True versus spurious long memory in cryptocurrencies

Rambaccussing, Dooruj; Mazibas, Murat Article True versus spurious long memory in cryptocurrencies Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Rambaccussing, Dooruj; Mazibas, Murat (2020) : True versus spurious long memory in cryptocurrencies, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 9, pp. 186-, https://doi.org/10.3390/jrfm13090186 This Version is available at: https://hdl.handle.net/10419/239308 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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/ Journal of Risk and Financial Management Communication True versus Spurious Long Memory in Cryptocurrencies Dooruj Rambaccussing * and Murat Mazibas School of Business, University of Dundee, Dundee DD1 3BH, UK; [email protected] *Correspondence: [email protected] Received: 22 July 2020; Accepted: 17 August 2020; Published: 19 August 2020   Abstract: We test whether the selected cryptocurrencies exhibit long memory behavior in returns and volatility. We use data on five most traded cryptocurrencies: Bitcoin, Litecoin, Ethereum, Bitcoin Cash, and XRP. Using recent tests of long memory developed against persistent and nonlinear alternatives, this paper finds that long memory is mostly rejected in returns. The tests fail to reject the null hypothesis of long memory in most cases across different volatility proxies and cryptocurrencies. The estimated memory parameters show that volatility is persistent, and when volatility is measured by log range, it is borderline nonstationary. Keywords: cryptocurrency; Bitcoin; persistence; long memory; fractional integration 1. Introduction It is not uncommon in applied research to encounter misspecified models where estimates from a model are found to be a good fit to the data when in reality the true model is generated from another data generating process. For instance, a researcher estimates a Generalized Autoregressive Conditional Heteroscedasticity (GARCH) process to model volatility when the observations are generated by a Fractionally Integrated Generalized Autoregressive Conditional Heteroscedasticity (FIGARCH) process. It is very likely that the GARCH parameter estimates will come out as statistically significant, although this is not the best modeling outcome. This paper examines the issue of spurious long memory in returns and volatility of five cryptocurrencies, namely Bitcoin, Ethereum, Bitcoin Cash, XRP, and Litecoin. In particular, this paper sheds light on cases when the true process follows a data generating process which is incorrectly interpreted as long memory using tests which account for the spectral nature of the processes. Cryptocurrencies have been a recent innovation in finance, with substantial market capitalization and enormous capital gain possibilities. The market capitalization and prices of Bitcoin, the most popular cryptocurrency, have witnessed increases of 262% and 232%, respectively, from January 2014 to December 2019. Rather unsurprisingly, cryptocurrencies have been the focus of policymakers, speculators, and economists. An interesting research question focuses on whether prices impound all information, which effectively translates into stating whether the efficient markets hypothesis holds. This hypothesis can be tested in a variety of ways, with a key cornerstone being the presence and exploitation of profitable trading strategies. Linked to this concept, the presence of long-range autocorrelations (or long memory) in cryptocurrencies opens up possibilities for making abnormal profits. Therefore, it is important to distinguish the nature of the persistence in returns. Failure to account for long memory may result in misspecifications with incorrect inferences. However, the opposite is also true when long memory is spuriously detected. In the context of the efficient markets hypothesis, this may lead to an erroneous conclusion that markets are inefficient, when in fact they are not. It is widely accepted that the presence of nonlinearities and structural breaks may also lead to such spurious detections, see for instance ( Wenger et al. 2018;Dehling et al. 2013;Iacone et al. 2014;Betken 2016 ). J. Risk Financial Manag. 2020,13, 186; doi:10.3390/jrfm13090186 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2020,13, 186 2 of 11 Other nonlinear models (Markov switching and bilinear models) having slow decay rates may also lead to spurious long memory ( Diebold and Inoue 2001;Granger and Hyung 2004 ). Finally, persistent (close to unity) processes can also exhibit serial correlation patterns similar to long memory due to observational equivalence. In these instances, estimators may incorrectly yield a statistically significant memory parameter. One of the added benefits of addressing this research question is the ability to pinpoint models which are more appropriate for modeling returns and volatility. For example, in the case of the latter, a statistically significant memory parameter would inform the practitioner that FIGARCH and Hyperbolic Generalized Generalized Autoregressive Process (HYGARCH) processes are better suited to model and forecast volatility rather than conventional GARCH processes. In this paper, we test the long memory hypothesis in both mean and volatility of the returns of five major cryptocurrencies: Bitcoin, Litecoin, Ethereum, Bitcoin Cash, and XRP. Long memory can be estimated in time and frequency domains. Frequency domain models tend to be nonparametric (or semiparametric) such as the GPH (Geweke and Porter-Hudak 1983), Local Whittle ( Robinson 1995 ), and Fourier (Moulines and Soulier 1999) approaches. Our contribution in this paper is testing for long memory using two novel and superior tests: the log periodogram bias test (Davidson and Sibbertsen 2009) and the skip-sampling test (Davidson and Rambaccussing 2015). The layout of the paper is as follows. Section 2provides a brief outline of the methodology. Section 3describes the related literature on cryptocurrencies. Section 4presents the empirical results. Section 5provides some concluding remarks. 2. Background Literature In this section, we review major findings on the empirical processes in cryptocurrencies focusing on the long-range dependence. Urquhart (2016) looks at the autocorrelation patterns in Bitcoin prices to conclude that the market for Bitcoin is not efficient, and Urquhart (2017) later extends to the use of price clustering to reach the conclusion of inefficient markets. Although the focus of our paper is not on the efficient markets hypothesis, the link between returns and autocorrelations is nonetheless important. Significant studies in the area include Nadarajah and Chu (2017) and Tiwari et al. (2018). Lookingexclusivelyatlongmemory, Bourietal. (2019)testthepersistenceinthepriceandvolatility of Bitcoin series and find that there is indeed long memory in the volatility series. For the purposes of volatility modeling, Katsiampa (2017) finds that GARCH processes may also include long memory and attempts to find the optimal conditional variance model. Long-range dependence in volatility has been expressed using the two main variants of long memory—the fractionally integrated GARCH model (Lahmiri et al. 2018) and the hyperbolic GARCH model. Phillip et al. (2018) perform a more comprehensive study by studying the returns of 224 cryptocurrencies, and they find that the stochastic volatility model exhibits anti-persistence, leverage, and heavy-tails. They, in turn, consider regime switching specifications in the return equation and long memory in volatility. Mensi et al. (2019) shows that after accounting for structural breaks, there is “dual” long memory in both returns and volatility. Nonlinearity, a potential source of spurious long memory, has also been investigated in the literature. Examples of related literature exploring nonlinearity include Cheah and Fry (2015). Accounting jointly for nonlinearity and long memory can be seen in Bouri et al. (2019), who model long memory in squared and absolute returns series after extracting potential break dates. 3. Long Memory Explained A fractionally integrated model, as a broader class of memory models, can be represented as: (1−L)dyt=εt. (1) where Lis the lag operator, dis the fractional differencing parameter, yt are observations recorded over time, and εt is an error term which is assumed to be independently and identically (iid) distributed. The parameter d defines the memory structure of the time series variable yt . When d= 0 the process J. Risk Financial Manag. 2020,13, 186 3 of 11 is short memory, d= 1 random walk, d< 0 antipersistent, and d> 0 is long memory. 1 / 2 ≤d< 3 / 2 corresponds to the nonstationary case. Equation (1) can be represented in spectral form (see Beran 1994 for a more involved discussion). d can also be estimated using a variety of parametric, semi-parametric, and nonparametric models. Short memory processes with persistent autoregressive terms can be mistaken for a long memory process. The main distinction between long memory and such processes is the behavior of the covariance of y. Denoting γj as the autocovariance of a stationary process yt , a short memory process has a summable autocovariance function ( ∞ P j=0γj<∞) , whereas a long memory has nonsummable autocovariances ( ∞ P j=0γj=∞ ) with a spectral density of infinity at the origin ( f(0)=∞ ). Similarly, the autocovariance has a convergence rate of γj=Oj2d−1 in the long memory case. The short memory is the case where convergence is faster γj=oj2d−1 , and there exists a parameter ρ > 0 such that γj=Oe−ρj. One of the variants which takes into account both the long and short memory structure is an Autoregressive Fractionally Integrated Moving Average (ARFIMA), which includes the fractional differencing operator (d) on the left hand side of the equation: θ(L)(1−L)dyt=Φ(L)εt. (2) When the true parameter dis equal to zero, Equation (1) becomes a simple autoregressive moving average model (ARMA), which implies a faster convergence. One of the main problems is the case when a highly persistent ARMA reveals a significant dwhen it is being estimated using semiparametric and nonparametric techniques. Figure 1illustrates simulated time series for an ARFIMA (0, d, 0) with d=0.4 and an autoregressive (AR) process with a coefficient of 0.9. Their corresponding correlograms show patterns in the autocovariance, which illustrate the above distinctions. As can be seen, it is hard to discern the type of process visually without looking at long autocorrelation patterns over time. The correlogram, which illustrates such patterns, shows that the ARFIMA autocorrelations take a long time to converge to zero. On the other hand, the autoregressive process converges geometrically 1 to zero. Moreover, the presence of highly persistent short-term dynamics will bias estimators such as the GPH dynamics. Other nonlinear models (and processes with structural shifts) can also imitate long memory processes. Diebold and Inoue (2001) provide a set of nonlinear processes, under which rejects the null of long memory. To illustrate certain cases where long memory may be mistaken for another process, we performed a Monte Carlo experiment. In the experiment we showed that when the true data generating process follows a nonlinear model, long memory estimators can show spurious estimates. Two processes were considered. The first model was a Markov-switching (MS) model with switching intercepts. In this case, the intercept (c) was allowed to switch between the two regimes S t , which was identified as a state variable. yst=cst+εt. (3) The intercept in regimes 1 and 2 were − 1 and − 5, respectively, and with constant variances in both regimes. The regimes were defined by a state function, which took a probability of 0.95 being in the first regime and switched back from the second regime to the first regime with a probability of 0.05. These probabilities were assigned to emulate possibilities of crashes, which are sudden but may go back to the previous regime rather quickly. 1 Geometric decay would be easily discerned in the bottom right plot in Figure 1if the number of lags was set at most at 30 (or at any low amount). J. Risk Financial Manag. 2020,13, 186 4 of 11 J. Risk Financial Manag. 2020, 13, x FOR PEER REVIEW 3 of 10 memory. 12≤<32 ⁄⁄ corresponds to the nonstationary case. Equation (1) can be represented in spectral form (see Beran 1994 for a more involved discussion).  can also be estimated using a variety of parametric, semi-parametric, and nonparametric models. Short memory processes with persistent autoregressive terms can be mistaken for a long memory process. The main distinction between long memory and such processes is the behavior of the covariance of y. Denoting  as the autocovariance of a stationary process , a short memory process has a summable autocovariance function (∑<∞)   , whereas a long memory has nonsummable autocovariances (∑=∞   ) with a spectral density of infinity at the origin ((0) = ∞). Similarly, the autocovariance has a convergence rate of =() in the long memory case. The short memory is the case where convergence is faster =(), and there exists a parameter >0 such that = (). One of the variants which takes into account both the long and short memory structure is an Autoregressive Fractionally Integrated Moving Average (ARFIMA), which includes the fractional differencing operator (d) on the left hand side of the equation: ()(1−)=Φ(). (2) When the true parameter d is equal to zero, Equation (1) becomes a simple autoregressive moving average model (ARMA), which implies a faster convergence. One of the main problems is the case when a highly persistent ARMA reveals a significant d when it is being estimated using semiparametric and nonparametric techniques. Figure 1 illustrates simulated time series for an ARFIMA (0, d, 0) with d = 0.4 and an autoregressive (AR) process with a coefficient of 0.9. Their corresponding correlograms show patterns in the autocovariance, which illustrate the above distinctions. As can be seen, it is hard to discern the type of process visually without looking at long autocorrelation patterns over time. The correlogram, which illustrates such patterns, shows that the ARFIMA autocorrelations take a long time to converge to zero. On the other hand, the autoregressive process converges geometrically1 to zero. Figure 1. Plot of Time Series. The top-left (ARFIMA d = 0.4) and bottom-left (AR 0.9) figures show the simulated series for an autoregressive fractionally integrated process with a fractional parameter d = 0.4 and an autoregressive process with a lag parameter equal to 0.9, respectively. The right-hand side shows the corresponding correlogram with 1000 lags. The correlogram shows that the autoregressive process (top right) declines geometrically to zero unlike the ARFIMA process (bottom right). Moreover, the presence of highly persistent short-term dynamics will bias estimators such as the GPH dynamics. Other nonlinear models (and processes with structural shifts) can also imitate long 1 Geometric decay would be easily discerned in the bottom right plot in Figure 1 if the number of lags was set at most at 30 (or at any low amount). -6 -4 -2 0 2 4 6 0 500 1000 1500 2000 Arfima d = 0.4 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 Arfima d = 0.4 -10 -8 -6 -4 -2 0 2 4 6 8 0 500 1000 1500 2000 AR 0.9 -0.2 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 AR 0.9 Figure 1. Plot of Time Series. The top-left (ARFIMA d=0.4) and bottom-left (AR 0.9) figures show the simulated series for an autoregressive fractionally integrated process with a fractional parameter d=0.4 and an autoregressive process with a lag parameter equal to 0.9, respectively. The right-hand side shows the corresponding correlogram with 1000 lags. The correlogram shows that the autoregressive process (top right) declines geometrically to zero unlike the ARFIMA process (bottom right). The second model considered was also a regime switching model with changes in persistence. In the first regime, the process was a unit root while the second regime was a stationary process, and the intercepts were different in both regimes. The processes in regime 1 and 2 were, respectively, yt=− 1 +yt−1+εt and yt=− 5 + 0.2 yt−1+εt . The probability of being in a regime was 0.95 and the probability of switching from regime 2 to regime 1 was given by 0.05. Figure 2shows plots of the simulated series. J. Risk Financial Manag. 2020, 13, x FOR PEER REVIEW 4 of 10 memory processes. Diebold and Inoue (2001) provide a set of nonlinear processes, under which rejects the null of long memory. To illustrate certain cases where long memory may be mistaken for another process, we performed a Monte Carlo experiment. In the experiment we showed that when the true data generating process follows a nonlinear model, long memory estimators can show spurious estimates. Two processes were considered. The first model was a Markov-switching (MS) model with switching intercepts. In this case, the intercept (c) was allowed to switch between the two regimes St, which was identified as a state variable. = +. (3) The intercept in regimes 1 and 2 were −1 and −5, respectively, and with constant variances in both regimes. The regimes were defined by a state function, which took a probability of 0.95 being in the first regime and switched back from the second regime to the first regime with a probability of 0.05. These probabilities were assigned to emulate possibilities of crashes, which are sudden but may go back to the previous regime rather quickly. The second model considered was also a regime switching model with changes in persistence. In the first regime, the process was a unit root while the second regime was a stationary process, and the intercepts were different in both regimes. The processes in regime 1 and 2 were, respectively, = −1+ + and = −5+0.2 +. The probability of being in a regime was 0.95 and the probability of switching from regime 2 to regime 1 was given by 0.05. Figure 2 shows plots of the simulated series. Figure 2. Plot of Simulated Series. The “MS Intercept” plot expresses Model 1, where the intercept switches between −5 and 1, with p11 and p12 equal to 0.99. The “MS Intercept + Autoregressive” relates to Model 2, which assumes that the model switches from a unit root to a stationary process. A Monte Carlo simulation was performed by simulating these two data generating processes 5000 times by drawing from a normal distribution. The sample was made up of 2000 observations. Table 1 shows the results for the Monte Carlo process. The results show that a process like a simple Markov-switching process with switches in intercept may replicate a long memory process, and lead to a statistically significant fractional differencing parameter. The local Whittle and GPH with the narrow bandwidth have the lower d (0.148), but they are still considered to be statistically significant. The second model also shows a significant d across a range of values from 0.282 to 0.741, which is fairly more substantial as the broader bandwidth is nonstationary. In a nutshell, both processes illustrate cases where some processes may emulate long memory processes. Other similar processes -10 -8 -6 -4 -2 0 2 4 0 500 1000 1500 2000 MS Intercept -10 -5 0 5 10 15 0 500 1000 1500 2000 MS Intercept + Autoregressive Figure 2. Plot of Simulated Series. The “MS Intercept” plot expresses Model 1, where the intercept switches between − 5 and 1, with p 11 and p 12 equal to 0.99. The “MS Intercept +Autoregressive” relates to Model 2, which assumes that the model switches from a unit root to a stationary process. A Monte Carlo simulation was performed by simulating these two data generating processes 5000 times by drawing from a normal distribution. The sample was made up of 2000 observations. Table 1shows the results for the Monte Carlo process. The results show that a process like a simple J. Risk Financial Manag. 2020,13, 186 5 of 11 Markov-switching process with switches in intercept may replicate a long memory process, and lead to a statistically significant fractional differencing parameter. The local Whittle and GPH with the narrow bandwidth have the lower d(0.148), but they are still considered to be statistically significant. The second model also shows a significant dacross a range of values from 0.282 to 0.741, which is fairly more substantial as the broader bandwidth is nonstationary. In a nutshell, both processes illustrate cases where some processes may emulate long memory processes. Other similar processes include SETAR (self exciting threshold autoregressive) models, bilinear models, and STOP Break processes (for further details, see Diebold and Inoue 2001). The results show that the long memory estimators can exceptionally be significant at some bandwidth levels. This rather unsurprising finding in the literature has led to an increase in the research for testing for long memory in the univariate ( Shimotsu and Phillips 2002;Qu 2011;Ohanissian et al. 2008 ) or multivariate settings (Sibbertsen et al. 2018) as a second stage procedure. Table 1. Monte Carlo Simulation results. The table reports the test results of the Monte Carlo exercise. Model 1 is the model with switching intercepts ( − 1, 5) with probabilities of 0.95 of staying in the first regime and 0.05 from switching from the second regime to the first. Model 2 is a model which accounts for switches in both intercept ( − 1, − 5) and autoregressive terms (1, 0.2). The models are computed using the following: Geweke and Porter-Hudak (GPH) approach with bandwidths 0.5 and 0.8, respectively, log periodogram regression with Fourier terms (as in Moulines and Soulier 1999), and the local Whittle estimator with bandwidth M =T 0.5 . The reported estimates are the mean over 5000 replications, and the figures in brackets report the standard deviation of the estimated parameters. GPH (M =T0.5) GPH (M =T0.8)Fourier Estimations Local Whittle Model 1 0.141 0.557 0.467 0.140 (0.109) (0.026) (0.125) (0.089) Model 2 0.284 0.741 0.447 0.282 (0.147) (0.051) (0.168) (0.135) 4. Tests of Long Memory In the presence of long memory, it becomes important to be able to discern when a series has a true associated d. An ideal testing strategy for long memory would follow two steps: (1) Test for the significance of d in the first stage (ideally with a narrow bandwidth GPH or local Whittle), and conditional on a nonrejection in (1), in the second stage, apply more formal tests from the literature. Such an approach would present a good compromise between size and power. To optimize the trade-off, we applied the log-periodogram test of Davidson and Sibbertsen (2009) and the skip-sampling test of Davidson and Rambaccussing (2015). They test for the null hypothesis of long memory. A brief explanation of both tests follows. The test of Davidson and Sibbertsen (2009) is a log-periodogram regression test with the null hypothesis that the time series is believed to be pure long memory containing no bias. To quantify the bias in d, the test estimates the potentially biased estimators under two different bandwidths, and tests whether the bias is statistically significant. The test proceeds by computing the Geweke and Porter-Hudak (1983) estimates from the periodograms with two different bandwidths (M1=T0.5 and M2=T0.8). The Geweke and Porter-Hudak approach involves estimating dfrom a periodogram based model. log(I(λk)) =β0−dlog4sin2λk 2+εk(4) where I (λk) is the periodogram at the harmonic frequencies 2 πj/T where j=1, 2, . . . M. The choice of the bandwidth M has been discussed in various papers. GPH recommend M =T 0.5 for unbiasedness, while Hurvich and Deo (1999) recommend T0.8 for efficiency. J. Risk Financial Manag. 2020,13, 186 6 of 11 Similar to the Hausman test, the bias and consistency are tested using a standard t-statistic: TS =ˆ d1−ˆ d2 SE ˆ d1−ˆ d2, (5) where the standard error (SE) makes use of a reduced form from Hurvich and Deo (1999). ˆ d1 is the parameter from the narrow bandwidth (M1) and ˆ d2is parameter from a broad bandwidth (M2). The second test we considered is the skip-sampling test, which exploits the property of aggregation. The test can include a range of alternatives including nonlinear models and weakly dependent data. If a series is indeed long memory, then recreating a series by subsequently sampling in their rightful ordering should preserve the autocorrelation structure of the original series. Under the null hypothesis of a true long memory process, recreating another series based on consecutive sampling (every kth observation) will ensure that the newly computed autocovariance will have similar memory structure for the newly computed series. TS =ˆ d−ˆ dN SE ˆ d1−ˆ dN, (6) where ˆ dN is the average memory parameter from Nskip samples. The standard error can be approximated by an asymptotic variance or a bootstrapped variance. In this paper, we focus on the former (and the results still hold under the latter). 5. Results 5.1. Data In the analysis, we used daily return data of the five most traded cryptocurrencies with the highest market capitalization as of 31 December 2019. The data, downloaded from coinmarketcap.com, covers the following sample sizes: Bitcoin (2190 days), Ethereum (1553 days), Litecoin (2435 days), Bitcoin Cash (883 days), and XRP (2340 days). We calculated the returns as the log difference of the closing prices. We employed three volatility proxies: absolute return, squared return, and the log range as measured in Phillip et al. (2019). Figure 3shows the plot of the daily closing prices for the period starting 2 January 2014 until 31 December 2019. Exceptionally, the series for Ethereum starts from 30 September 2015, and that of Bitcoin Cash from 1 August 2017. The closing prices show the predominance of Bitcoin, which had lower closing prices at the start of the sample, picked up in mid-2016, and reached its highest value at the end of 2018, and with moderate fluctuations from then on. Other currencies, such as XRP, started picking up from early 2017, but showed signs of decline by the end of 2019. Examining the period 2017–2019, Bitcoin would have been the highest return, averaging 0.1% daily, which is substantially higher than the other currencies, which have been mostly negative for the period. Moreover, the standard deviation of returns turns out to be the lowest over the period, hence having the highest Sharpe ratio. The smaller currencies—XRP and Bitcoin Cash—turn out to be more volatile. Table 2also reports the summary statistics for the five currencies. The results show that the daily returns are fairly small, with Ethereum performing better than the remaining currencies. The standard deviation differs across the various currencies with less variability encountered with Bitcoin. J. Risk Financial Manag. 2020,13, 186 7 of 11 J. Risk Financial Manag. 2020, 13, x FOR PEER REVIEW 6 of 10 dependent data. If a series is indeed long memory, then recreating a series by subsequently sampling in their rightful ordering should preserve the autocorrelation structure of the original series. Under the null hypothesis of a true long memory process, recreating another series based on consecutive sampling (every kth observation) will ensure that the newly computed autocovariance will have similar memory structure for the newly computed series.  =   −   (  −  ), (6) where   is the average memory parameter from N skip samples. The standard error can be approximated by an asymptotic variance or a bootstrapped variance. In this paper, we focus on the former (and the results still hold under the latter). 5. Results 5.1. Data In the analysis, we used daily return data of the five most traded cryptocurrencies with the highest market capitalization as of 31 December 2019. The data, downloaded from coinmarketcap.com, covers the following sample sizes: Bitcoin (2190 days), Ethereum (1553 days), Litecoin (2435 days), Bitcoin Cash (883 days), and XRP (2340 days). We calculated the returns as the log difference of the closing prices. We employed three volatility proxies: absolute return, squared return, and the log range as measured in Phillip et al. (2019). Figure 3 shows the plot of the daily closing prices for the period starting 2 January 2014 until 31 December 2019. Exceptionally, the series for Ethereum starts from 30 September 2015, and that of Bitcoin Cash from 1 August 2017. The closing prices show the predominance of Bitcoin, which had lower closing prices at the start of the sample, picked up in mid-2016, and reached its highest value at the end of 2018, and with moderate fluctuations from then on. Other currencies, such as XRP, started picking up from early 2017, but showed signs of decline by the end of 2019. Figure 3. Plot of Time Series. Plot of closing prices for Bitcoin, Bitcoin Cash, Ethereum, Litecoin, and XRP. Figure 3. Plot of Time Series. Plot of closing prices for Bitcoin, Bitcoin Cash, Ethereum, Litecoin, and XRP. Table 2. Summary Statistics. The table reports the summary statistics for the cryptocurrencies in the sample. Bitcoin Bitcoin Cash XRP Litecoin Ethereum Mean 0.00102 −0.0007 0.00149 0.00098 0.00332 Std dev 0.03897 0.07881 0.07271 0.06444 0.06104 Skewness −0.340 0.612 2.076 1.720 0.271 Kurtosis 8.459 10.651 32.91 28.618 7.052 Jarque–Bera 2761 2206 88925 67783 1082 N 2190 882 2340 2435 1553 5.2. Findings We examined the presence of long memory in the price processes of the five most common cryptocurrencies and report the results of the estimated long memory tests in Table 3. Panel A shows the test results for the returns, while Panel B for the estimated volatility proxies. In interpreting the test results, we first checked the conventional p-value of the GPH estimator. The parameters show low estimated d values ranging from 0.057 (XRP) to 0.259 (Ethereum), and the corresponding p-values do not reject the null that the parameter is no different from zero. In this case, it is not convenient to proceed to the second stage and interpret either the bias or the skip-sampling test as the null of long memory is already rejected at the first stage. J. Risk Financial Manag. 2020,13, 186 8 of 11 Table 3. Long Memory Tests. The table reports the long memory test results for the five most common cryptocurrencies. Panel A exhibits the GPH estimates, bias test, and skip-sampling tests with different rates of h =4, 8 months for returns. Panel B shows the results for three measures (i.e., proxies) of volatility: daily low high range (LHR), squared returns (SR), and absolute returns (AR). The two skip-sampling rates were chosen in favor of producing results with a narrow bandwidth. T is the sample size in number of days. Bitcoin Ethereum Litecoin Bitcoin Cash XRP T=2190 T =1553 T =2435 T =883 T =2340 Estimate p-Value Estimate p-Value Estimate p-Value Estimate p-Value Estimate p-Value Panel A: Return GPH ( ˆ d)0.153 0.165 0.259 ** 0.036 0.170 0.111 0.121 0.405 0.057 0.589 Bias Test −0.454 0.675 −1.407 0.920 −0.122 0.549 0.806 0.210 −0.886 0.812 Skip-Sampling (h =4) −0.522 0.699 −0.816 0.793 0.946 0.172 −1.084 0.861 0.624 0.266 Skip-Sampling (h =8) 0.378 0.353 0.168 0.433 0.044 0.482 −0.486 0.686 0.544 0.293 Panel B: Volatility Volatility (LHR) GPH ( ˆ d)0.536 *** 0.000 0.505 *** 0.000 0.584 *** 0.000 0.541 *** 0.001 0.486 *** 0.000 Bias Test −0.590 0.722 0.044 0.483 * 0.084 0.467 0.322 0.374 2.356 *** 0.009 Skip-Sampling (h =4) −0.073 0.529 0.431 0.333 −0.768 0.779 −0.451 0.674 −0.406 0.658 Skip-Sampling (h =8) 0.193 0.424 0.560 0.288 −0.423 0.664 −0.307 0.645 −0.560 0.712 Volatility (SR) GPH ( ˆ d)0.326 *** 0.004 0.202* 0.099 0.359 *** 0.000 0.174 0.235 0.191 * 0.077 Bias Test 1.524 * 0.064 −1.865 0.969 0.365 0.357 3.119 *** 0.001 0.868 0.193 Skip-Sampling (h =4) −0.860 0.805 0.637 0.262 0.142 0.443 0.585 0.279 −0.737 0.770 Skip-Sampling (h =8) −1.055 0.854 0.370 0.356 0.539 0.295 0.289 0.386 −0.752 0.774 Volatility (AR) GPH ( ˆ d)0.375 *** 0.001 0.378 *** 0.003 0.466 *** 0.000 0.336 ** 0.026 0.381 *** 0.001 Bias Test −2.229 0.987 −1.535 0.938 2.086 ** 0.018 1.957 ** 0.025 −1.185 0.882 Skip-Sampling (h =4) −1.027 0.848 0.423 0.336 −1.229 0.890 −0.013 0.505 0.070 0.472 Skip-Sampling (h =8) −1.081 0.860 0.848 0.198 −0.482 0.685 −0.257 0.601 −0.421 0.663 ‘***’, ‘**’, ‘*’ implies significance at the 1%, 5% and 10% respectively.