy price forecasting. Finally, compared to large fluctuations, the cross-correlation behavior of small fluctuations is still more persistent, indicating that short-term shocks have a longerlasting effect on the market than do large shocks. Ultimately, this means that investors and fund managers must exercise caution when considering the energy market as a shelter during volatile times. The Figure 9. Dynamic changes of Fossil Fuels and Renewable Energy Market Indices. 20 A. RASOOL MALIK, F. ASLAM, AND P. FERREIRA conclusions of this study have a number of significant implications for researchers, investors, and legislators. The nonlinear dependence of the cross-correlations indicates that changes to the CBECI will affect the volatility and return of energy prices. Investors can also use CBECI-related portfolio management techniques by considering energy prices in response to changes in the CBECI. For academia, common linear models, such as OLS, are not suitable for assessing the cross-correlation between variables, such as CBECI and energy markets. Finally, time-varying dynamic changes were observed between energy consumption and CBECI. This implies that portfolio managers should consider these dynamic changes, because their relationships and persistence vary with the situation. The study focused on the power consumption index of Bitcoin, but other cryptocurrencies with growing popularity and market capitalization can be used instead of Bitcoin, such as the Ethereum electricity index. To reveal microstructures, intraday data can be used in future studies. Authors’contributions Ayesha Rasool and Paulo Ferreira contributed to the study’s conception and design. Material preparation, data collection, and analysis were performed by Paulo Ferreira and Faheem Aslam. Ayesha Rasool wrote the first draft of the manuscript while Faheem Aslam supervised the process. Finally, all authors reviewed, edited, and commented on previous versions of the manuscript. All authors read and approved the final manuscript. Disclosure statement No potential competing interest was reported by the authors. Notes 1. Due to the decentralised nature of the network, CBCIE calculation is based on several assumptions including hypothetical lower-bound (floor) and upper-bound (ceiling) estimates. These two boundaries encompass a bestguess estimate, a more accurate indication of the actual power demand. For details, visit https://ccaf.io/cbnsi/ cbeci/methodology. 2. The detail documentation is available at https://www.rdocumentation.org/packages/MFDFA/versions/1.1/topics/ MFDFA. Funding Paulo Ferreira acknowledges financial support from Fundac¸~ ao para a Ci^ encia e a Tecnologia (grant UIDB/05064/ 2020). About the authors Ayesha Rasool Malik, a BS Accounting and Finance Graduate, advanced her expertise with master’s in finance from COMSATS University, Islamabad. Her research focused on energy prices, fintech and cryptocurrencies. With her strong background in financial analysis, she is dedicated to leveraging my skills to drive impactful financial strategies and solutions. Dr. Faheem Aslam is an Associate Professor of Finance at Business School, Al Akhawayn University, Morocco. He earned his Master's and Ph.D. Degrees from Hanyang University Business School, Seoul, South Korea. His work extends beyond theoretical models and equations, making a significant impact on social wellbeing. He leverages cutting-edge finance and economics techniques, such as financial networks, multifractality, volatility spillovers, and machine learning, to identify market inefficiencies, sources of financial instability, changes in financial networks, and the spillover effects of black swan events. Paulo Ferreira is an Economist and holds a PhD in Management. With several peer reviewed published papers in international reviews, as well as technical books, his research is focused on the analysis of financial markets, although with research also in other research fields. Actually he is Full Professor and Pro-President for Research, Innovation and Technology Transfer at the Polytechnic Portalegre University, but with teaching experience in other higher education institutions. He is researcher at VALORIZA - Research Center for Endogenous Resource Valorization (Portalegre). COGENT ECONOMICS & FINANCE 21 ORCID Faheem Aslam http://orcid.org/0000-0001-7308-096X Paulo Ferreira http://orcid.org/0000-0003-1951-889X Data availability statementof data The data set used in the study is available on request to the corresponding author ( [email protected]). References Adcock, R., & Gradojevic, N. (2019). Non-fundamental, non-parametric Bitcoin forecasting. Physica A: Statistical Mechanics and Its Applications,531, 121727. https://doi.org/10.1016/j.physa.2019.121727 Aslam, F., Ali, I., Amjad, F., Ali, H., & Irfan, I. (2023). 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