Are GARCH and DCC values of 10 cryptocurrencies affected by COVID-19?

Yan, Kejia; Yan, Huqin; Gupta, Rakesh Article Are GARCH and DCC values of 10 cryptocurrencies affected by COVID-19? Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Yan, Kejia; Yan, Huqin; Gupta, Rakesh (2022) : Are GARCH and DCC values of 10 cryptocurrencies affected by COVID-19?, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 15, Iss. 3, pp. 1-25, https://doi.org/10.3390/jrfm15030113 This Version is available at: https://hdl.handle.net/10419/258836 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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Journal of Risk and Financial Management 15: 113. https://doi.org/10.3390/ jrfm15030113 Academic Editor: Thanasis Stengos Received: 11 January 2022 Accepted: 28 February 2022 Published: 1 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Journal of Risk and Financial Management Article Are GARCH and DCC Values of 10 Cryptocurrencies Affected by COVID-19? Kejia Yan 1,*, Huqin Yan 2and Rakesh Gupta 1 1Department of Accounting, Finance and Economics, Griffith University, Nathan 4111, Australia; r[email protected] 2Xiamen National Accounting Institute, Xiamen 361005, China; [email protected] *Correspondence: [email protected] Abstract: This paper examines the dynamic conditional correlations among 10 cryptocurrencies and the possibility of hedging investment strategies among multiple cryptocurrencies over the period affected by COVID-19 from 2017 to 2022. After studying the relationship between Bitcoin, Ethereum, and the other eight cryptocurrencies, four main results were obtained in this paper: first, from the pre-COVID-19 period to the COVID-19 period, almost all of the cryptocurrencies’ return growth rates increased, and COVID-19 had a positive effect on the returns of cryptocurrencies. Second, all of the cryptocurrencies’ return indices had features of volatility clustering and memory persistence in the long run; from pre-COVID-19 to COVID-19, these cryptocurrencies’ GARCH values decreased, but the correlations among the varying GARCH values increased. Third, the varying correlations between the return indices of Bitcoin, Ethereum, and the other cryptocurrencies were very strong; from pre-COVID-19 to COVID-19, the average dynamic correlations between Bitcoin and the others increased. Fourth, Tether can be used as a hedge cryptocurrency against the other cryptocurrencies as COVID-19 enhanced its hedging feature. Keywords: cryptocurrencies; dynamic conditional correlation; generalized autoregressive conditional heteroscedasticity; COVID-19 pandemic 1. Introduction Cryptocurrencies have become a popular economic and financial topic. When a cryptocurrency is defined as a digital currency, it is very different from a fiat currency because cryptocurrencies are not issued by any judicial body (IFRSIC 2019). Generally, a cryptocurrency does not have any original intrinsic value; however, it has an extrinsic value that is totally dependent on the expectation that future investors will be willing to pay for it in the cryptocurrency market. Many researchers believe that cryptocurrencies will become a mainstream financial instrument in future global financial markets in addition to common stocks, commodities, and precious metals or foreign exchange instruments (Soylu et al. 2020). The risk involved in cryptocurrencies is obvious. Because of their higher volatilities (Caporale and Zekokh 2019;Siswantoro et al. 2020), cryptocurrencies cannot be accepted as a common standard for measuring the relative worth of goods and services, even though many researchers admit that cryptocurrencies are a medium of exchange. Accordingly, some researchers do not accept that cryptocurrencies are currencies; they prefer to maintain that cryptocurrencies behave more like an investment instrument than a currency (˙ Içellio˘glu and Öner 2019). However, some researchers have suggested that the higher volatilities may be Granger causes of the higher liquidities. B˛edowska-Sójka et al. (2019) verified the relationship between the volatility and liquidity of cryptocurrencies by investigating the daily and weekly data of the 12 most popular cryptocurrencies during the period of 2013–2017 J. Risk Financial Manag. 2022,15, 113. https://doi.org/10.3390/jrfm15030113 https://www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2022,15, 113 2 of 25 and found that the cryptocurrencies with higher volatilities are Granger causes of high liquidities and can attract investors and lead to higher interest from investors. In terms of changes in the value of cryptocurrencies, this volatility seems to have intensified during the COVID-19 pandemic (Siswantoro et al. 2020). As the year 2022 progresses, the epidemic has slowed down in many countries as vaccines become more widely available. Simultaneously, the dynamic conditional correlation (DCC) changes in cryptocurrencies before and after COVID-19 have become a major point of contention for investors. From a portfolio perspective, if the dynamic conditional correlation among cryptocurrencies increases, then holding multiple cryptocurrencies at the same time will increase the portfolio risk. Conversely, if the dynamic conditional correlation among cryptocurrencies decreases, then there is an opportunity to hedge risk. This study fills the research gap by identifying the volatility of cryptocurrencies and the dynamic conditional correlation among different cryptocurrencies since the beginning of the COVID-19 pandemic. After empirical analysis using sample data from 8 September 2017 to 14 February 2022 and studying the relationship between Bitcoin, Ethereum, and the other eight cryptocurrencies, including Tether, Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, we confirmed that from the pre-COVID-19 period to the COVID-19 period almost all of the 10 cryptocurrencies’ return growth rates increased. Moreover, the researched 10 cryptocurrencies’ return indices had features of volatility clustering or memory persistence in the long run, and all of the 10 cryptocurrencies’ GARCH values decreased from the pre- COVID-19 period to the COVID-19 period. The correlations among the varying GARCH time series of the 10 cryptocurrencies were quite high, and the correlations among the varying GARCH time series of the 10 cryptocurrencies increased from the pre-COVID-19 period to the COVID-19 period. This study also found that, except for Tether, the varying correlations between the return indices of Bitcoin, Ethereum, and the other cryptocurrencies were very strong; the correlations between the return indices of Ethereum and the other cryptocurrencies were higher than for Bitcoin and the others. Except for Tether, the average DCC values between Bitcoin, Ethereum, and the other cryptocurrencies increased; since the COVID-19 pandemic began, the correlations among the 10 cryptocurrencies’ return indices, except for Tether’s, have become higher than before. Finally, the correlations between the return indices of Tether and the other nine cryptocurrencies were negative, and Tether can be a hedge cryptocurrency for the other cryptocurrencies. 2. Literature Review The volatilities of cryptocurrencies exhibit the characteristics of significant time varying and clustering. When large fluctuations in returns tend to be followed by relatively large fluctuations, smaller fluctuations in returns tend to be followed by relatively small fluctuations. This is accompanied by the realization that the bad news has a much bigger impact on the cryptocurrency market volatility than the good news (Palamalai et al. 2020). The characteristics of long memory or persistence in volatility have also been discussed by some researchers. Abakah et al. (2020) analyzed the volatility persistence in 12 main cryptocurrencies, including Bitcoin, Bitshare, Bytecoin, Dash, Ethereum, Litecoin, Monero, Nem, Ripple, Siacoin, Stellar, and Tether, by considering the possibility of structural breaks and found that the volatilities represented in both absolute and squared returns display long memory features, but after accounting for structural breaks, the degree of persistence in the cryptocurrency market is reduced. Different cryptocurrencies have different volatility clustering structures and different spillover patterns, and the market price bubbles are associated with the volatilities of cryptocurrencies. Bitcoin, Ethereum, and Litecoin are the most relevant cryptocurrencies in general, serving as connection hubs for the linking of many other cryptocurrencies. However, their roles have been challenged lately, potentially owing to the increased usage of other cryptocurrencies over time. Sensoy et al. (2020) examined the high-frequency return and volatility of major cryptocurrencies, including Bitcoin, Bitcoin Cash, Dash, EOS, Ethereum, Ethereum Classic, Iota, Litecoin, OmiseGO, Monero, Ripple, and Zcash, J. Risk Financial Manag. 2022,15, 113 3 of 25 using the 15-min time series from 10 August 2017 to 23 June 2018 and found that volatility clustering structures of the returns are distinct among the different cryptocurrencies. Enoksen et al. (2020) also studied which variables can predict bubbles in the prices of eight major cryptocurrencies by using the data from 27 December 2013 to 25 February 2019 and found that the multiple bubble periods were located in 2017 and early 2018. They mentioned that the cryptocurrencies’ higher volatilities, trading volume, and transactions were positively associated with the presence of bubbles across the cryptocurrencies. In fact, the relationship between cryptocurrencies and COVID-19 is a very topical subject (Iqbal et al. 2021). García-Medina and Hernández C (2020) investigated the effects of the financial turbulence of 2020 on the cryptocurrency market by considering the hourly price and volume of transactions from December 2019 to April 2020, finding that the volatility clustering increased dramatically in March 2020. Corbet et al. (2020) analyzed the largest cryptocurrencies’ time-varying correlations by using daily data from 2019 to 2020 and found that the cryptocurrencies’ returns were significantly influenced by the negative sentiment around COVID-19 during 2020, and the trading volumes and returns of cryptocurrencies significantly increased. James et al. (2021) examined the distribution extremities and erratic behaviors of 51 cryptocurrencies using a structural break method to evaluate the impact of COVID-19 on the cryptocurrency market when the time period was divided into the pre-COVID-19 period from 30 June 2018 to 31 December 2019 and the COVID-19 period from 1 January 2020 to 24 June 2020. They found that during the pre-COVID-19 period, the cryptocurrency market exhibited considerable homogeneity with respect to the structural breaks in volatility, whereas during the COVID-19 period the homogeneity of volatility was disrupted by the pandemic and the self-similarity was reduced. Since COVID-19 began in January 2020, and after the volatility clustering increased dramatically in March 2020 (García-Medina and Hernández C 2020), the trading volumes and returns of cryptocurrencies have significantly increased (Corbet et al. 2020), with an unexpected shift toward positive average return among the distribution extremities (James et al. 2021), and most cryptocurrencies absorbed the small shocks of COVID-19 by registering positive gains (Iqbal et al. 2021). In terms of financial strategies, after analyzing the correlations and the characteristics of hedging among cryptocurrencies, some scholars announced that the correlations between Bitcoin and the other cryptocurrencies are strong, and no hedging abilities exist among cryptocurrencies (Kyriazis et al. 2019). Ciaian et al. (2018) examined the interdependencies between Bitcoin and the other 16 alternative cryptocurrencies in the short run and long run by applying time series analytical mechanisms for the daily data during 2013–2016 and found that the correlations between the prices of Bitcoin and the other 16 alternative cryptocurrencies are indeed significantly strong in both the short run and the long run. However, it is worth examining whether such an opportunity is arising in the post- COVID-19 pandemic period. To illustrate, the unique characteristics of Tether have been isolated from the other cryptocurrencies, and some researchers have proven that Tether has different characteristics from the other cryptocurrencies. Tether exhibits unusually docile profiles for extreme behaviors (James et al. 2021). Dilek et al. (2020) studied how the changes in gold and oil prices affected the daily price movements of various cryptocurrencies, including Bitcoin, Ethereum, Tether, Litecon, and EOS, for the period from 1 August 2017 to 3 April 2019 and found that there were no cointegration relationships between the cryptocurrencies and the macroeconomic factors, including gold and oil prices, except for Tether. Huynh et al. (2020) investigated the volatility spillover effects among 14 cryptocurrencies by using the daily dataset covering the period from April 2013 to April 2019, finding that only Tether had average negative return while all the other cryptocurrencies exhibited positive values. From the above literature review, we found deficits in the research on cryptocurrencies that we needed to pay more attention to in our research. J. Risk Financial Manag. 2022,15, 113 4 of 25 Firstly, although many researchers have studied the varying volatilities of cryptocurrencies (Palamalai et al. 2020;Abakah et al. 2020;Enoksen et al. 2020) and the impacts of COVID-19 on the cryptocurrencies’ volatility (Ardia et al. 2019;García-Medina and Hernández C 2020;James et al. 2021), the average decreasing features from the pre-COVID-19 period to the COVID-19 period have not been summarized by anyone, and we will discuss this issue. Actually, volatility clustering is a basic in-sample characteristic of cryptocurrencies (Ardia et al. 2019); based on a GARCH(1,1) model, the characteristics of clustering, spillover, asymmetry, and long memory in volatility share the same feature, which is dependent on the coefficient of GARCH. If we do not consider the reasons for the time series’ volatility, we can find the characteristics of volatility by investigating the models of GARCH. Secondly, although the volatility connectedness of cryptocurrencies has been discussed by some researchers (Le et al. 2021), the structure changes between the pre-COVID-19 and COVID-19 periods have not been discussed. Because the sample observations of the previous researchers for the COVID-19 period are not enough, it is necessary to reassess the result. Thirdly, even though some researchers have discussed the time-varying correlations (Corbet et al. 2020) and returns (Iqbal et al. 2021), seldom have researchers discussed how COVID-19 impacts on the cryptocurrencies’ correlation and return together. For cryptocurrencies, higher positive correlations will represent the homogeneity among them, but low or negative correlations will represent the hedging abilities among them. The dynamic conditional correlation (DCC) models are usually used to represent the dynamic relationship for a normality time series. It is necessary to analyze the correlations of the cryptocurrencies dynamically. Finally, even though some researchers have proven that no hedging abilities exist among the cryptocurrencies (Kyriazis et al. 2019), it is still necessary to discuss the characteristics of Tether (Dilek et al. 2020;Huynh et al. 2020;James et al. 2021). We will discuss if Tether can be a hedge cryptocurrency for the other cryptocurrencies. 3. Data For this paper, the sample data were collected from the world’s largest open access cryptocurrencies database. The prices of the cryptocurrencies are represented by US Dollars (USD), and the data period covers 8 September 2017 to 14 February 2022, which contains 1621 daily observations. The abbreviations BTC, ETH, TET, XRP, LTC, BCH, XLM, XMR, EOS, and NEO are used to represent the 10 top cryptocurrencies, which are ranked on the cryptocurrency market list between 1st and 58th within all 10,707 cryptocurrencies (Investing 2022). Table 1lists the ranking, price, market cap, and 24 h trading volume of the 10 cryptocurrencies in the global market on 18 February 2022. To compare the impacts of COVID-19 on the return indices of cryptocurrencies between the periods before and after COVID-19, the full time period was divided into a pre-COVID- 19 period from 8 September 2017 to 31 December 2019 with 845 observations and a COVID- 19 period from 1 January 2020 to 18 February 2022 with 776 observations. Statistically, by 18 February 2022, the total number of cryptocurrencies in the world had reached 10,707, the total market capitalization had reached USD 1850 billion, and the 24-hour exchange volume had reached USD 56.906 billion. Comparatively, the total market capitalization of these 10 cryptocurrencies reached USD 1261.318 billion with 68.18% of the total cryptocurrency market, and the 24-h exchange volume reached USD 33.76 billion with 59.32% of the world total cryptocurrency market. These 10 top cryptocurrencies represented the characteristics of the total cryptocurrency market. Each cryptocurrency’s market ranking was based on the ratio of the market cap in the whole market. It was clear that Bitcoin, Ethereum, and Tether were the three highest ranking cryptocurrencies, with market cap ratios of 41.69, 18.77, and 4.26%. J. Risk Financial Manag. 2022,15, 113 5 of 25 Table 1. Ranking, price, market cap, and 24 h trading volume of the 10 cryptocurrencies in the global market on 18 February 2022. Ranking Cryptocurrency Abbreviation Price (USD) Market Capitalization 24 h Trading Volume Market Cap (USD) Ratio (%) 24 h Volume (USD) Ratio (%) 1 Bitcoin BTC 40782 771.32B 41.69% 17.460000B 30.68% 2 Ethereum ETH 2899.3 347.288B 18.77% 13.210000B 23.21% 3 Tether (USDT) TET 1.009 78.73B 4.26% 2.616600B 4.60% 6 Ripple XRP 0.78793 37.74B 2.04% 0.139910B 0.25% 20 Litecoin LTC 117 8.15B 0.44% 0.115070B 0.20% 28 Bitcoin Cash BCH 313.8 5.96B 0.32% 0.077834B 0.14% 31 Stellar XLM 0.20498 5.11B 0.28% 0.034826B 0.06% 45 Monero XMR 164.38 2.98B 0.16% 0.028847B 0.05% 48 EOS EOS 2.3559 2.31B 0.12% 0.051469B 0.09% 58 NEO NEO 24.59 1.73B 0.09% 0.020801B 0.04% Sum of the 10 cryptocurrencies 1261.318B 68.18% 33.755357B 59.32% Total 10,707 cryptocurrencies 1850B 100.00% 56.906B 100.00% Note: B represents USD 1 billion. Comparatively, similar to the market cap, Bitcoin had the highest ratio of 24 h trading volume in the total market. The 24 h trading volume ratio of Bitcoin was as high as 30.68%, which was much greater than the 24 h trading volume ratios of Ethereum at 23.21% and Tether at 4.60%. They were the three most important cryptocurrencies in the market. As opposed to the stock market, cryptocurrencies are exchanged every day in the cryptocurrency market. All the continuous daily data were collected every day during the sample observation period. EViews and MATLAB software were used for the empirical analysis. Assume that the time variable is t∈{1, 2, . . . , T} . The terminal point T is the total number of daily observations. When the variable i∈ {BTC,ETH,XRP,TET,LTC,BCH,EOS, XLM,XMR,NEO} , for the ith cryptocurrency, if the variable pi,t is the daily closing price at the time point t, then the return index variable ri,twill be ri,t=pi,t pi,t−1,when t =2,3, . . . , T. (1) Assume ri,t= 1, when t= 1. The curve of the return index ri,t will fluctuate around the line of one. The 10 cryptocurrencies’ return indices will be the basic variable of our research. 4. Methodology 4.1. Ljung–Box Autocorrelation Test Assume that the variable rt is an independent and identically distributed (IID) time series, and the variable ρl represents the autocorrelation coefficient (AC) between the variable rt and its lagged variable rt−l when l= 1,2, . . . , m .Box and Pierce (1970) defined a statistic variable Q∗(m) to test if a time series rt is not an autocorrelation series. The null hypothesis is H0:ρ1=. . . =ρm=0; the alternative hypothesis is Ha:ρl6=0. J. Risk Financial Manag. 2022,15, 113 6 of 25 Ljung and Box (1978) changed the statistic variable Q∗(m) to a new statistic variable Q(m) . The conditions of denying the null hypothesis H0 are Q(m)>χ2 α(m) , the probability confidence interval is 1 −α, when the statistic variable Q(m)is defined as Q(m)=T(T+2) m ∑ l=1 ρ2 l T−l, lim T→∞Q(m)∼χ2 α(m). (2) 4.2. ADF Unit Root Test The unit root test is aimed at testing if a time series is stationary. The general model of AR(p) is rt=ϕ0+ϕ1rt−1+ϕ2rt−2+· · · +ϕprt−p+at. (3) If the time series rt is an autocorrelation, then the parameters of ϕ1 , . . . , ϕp are partial autocorrelations (PAC). The time series rt is stationary if and only if that model AR(p) has characteristics when p= 1 then |ϕ1|< 1, and E(rt)=µt , E(at)= 0, Var(rt)=Var(at)= σ2 a<∞ , Cov(at,at−s)= 0 for any lag order s= 1,2, . . . , t− 1. Inversely, if ϕ1= 1, then the time series rtis not stationary. The Dickey–Fuller (DF) test (Dickey and Fuller 1979) and the augmented Dickey– Fuller (ADF) test (Dickey and Fuller 1981) are usually used as the stationary test or unit root test. When the null hypothesis is H0 : θ=ϕ1− 1 = 0, then there are three ADF test models, such as Model 3 : ∆rt=α+βt+θrt−1+ p ∑ l=1 γl∆rt−l+ηt, (4) Model 2 : ∆rt=α+θrt−1+ p ∑ l=1 γl∆rt−l+ηt, (5) Model 1 : ∆rt=θrt−1+ p ∑ l=1 γl∆rt−l+ηt. (6) When the ADF test is applied to the time series rt , it is better to apply Model 3 first, then Model 2 and Model 1 (Wooldridge 2000). If the level time series is stationary, then it will be a variable of I(0) ; if a 1-order or 2-order difference time series is stationary, then it will be a variable of I(1)or I(2). 4.3. AR(1)-GARCH(1,1) Model The generalized auto-regressive conditional heteroscedasticity (GARCH) model is a method to deal with the single-variable time series. Assume variable ri,t represents a return time series of the ith cryptocurrency at any time t ; Fi,t−1 represents the information set when the discrete time set is t= 1,2, . . . , T . Then, the autoregressive (AR) model AR(1) can be defined as ri,t=ϕi,0 +ϕi,1ri,t−1+ai,t,ri,t|Fi,t−1∼Nµi,t,σ2 i,t. (7) The expected values of ri,tand ai,tare Et−1(ri,t|Fi,t−1) = µi,t=ϕi,0 +ϕi,1ri,t−1,ai,t|Fi,t−1∼N0, σ2 i,t. (8) If the parameter ωi> 0, αi≥ 0, βi≥ 0, and αi+βi < 1, then the GARCH(1, 1) model can be defined as σ2 i,t=ωi+αia2 i,t−1+βiσ2 i,t−1,ai,t=σi,tεi,t,εi,t|Fi,t−1∼N(0,1). (9) J. Risk Financial Manag. 2022,15, 113 7 of 25 If the long static variance is σ2 i,a, then it will satisfy the condition of σ2 i,a=ωi 1−(αi+βi). (10) Generally, if a time series is a partial autocorrelation, it is good to choose the AR(p) model; the residual item can be used in the GARCH model. Inversely, if a time series is not an autocorrelation, some researchers prefer to directly use both the absolute and the squared values of the returns in the GARCH model (Abakah et al. 2020). If a time series is not an autocorrelation but the AR(p) model is chosen and the residual item is used in the GARCH model, it does not matter for the GARCH model. 4.4. DCC(1,1) Model Assume there are two time series, ri,t , rj,t , after applying the two AR(1) models, there are two residual time variables, ai,t , aj,t . For these two residual variables, assume variable Ht represents the dynamic conditional covariance matrix, variable Rt represents the dynamic conditional correlation (DCC) matrix, variable Dt represents the diagonal matrix from the covariance matrix Ht , and the variable D−1 t represents the inverse matrix of the matrix Dt. Then the relationship between the matrices of Ht,Rt,Dt, and D−1 tis Ht=DtRtDt,Rt=D−1 tHtD−1 t. (11) After using the two GARCH(1, 1) models, there are two normalized residual variables, εi,t , εj,t . For these two residual variables, assume variable Qt represents the covariance matrix, variable Ct represents the correlation matrix, variable Gt represents the diagonal matrix of the covariance matrix Qt , and variable Q−1 t represents the inverse matrix of the matrix Qt. The relationships between the matrices of Qt,Ct,Gtand G−1 tare Qt=GtCtGt,Ct=G−1 tQtG−1 t. (12) For a 2-order matrix Rt,Ht, and Qt, assume Rt=ρi,tρij,t ρji,tρj,t,Ht=σi,tσij,t σji,tσj,t,Qt=qi,tqij,t qji,tqj,t, (13) σij,t=σi,tρij,tσj,t,σji,t=σi,tρji,tσj,t. (14) Because both matrix Rt and Ct are isomorphisms (Engle 1982,2002), when Rt=Ct , then the covariance matrix can be represented as Qt=GtCtGt=GtRtGt=GtD−1 tHtD−1 tGt. (15) Byusing therelationshipsof ai,t=σi,tεi,t and aj,t=σj,tεj,t from AR(1) and GARCH(1,1) , then the DCC(1,1) model can be defined (Engle 1982,2002) as q2 i,t=(1−α−β)ρ2 i,0 +αε2 i,t−1+βq2 i,t−1, (16) q2 j,t=(1−α−β)ρ2 j,0 +αε2 j,t−1+βq2 j,t−1, (17) qij,t=(1−α−β)ρij,0 +αεi,t−1εj,t−1+βqij,t−1, (18) qji,t=(1−α−β)ρji,0 +αεj,t−1εi,t−1+βqji,t−1. (19) Here, the correlations of ρi,0,ρj,0,ρij,0,ρji,0 are static correlations, which are defined as ρi,0 =ρj,0 =1, ρij,0 =ρji,0 =ρεi,εj. (20) J. Risk Financial Manag. 2022,15, 113 8 of 25 Then, the dynamic conditional correlations can be defined as ρij,t=qij,t qi,tqj,t ,ρji,t=qji,t qj,tqi,t , where ρij,t=ρji,t. (21) Because the time variable t is considered, the correlation variables ρij,t and ρji,t are varying correlations. 4.5. Maximum Likelihood Estimation of Parameters The maximum likelihood estimation (MLE) is used to estimate the parameters of the models of AR(1), GARCH(1,1), and DCC(1,1). According to the suggestion of Engle (2002), the log-likelihood equation of MLE is defined (Engle 2002) as L=LVolatility +LCorrelation , which is based on Gaussian normal distribution’s probability density function. For estimating the parameters of the AR(1) and GARCH(1,1) models, Gaussian density function is stated as LVolatility = T ∑ t=1(−1 2" ln(2π)+lnσ2 i,t+a2 i,t σ2 i,t!+ ln(2π)+lnσ2 j,t+a2 j,t σ2 j,t!#). (22) For estimating the parameters of the DCC(1,1) models, the correlation method defined by Engle (2002) is stated as LCorrelation =− T ∑ t=1(ln(2π)+1 2lnq2 i,tq2 j,t−qij,tqji,t+1 2 q2 j,tε2 i,t−qji,tεi,t,εj,t−qij,tεi,t,εj,t+q2 i,tε2 j,t q2 i,tq2 j,t−qij,tqji,t!). (23) 5. Descriptive Statistics and Tests 5.1. Average Growth Rates of the 10 Cryptocurrencies for the Three Periods For the full period, there were nine cryptocurrencies that each had a positive return growth rate; the average return growth rates of Bitcoin, Ethereum, Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO were 0.2295, 0.2813, 0.3111, 0.2097, 0.2007, 0.4058, 0.1835, 0.3116, and 0.2299%, respectively; inversely, only Tether had a negative return growth rate as low as − 0.0006%. Stellar, EOS, and Ripple had the highest growth rates; Tether had the lowest growth rate. Table 2lists the descriptive statistics of the 10 cryptocurrencies’ return indices for the full period. Table 2. Descriptive statistics of the 10 cryptocurrencies’ return indices for the full period. Stats rBTC,trETH,trTET,trXRP,trLTC,trBCH,trXLM,trXMR,trEOS,trNEO,t Mean 1.0023 1.0028 1.0000 1.0031 1.0021 1.0020 1.0041 1.0018 1.0031 1.0023 Growth 0.2295% 0.2813% −0.0006% 0.3111% 0.2097% 0.2007% 0.4058% 0.1835% 0.3116% 0.2299% Median 1.0015 1.0015 1.0000 1.0001 0.9995 0.9988 1.0000 1.0026 1.0000 1.0010 Maximum 1.2255 1.2596 1.0352 1.8558 1.6106 1.5291 1.8977 1.4080 1.5618 1.6605 Minimum 0.6082 0.5545 0.9787 0.5822 0.6146 0.5501 0.6438 0.5854 0.5801 0.5996 Std. Dev. 0.0418 0.0527 0.0034 0.0705 0.0601 0.0699 0.0745 0.0558 0.0713 0.0701 Skewness −0.2508 −0.2809 0.7773 2.6610 1.0338 1.1259 2.6139 −0.1714 1.0374 0.8436 Kurtosis 10.23 8.77 22.64 28.64 15.55 14.29 27.45 10.46 11.95 11.87 Jarque–Bera 3544 2268 26,225 46,333 10,935 8947 42,217 3767 5696 5501 Probability 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Obs 1621 1621 1621 1621 1621 1621 1621 1621 1621 1621 For the pre-COVID-19 period, there were nine cryptocurrencies that each had a positive return growth rate; the average return growth rates of Bitcoin, Ethereum, Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO were 0.1523, 0.0338, 0.2064, 0.1154, 0.1482, J. Risk Financial Manag. 2022,15, 113 15 of 25 Table 9. Comparison of average GARCH values of the 10 cryptocurrencies’ return indices for the three periods. Return Index Pre_COVID-19 (1) COVID-19 (2) Full Period (3) (2)–(1) (2)–(3) σBTC,t0.041508 0.039749 0.040666 −0.001759 −0.000917 σETH,t0.051479 0.051481 0.051480 0.000002 0.000001 σTET,t0.004809 0.000730 0.002857 −0.004079 −0.002127 σXRP,t0.062983 0.064909 0.063905 0.001926 0.001004 σLTC,t0.058295 0.056452 0.057413 −0.001843 −0.000961 σBCH,t0.072249 0.062369 0.067519 −0.009880 −0.005150 σXLM,t0.073247 0.064930 0.069265 −0.008317 −0.004335 σXMR,t0.056321 0.053340 0.054894 −0.002981 −0.001554 σEOS,t0.073962 0.063003 0.068715 −0.010959 −0.005712 σNEO,t0.071897 0.063170 0.067719 −0.008727 −0.004549 During the COVID-19 period, the average GARCH value of Tether was 0.000730; however, the average GARCH values of the other nine cryptocurrencies were between 0.039749 and 0.064930. This meant that the volatility of Tether had less fluctuation than the other nine cryptocurrencies. Second, when comparing the GARCH values between both periods of pre-COVID-19 and COVID-19, it was proven that the GARCH values of 8 out of 10 cryptocurrencies, including Bitcoin, Tether, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, decreased from the pre-COVID-19 period to the COVID-19 period. From the pre-COVID-19 period to the COVID-19 period, the average GARCH values of Bitcoin, Tether, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO decreased in differences of − 0.001759, − 0.004079, − 0.001843, − 0.009880, − 0.008317, − 0.002981, − 0.010959, and −0.008727, respectively. The differences in the GARCH values of Ethereum and Ripple between the pre-COVID- 19 and COVID-19 periods were positive, but the differences were quite small at 0.000002 and 0.001926. It was proven that COVID-19 caused the cryptocurrencies’ volatilities in the COVID-19 period to fluctuate less than in the pre-COVID-19 period. Since 2020, the volatilities of most of the cryptocurrencies have decreased. This means that most of the cryptocurrencies fluctuate less than before the beginning of COVID-19. Third, from the correlations among the varying GARCH time series of the 10 cryptocurrencies, we found that the correlations were quite high. This result was similar to that of Le et al. (2021). Table 10 lists the results of the correlations among the varying GARCH values of the 10 cryptocurrencies’ return indices for the full period. For the full period, the average correlations between each of the 10 varying GARCH time series and the other 9 varying GARCH time series were 0.6470024, 0.6425153, 0.3233569, 0.4865397, 0.6103941, 0.6115659, 0.519619, 0.6720849, 0.6497457, and 0.6058991, respectively, for Bitcoin, Ethereum, Tether, Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO. These correlations were quite high. These high correlations revealed that the volatilities of all 10 cryptocurrencies fluctuated in a similar fashion. J. Risk Financial Manag. 2022,15, 113 16 of 25 Fourth, the correlations among the varying GARCH time series of the 10 cryptocurrencies increased from the pre-COVID-19 period to the COVID-19 period. Table 10. Correlations among the varying GARCH values of the 10 cryptocurrencies’ return indices for the full period. Correlation σBTC,tσETH,tσTET,tσXRP,tσLTC,tσBCH,tσXLM,tσXMR,tσEOS,tσNEO,t σBTC,t1.000000 0.776808 0.281069 0.424849 0.689947 0.634587 0.523330 0.792656 0.653542 0.693236 σETH,t0.776808 1.000000 0.215460 0.502605 0.716635 0.659216 0.467497 0.814021 0.656849 0.616062 σTET,t0.281069 0.215460 1.000000 0.134521 0.208981 0.282569 0.292132 0.260092 0.292969 0.265776 σXRP,t0.424849 0.502605 0.134521 1.000000 0.630485 0.394366 0.490050 0.439728 0.466209 0.382584 σLTC,t0.689947 0.716635 0.208981 0.630485 1.000000 0.578735 0.438103 0.676326 0.615683 0.549046 σBCH,t0.634587 0.659216 0.282569 0.394366 0.578735 1.000000 0.493803 0.739067 0.746646 0.586670 σXLM,t0.523330 0.467497 0.292132 0.490050 0.438103 0.493803 1.000000 0.466423 0.564658 0.460194 σXMR,t0.792656 0.814021 0.260092 0.439728 0.676326 0.739067 0.466423 1.000000 0.764007 0.768529 σEOS,t0.653542 0.656849 0.292969 0.466209 0.615683 0.746646 0.564658 0.764007 1.000000 0.736894 σNEO,t0.693236 0.616062 0.265776 0.382584 0.549046 0.586670 0.460194 0.768529 0.736894 1.000000 Minimum 0.281069 0.215460 0.134521 0.134521 0.208981 0.282569 0.292132 0.260092 0.292969 0.265776 Maximum 0.792656 0.814021 0.292969 0.630485 0.716635 0.746646 0.564658 0.814021 0.764007 0.768529 Average 0.6470024 0.6425153 0.3233569 0.4865397 0.6103941 0.6115659 0.519619 0.6720849 0.6497457 0.6058991 Table 11 lists the results of the correlations among the varying GARCH values of the 10 cryptocurrencies’ return indices for the pre-COVID-19 period. Table 11. Correlations among the varying GARCH values of the 10 cryptocurrencies’ return indices for the pre-COVID-19 period. Correlation σBTC,tσETH,tσTET,tσXRP,tσLTC,tσBCH,tσXLM,tσXMR,tσEOS,tσNEO,t σBTC,t1.000000 0.707955 0.330240 0.509360 0.675031 0.603496 0.533173 0.819559 0.686039 0.708381 σETH,t0.707955 1.000000 0.318743 0.630937 0.677730 0.591474 0.402037 0.805435 0.651593 0.565191 σTET,t0.330240 0.318743 1.000000 0.196235 0.228384 0.274286 0.317141 0.339486 0.320246 0.253864 σXRP,t0.509360 0.630937 0.196235 1.000000 0.701694 0.387791 0.437912 0.536644 0.554319 0.388788 σLTC,t0.675031 0.677730 0.228384 0.701694 1.000000 0.454941 0.357114 0.597625 0.561834 0.455023 σBCH,t0.603496 0.591474 0.274286 0.387791 0.454941 1.000000 0.409046 0.711409 0.644365 0.463473 σXLM,t0.533173 0.402037 0.317141 0.437912 0.357114 0.409046 1.000000 0.462733 0.614799 0.428417 σXMR,t0.819559 0.805435 0.339486 0.536644 0.597625 0.711409 0.462733 1.000000 0.750222 0.739181 σEOS,t0.686039 0.651593 0.320246 0.554319 0.561834 0.644365 0.614799 0.750222 1.000000 0.674533 σNEO,t0.708381 0.565191 0.253864 0.388788 0.455023 0.463473 0.428417 0.739181 0.674533 1.000000 Minimum 0.330240 0.318743 0.196235 0.196235 0.228384 0.274286 0.317141 0.339486 0.320246 0.253864 Maximum 0.819559 0.805435 0.339486 0.701694 0.701694 0.711409 0.614799 0.819559 0.750222 0.739181 Average 0.6573234 0.6351095 0.3578625 0.534368 0.5709376 0.5540281 0.4962372 0.6762294 0.645795 0.5676851 Table 12 lists the results of the correlations among the varying GARCH values of the 10 cryptocurrencies’ return indices for the COVID-19 period. J. Risk Financial Manag. 2022,15, 113 17 of 25 From the pre-COVID-19 period to the COVID-19 period, the average correlations between each of the 10 varying GARCH time series and the other 9 varying GARCH time series increased 0.0070144, 0.0859532, 0.00509, 0.1442054, 0.1644883, 0.0677548, 0.0246983, 0.0081004, and 0.112502, respectively, for Bitcoin, Ethereum, Tether, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, except for Ripple. These positive differences proved that from the pre-COVID-19 period to the COVID-19 period, the correlations among the varying GARCH value time series increased. Table 12. Correlations among the varying GARCH values of the 10 cryptocurrencies’ return indices for the COVID-19 period. Correlation σBTC,tσETH,tσTET,tσXRP,tσLTC,tσBCH,tσXLM,tσXMR,tσEOS,tσNEO,t σBTC,t1.000000 0.858839 0.477586 0.329774 0.728852 0.673157 0.508551 0.767655 0.610086 0.688878 σETH,t0.858839 1.000000 0.449101 0.392509 0.828315 0.785365 0.594386 0.827712 0.699520 0.774880 σTET,t0.477586 0.449101 1.000000 0.105073 0.341076 0.409906 0.243316 0.307797 0.138836 0.156834 σXRP,t0.329774 0.392509 0.105073 1.000000 0.531854 0.435479 0.601903 0.350426 0.391736 0.414526 σLTC,t0.728852 0.828315 0.341076 0.531854 1.000000 0.812879 0.601509 0.820240 0.724326 0.762379 σBCH,t0.673157 0.785365 0.409906 0.435479 0.812879 1.000000 0.624312 0.790189 0.864738 0.789139 σXLM,t0.508551 0.594386 0.243316 0.601903 0.601509 0.624312 1.000000 0.487252 0.479866 0.498825 σXMR,t0.767655 0.827712 0.307797 0.350426 0.820240 0.790189 0.487252 1.000000 0.785721 0.872285 σEOS,t0.610086 0.699520 0.138836 0.391736 0.724326 0.864738 0.479866 0.785721 1.000000 0.844125 σNEO,t0.688878 0.774880 0.156834 0.414526 0.762379 0.789139 0.498825 0.872285 0.844125 1.000000 Minimum 0.329774 0.392509 0.105073 0.105073 0.341076 0.409906 0.243316 0.307797 0.138836 0.156834 Maximum 0.858839 0.858839 0.477586 0.601903 0.828315 0.864738 0.624312 0.872285 0.864738 0.872285 Average 0.6643378 0.7210627 0.3629525 0.455328 0.715143 0.7185164 0.563992 0.7009277 0.6538954 0.6801871 This meant that COVD-19 increased the correlations among the different cryptocurrencies’ dynamic volatilities. It was proven that the trends of cryptocurrencies’ dynamic volatilities moved in a similar pattern. Fifth, for the pre-COVID-19 period, the highest GARCH values occurred during 2017– 2018. For the COVID-19 period, the highest GARCH values occurred during March 2020. Although the highest GARCH values were not avoidable during the COVID-19 period, the average GARCH values decreased, and the correlations among the varying GARCH time series of the 10 cryptocurrencies increased. 6.2. DCC(1,1) Models Generally, a dynamic conditional correlation (DCC) was calculated from two varying time series variables. Because Bitcoin and Ethereum were the two most representative cryptocurrencies, we built the empirical models of the DCC(1,1) between the return indices of Bitcoin and Ethereum and the other cryptocurrencies. Table 13 lists the results of the DCC(1,1) models built between the return indices of Bitcoin and Ethereum and the other cryptocurrencies’ return indices for the full period. For all of the DCC(1,1) models, the t-statistic values proved that all of their coefficients represented by α and β were statistically substantial at the probability level of 1%. Substantially, these 18 DCC(1,1) models were used to analyze the characteristics of the dynamic varying correlations. Figure 3depicts the curves of the DCC between Bitcoin and the other nine cryptocurrencies for the full period. J. Risk Financial Manag. 2022,15, 113 18 of 25 Table 13. DCC(1,1) models between the return indices of Bitcoin and Ethereum and the other cryptocurrencies’ return indices for the full period. DCC(1,1) α β LLH SIC DCC(1,1) α β LLH SIC ρ(εBTC,t,εETH,t)0.059207 *** (0.0000) 0.931559 *** (0.0000) −2771815 3419 ρ(εETH,t,εBTC,t)0.059192 *** (0.0000) 0.931558 *** (0.0000) −2770250 17 ρ(εBTC,t,εTET,t)0.058019 *** (0.0000) 0.927671 *** (0.0000) −752884 928 ρ(εETH,t,εTET,t)0.039144 *** (0.0000) 0.975561 *** (0.0000) −5206703 6424 ρ(εBTC,t,εXRP,t)0.059859 *** (0.0000) 0.932308 *** (0.0000) −22687280 27991 ρ(εETH,t,εXRP,t)0.059832 *** (0.0000) 0.932277 *** (0.0000) −19675829 24276 ρ(εBTC,t,εLTC,t)0.059641 *** (0.0000) 0.932072 *** (0.0000) −8820785 10883 ρ(εETH,t,εLTC,t)0.058988 *** (0.0000) 0.931386 *** (0.0000) −2816160 3474 ρ(εBTC,t,εBCH,t)0.059800 *** (0.0000) 0.932236 *** (0.0000) −14754445 18204 ρ(εETH,t,εBCH,t)0.059671 *** (0.0000) 0.932067 *** (0.0000) −8867302 10940 ρ(εBTC,t,εXLM,t)0.059763 *** (0.0000) 0.932193 *** (0.0000) −14013995 17290 ρ(εETH,t,εXLM,t)0.059753 *** (0.0000) 0.932179 *** (0.0000) −4032724 17313 ρ(εBTC,t,εXMR,t)0.058533 *** (0.0000) 0.930968 *** (0.0000) −1154885 1424 ρ(εETH,t,εXMR,t)0.060000 *** (0.0000) 0.932472 *** (0.0000) −765969 945 ρ(εBTC,t,εEOS,t)0.59694 *** (0.0000) 0.932117 *** (0.0000) −10202643 12588 ρ(εETH,t,εEOS,t)0.059535 *** (0.0000) 0.931893 *** (0.0000) −6670654 8230 ρ(εBTC,t,εNEO,t)0.059659 *** (0.0000) 0.932085 *** (0.0000) −8168736 10078 ρ(εETH,t,εNEO,t)0.059562 *** (0.0000) 0.931949 *** (0.0000) −6566249 8101 Note: The symbols *** indicates that the result is statistically substantial under the probability thresholds of 1%; The initial values of parameters ρi,0 and ρj,0 are defined as one; the initial values of parameters ρij,0 and ρji,0 are defined as the static Pearson correlation coefficient between εi,t and εj,t ; AIC is Akaike information criterion; LLH is log-likelihood. J. Risk Financial Manag. 2022, 15, x FOR PEER REVIEW 19 of 26 Figure 3. Curves of DCC between Bitcoin and the other nine cryptocurrencies for the full period. During the full period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, were 0.63941, 0.773738, 0.699301, 0.624351, 0.708394, 0.690766, and 0.658295, respectively; otherwise, the average varying correlations between the return indices of Ethereum and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, were 0.725803, 0.803866, 0.740293, 0.681546, 0.714453, 0.740066, and 0.730576, respectively; in comparison, the differences between both groups of correlations were 0.086393, 0.030128, 0.040992, 0.057195, 0.006059, 0.049300, and 0.072281. It was clear that the average values of the DCC between Ethereum and the other cryptocurrencies were higher than the average values of the DCC between Bitcoin and the other cryptocurrencies. This means that Ethereum has become a more important representative cryptocurrency than Bitcoin or that Ethereum has a higher impact on the other cryptocurrencies than Bitcoin. Figure 3. Curves of DCC between Bitcoin and the other nine cryptocurrencies for the full period. Figure 4depicts the curves of the DCC between Ethereum and the other eight cryptocurrencies for the full period. J. Risk Financial Manag. 2022,15, 113 19 of 25 J. Risk Financial Manag. 2022, 15, x FOR PEER REVIEW 20 of 26 Figure 4. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the full period. Third, except for Tether, when comparing the changes in the DCC mean values between the pre-COVID-19 period and the COVID-19 period, since the COVID-19 pandemic began, the average DCC values between Bitcoin and the other cryptocurrencies have increased. Figure 5 depicts the curves of the DCC between Bitcoin and the other nine cryptocurrencies for the pre-COVID-19 period. Figure 6 depicts the curves of the DCC between Ethereum and the other eight cryptocurrencies for the pre-COVID-19 period. Figure 7 depicts the curves of the DCC between Bitcoin and the other nine cryptocurrencies for the COVID-19 period. Figure 8 depicts the curves of the DCC between Ethereum and the other eight cryptocurrencies for the COVID-19 period. During the pre-COVID-19 period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, EOS, and NEO, were 0.633941, 0.754496, 0.672014, 0.615718, 0.685013, and 0.642678, respectively. During the COVID-19 period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, EOS, and NEO, were 0.645365, 0.794691, 0.729014, 0.633752, 0.697030, and 0.675301, respectively. From the pre-COVID-19 period to the COVID-19 period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, EOS, and NEO, increased by differences of 0.011424, 0.040195, 0.057000, 0.018034, 0.012017, and 0.032623, respectively. Figure 4. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the full period. Table 14 lists the comparison results of the mean values of DCC(1,1) between the return indices of Bitcoin and Ethereum and the other cryptocurrencies for the three periods. Table 14. Mean values of DCC(1,1) between the return indices of Bitcoin and Ethereum and the other cryptocurrencies for the three periods. DCC(1,1) Full Period Pre-COVID COVID DCC(1,1) Full Period Pre-COVID COVID ρ(rBTC,t,rETH,t)0.778316 0.77948 0.777049 ρ(rETH,t,rBTC,t)0.778307 0.779468 0.777042 ρ(rBTC,t,rTET,t)−0.017080 0.013162 −0.050011 ρ(rETH,t,rTET,t)0.005532 0.072424 −0.067308 ρ(rBTC,t,rXRP,t)0.639410 0.633941 0.645365 ρ(rETH,t,rXRP,t)0.725803 0.747717 0.70194 ρ(rBTC,t,rLTC,t)0.773738 0.754496 0.794691 ρ(rETH,t,rLTC,t)0.803866 0.801298 0.806663 ρ(rBTC,t,rBCH,t)0.699301 0.672014 0.729014 ρ(rETH,t,rBCH,t)0.740293 0.722064 0.760142 ρ(rBTC,t,rXLM,t)0.624351 0.615718 0.633752 ρ(rETH,t,rXLM,t)0.681546 0.677399 0.686063 ρ(rBTC,t,rXMR,t)0.708394 0.727672 0.687401 ρ(rETH,t,rXMR,t)0.714453 0.753233 0.672226 ρ(rBTC,t,rEOS,t)0.690766 0.685013 0.69703 ρ(rETH,t,rEOS,t)0.740066 0.744708 0.735012 ρ(rBTC,t,rNEO,t)0.658295 0.642678 0.675301 ρ(rETH,t,rNEO,t)0.730576 0.731375 0.729706 First, except for Tether, during the full time period the varying correlations between the return indices of Bitcoin and Ethereum and the other eight cryptocurrencies from the descriptive statistics were positive and quite high. For the full period, except for Tether, the mean values of the DCC between Bitcoin and the other eight cryptocurrencies were between 0.6243 and 0.7783; the mean values of the DCC between Ethereum and the other cryptocurrencies were between 0.7393 and 0.8038. J. Risk Financial Manag. 2022,15, 113 20 of 25 This result was similar to the research of Ciaian et al. (2018) and Lahajnar and Rozanec (2020) and proved that the correlations between Bitcoin and the other cryptocurrencies were strong. Second, the correlations between Ethereum and the other cryptocurrencies were higher than the correlations between Bitcoin and the other cryptocurrencies. Figure 4depicts the curves of the DCC between Ethereum and the other eight cryptocurrencies for the full period. Figure 3depicts the curves of the DCC between Bitcoin and the other nine cryptocurrencies for the full period. During the full period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, were 0.63941, 0.773738, 0.699301, 0.624351, 0.708394, 0.690766, and 0.658295, respectively; otherwise, the average varying correlations between the return indices of Ethereum and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, were 0.725803, 0.803866, 0.740293, 0.681546, 0.714453, 0.740066, and 0.730576, respectively; in comparison, the differences between both groups of correlations were 0.086393, 0.030128, 0.040992, 0.057195, 0.006059, 0.049300, and 0.072281. It was clear that the average values of the DCC between Ethereum and the other cryptocurrencies were higher than the average values of the DCC between Bitcoin and the other cryptocurrencies. This means that Ethereum has become a more important representative cryptocurrency than Bitcoin or that Ethereum has a higher impact on the other cryptocurrencies than Bitcoin. Third, except for Tether, when comparing the changes in the DCC mean values between the pre-COVID-19 period and the COVID-19 period, since the COVID-19 pandemic began, the average DCC values between Bitcoin and the other cryptocurrencies have increased. Figure 5depicts the curves of the DCC between Bitcoin and the other nine cryptocurrencies for the pre-COVID-19 period. J. Risk Financial Manag. 2022, 15, x FOR PEER REVIEW 21 of 26 Figure 5. Curves of the DCC between Bitcoin and the other nine cryptocurrencies for the pre- COVID-19 period. Figure 6. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the pre- COVID-19 period. Figure 5. Curves of the DCC between Bitcoin and the other nine cryptocurrencies for the pre-COVID- 19 period. J. Risk Financial Manag. 2022,15, 113 21 of 25 Figure 6depicts the curves of the DCC between Ethereum and the other eight cryptocurrencies for the pre-COVID-19 period. J. Risk Financial Manag. 2022, 15, x FOR PEER REVIEW 21 of 26 Figure 5. Curves of the DCC between Bitcoin and the other nine cryptocurrencies for the pre- COVID-19 period. Figure 6. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the pre- COVID-19 period. Figure 6. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the pre- COVID-19 period. Figure 7depicts the curves of the DCC between Bitcoin and the other nine cryptocurrencies for the COVID-19 period. J. Risk Financial Manag. 2022, 15, x FOR PEER REVIEW 22 of 26 Figure 7. Curves of the DCC between Bitcoin and the other nine cryptocurrencies for the COVID-19 period. Figure 8. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the COVID- 19 period. This means that the correlations between Bitcoin and the other cryptocurrencies have enhanced since the beginning of 2020. However, these correlations were not proven for Ethereum. Fourth, except for Tether, from the correlations among the varying DCC values between Bitcoin and Ethereum, and between Bitcoin, Ethereum, and the other cryptocurrencies, we determined that the correlations among these cryptocurrencies were similar to those of Bitcoin and Ethereum. The correlations among the varying DCC value time series between Bitcoin and Ethereum and the varying DCC value time series between Bitcoin and Ripple, Litecoin, Figure 7. Curves of the DCC between Bitcoin and the other nine cryptocurrencies for the COVID-19 period. Figure 8depicts the curves of the DCC between Ethereum and the other eight cryptocurrencies for the COVID-19 period. J. Risk Financial Manag. 2022,15, 113 22 of 25 J. Risk Financial Manag. 2022, 15, x FOR PEER REVIEW 22 of 26 Figure 7. Curves of the DCC between Bitcoin and the other nine cryptocurrencies for the COVID-19 period. Figure 8. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the COVID- 19 period. This means that the correlations between Bitcoin and the other cryptocurrencies have enhanced since the beginning of 2020. However, these correlations were not proven for Ethereum. Fourth, except for Tether, from the correlations among the varying DCC values between Bitcoin and Ethereum, and between Bitcoin, Ethereum, and the other cryptocurrencies, we determined that the correlations among these cryptocurrencies were similar to those of Bitcoin and Ethereum. The correlations among the varying DCC value time series between Bitcoin and Ethereum and the varying DCC value time series between Bitcoin and Ripple, Litecoin, Figure 8. Curves of the DCC between Ethereum and the other eight cryptocurrencies for the COVID- 19 period. During the pre-COVID-19 period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, EOS, and NEO, were 0.633941, 0.754496, 0.672014, 0.615718, 0.685013, and 0.642678, respectively. During the COVID-19 period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, EOS, and NEO, were 0.645365, 0.794691, 0.729014, 0.633752, 0.697030, and 0.675301, respectively. From the pre-COVID-19 period to the COVID-19 period, except for Tether, the average varying correlations between the return indices of Bitcoin and the other cryptocurrencies, including Ripple, Litecoin, Bitcoin Cash, Stellar, EOS, and NEO, increased by differences of 0.011424, 0.040195, 0.057000, 0.018034, 0.012017, and 0.032623, respectively. This means that the correlations between Bitcoin and the other cryptocurrencies have enhanced since the beginning of 2020. However, these correlations were not proven for Ethereum. Fourth, except for Tether, from the correlations among the varying DCC values between Bitcoin and Ethereum, and between Bitcoin, Ethereum, and the other cryptocurrencies, we determined that the correlations among these cryptocurrencies were similar to those of Bitcoin and Ethereum. The correlations among the varying DCC value time series between Bitcoin and Ethereum and the varying DCC value time series between Bitcoin and Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO were 0.761078, 0.727885, 0.808231, 0.680740, 0.787096, 0.715787, and 0.839999, respectively, which were quite high. The correlations among the varying DCC value time series between Ethereum and Bitcoin and the varying DCC value time series between Ethereum and Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO were 0.587975, 0.620270, 0.739476, 0.732984, 0.627493, 0.666690, and 0.541859, respectively, which were also quite high. It was clear that the trend changes in the DCC value time series between Bitcoin, Ethereum, and the other cryptocurrencies were similar. Fifth, we determined the differences of the other cryptocurrencies from Tether, whose characteristics were quite different. J. Risk Financial Manag. 2022,15, 113 23 of 25 For the full period, the average DCC values between the return indices of Bitcoin and Tether were negative, being − 0.01701 for the full period, 0.0131 for the pre-COVID-19 period, and − 0.0500 for the COVID-19 period. For the full period, the correlations among the DCC value time series between Bitcoin and Tether and the DCC value time series between Bitcoin and the other cryptocurrencies, including Ethereum, Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO, were negative at − 0.081714, − 0.029559, −0.099312, −0.145631, −0.067753, −0.054917, −0.094690, and −0.069847, respectively. For the full period, the correlations between the return index of Tether and the other cryptocurrencies, including Bitcoin, Ethereum, Ripple, Bitcoin Cash, Stellar, Monero, and EOS, were negative at − 0.021680, − 0.026539, − 0.032471, − 0.034959, − 0.034237, − 0.022461, and −0.022924. For the pre-COVID-19 period, the correlations between the return index of Tether and the other cryptocurrencies, including Bitcoin, Ethereum, Ripple, Bitcoin Cash, Stellar, Monero, and EOS, were 0.010457, 0.001710, − 0.018437, − 0.014268, − 0.022881, 0.005238, and − 0.003722, respectively. Although the static correlations between the return indices of Tether and the other cryptocurrencies were not always negative, the maximum values of the static correlations were less than 0.010457. For the COVID-19 period, the correlations between the return index of Tether and the other cryptocurrencies, including Bitcoin, Ethereum, Ripple, Bitcoin Cash, Stellar, Monero, and EOS, were negative at − 0.243461, − 0.221253, − 0.155334, − 0.214276, − 0.148999, − 0.219096, and − 0.175636, respectively. This means that the correlations between the return index of Tether and most of the other cryptocurrencies were negative. From the pre-COVID-19 period to the COVID-19 period, on average, the correlations between Tether and the other cryptocurrencies changed from negative or very small to negative; Tether became a highly hedging cryptocurrency against the other cryptocurrencies. Because the correlations between the return indices of Tether and the other cryptocurrencies were mostly negative or very low, Tether can be a hedge cryptocurrency against the other cryptocurrencies. This result was totally different from the research of Kyriazis et al. (2019) because they confirmed that no hedging abilities existed among cryptocurrencies. COVID-19 has enhanced the degree of negative correlations, or it has increased the hedging characteristics between Tether and the other cryptocurrencies. 7. Summary and Further Studies This paper focused on studying the relationship between Bitcoin, Ethereum, and the other eight cryptocurrencies, including Tether, Ripple, Litecoin, Bitcoin Cash, Stellar, Monero, EOS, and NEO. The observation sample data covered the full time period from 8 September 2017 to 14 February 2022, with 1621 observations, and covered the time when the full period was divided into the pre-COVID-19 period from 8 September 2017 to 31 December 2019, with 845 observations, and the COVID-19 period from 1 January 2020 to 14 February 2022, with 776 observations. After an empirical analysis, we arrived at four main results. First, the descriptive statistics and tests proved that, from the pre-COVID-19 period to the COVID-19 period, almost all of the 10 cryptocurrencies’ growth rates increased; thus, COVID-19 had a positive effect on the returns of cryptocurrencies. Second, from the empirical results of the GARCH(1,1) models, we proved that, for all of the 10 GARCH(1,1) models, the values of the coefficient βi were greater than 0.641374, which means that these 10 cryptocurrencies’ return indices had features of volatility clustering or memory persistence in the long run. This result was similar to those of Soylu et al. (2020), Palamalai et al. (2020), Abakah et al. (2020), and Sensoy et al. (2020). Tether had the lowest GARCH values, but the other nine cryptocurrencies had higher GARCH values than Tether; all of the 10 cryptocurrencies’ GARCH values decreased from the pre-COVID-19 period to the COVID-19 period. The correlations among the varying GARCH time series of the 10 cryptocurrencies were quite high and were similar to the findings of Le et al. (2021). The correlations among the varying GARCH time series of the 10 cryptocurrencies increased J. Risk Financial Manag. 2022,15, 113 24 of 25 from the pre-COVID-19 period to the COVID-19 period. The trends of cryptocurrencies’ dynamic volatilities moved in a similar pattern: for the pre-COVID-19 period, the highest GARCH values occurred during 2017–2018; for the COVID-19 period, the highest GARCH values occurred during March 2020. Third, from the empirical results of the DCC(1,1) models, we proved that, except for Tether, the varying correlations between the return indices of Bitcoin, Ethereum, and the other cryptocurrencies were very strong, similar to the findings of Ciaian et al. (2018) and Lahajnar and Rozanec (2020). They proved that the correlations between Bitcoin and the other cryptocurrencies were strong; the correlations between Ethereum and the others were higher than between Bitcoin and the others. Since the COVID-19 pandemic began, the average values of the DCC between Bitcoin and the other cryptocurrencies, except Tether, have increased; except for Tether, since the COVID-19 pandemic began, the correlations among the 10 cryptocurrencies’ return indices have become higher than before. Fourth, the characteristics of Tether were quite different from those of the other cryptocurrencies: during the COVID-19 period, the static correlations between the return indices of Tether and the other nine cryptocurrencies were negative; during the pre-COVID- 19 period and the full period, the static correlations between the return indices of Tether and the other cryptocurrencies were not always negative but were very low at less than 0.010457. Tether can act as a hedge cryptocurrency for the other cryptocurrencies, and this result differed from the research of Kyriazis et al. (2019) because they found that no hedging abilities existed among cryptocurrencies. Author Contributions: Conceptualization, K.Y. and H.Y.; methodology, K.Y. and H.Y.; software, K.Y. and H.Y.; validation, K.Y. and H.Y.; formal analysis, K.Y. and H.Y.; investigation, K.Y. and H.Y.; resources, K.Y. and H.Y.; data curation, K.Y. and H.Y.; writing—original draft preparation, K.Y. and H.Y.; writing—review and editing, H.Y. and H.Y.; visualization, K.Y. and H.Y.; supervision, H.Y. and R.G.; project administration, K.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The open data source cn.investing.com (accessed on 17 February 2022) is used as data sources for this paper. Conflicts of Interest: The authors declare no conflict of interest. References Abakah, Emmanuel Joel Aikins, Luis Alberiko Gil-Alana, Godfrey Madigu, and Fatima Romero-Rojo. 2020. 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