An α -Stable Approach to Modelling Highly Speculative Assets and Cryptocurrencies
AAn Alpha-Stable Paretian approach to modelling highly speculative assets and cryptocurrencies
Taurai Muvunza ∗ Tsinghua-Berkeley Shenzhen InstituteData Science and Information Technology Center
Abstract
We investigate the behaviour of cryptocurrencies’ return data. Using return data forbitcoin, ethereum and ripple which account for over 70% of the cyrptocurrency market,we demonstrate that α -stable distribution models highly speculative cryptocurrenciesmore robustly compared to other heavy tailed distributions that are used in financialeconometrics. We find that the maximum likelihood method proposed by DuMouchel(1971) produces estimates that fit the cryptocurrency return data much better thanthe quantile based approach of McCulloch (1986) and sample characteristic method byKoutrouvelis (1980). The empirical results show that the leptokurtic feature presentedin cryptocurrency return data can be captured by an α -stable distribution. This paperscovers predominant literature in cryptocurrencies and stable distributions. Keywords: cryptocurrency, bitcoin, α -stable distribution, heavy tails. ∗ Address: Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate School,University Town, Nanshan District, Shenzhen, 518055, China; Email: [email protected]. Wewould like to thank participants of The 4th PKU-NUS International Conference in Quantitative Financeand Economics for their valuable comments. a r X i v : . [ q -f i n . M F ] F e b Introduction
Bitcoin, ethereum and ripple are members of a family of cryptocurrencies whose returnbehaviour exhibit features that are inconsistent with traditional commodities and stocks.Thereturn distribution of cryptocurrencies is characterised by skewness, a higher peak and heavytails in contrast to those of normal distribution. Buttressing the distribution of price changesfor any asset is vital for risk analysis and portfolio management. In this paper we model thecryptocurrencies with α -stable distribution, and compare the goodness of fit to other distri-butions that are commonly used in financial econometrics. Our results show that α -stableis the best distribution for the return data of cryptocurrencies. This paper is structured asfollows; the first section reviews literature on cryptocurrencies, section 2 describes the data,section 3 introduces the emperical model used and covers literature on application of stabledistributions. We review parameter estimation techniques in section 4 before discussing re-sults and goodness of fit test in section 5. We conclude in section 6.Cryptocurrencies have received a lot of attention in mainstream media but their returnbehaviour has not been fully examined. While structural breaks in returns and volatility ofbitcoin are frequent (Thies and Molnar, 2018) this paper proposes a distribution that can beutilised in the cryptocurrency market given any return series of the currencies under study.Bitcoin has characteristics that are similar to gold and the dollar (see Dyhrberg, 2016), oneextreme being the pure store of value and the other extreme being pure medium of exchange.As a store of value such as gold, cryptocurrencies should not generate cashflow but rather re-tain their value (Cheah and Fry, 2015). However, bitcoin for instance is characterised by highvolatility which makes it possible to earn ernomous returns. The weak correlation betweencryptocurrencies and equity markets (Bouri abd Molnar, 2017) enhances the attractivenessof the currencies. When pulled together with traditional assets, bitcoin increases the valueof a portfolio (Trimborn and HÃďrdle, 2017) and can further serve as a hedge, safe havenor diversifier for other equity indices (see Bouri, 2017). Much of the academic literatureon cryptocurrencies focuses on price volatility and legal aspects, and a more comprehensiveanalysis related to the behaviour of returns is necessary.In his work "Two Concepts of Money", Goodhart (1998) stressed that severe transactioncosts in barter could lead to an evolution in search for cost minimization procedures withina private sector system, with which the government has no control at all. Cryptocurren-cies have emerged as pioneers of this evolution. Bitcoin was first introduced by pseudonymSatoshi Nakamoto, who argued that traditional financial institutions that are establishedon trust-based models have high transaction costs since they cannot avoid meditating dis-putes as third parties. Nakamoto (2008) further proposed that an electronic system basedon cryptographic proof instead of trust is one of the ways third party costs can be avoided.Bitcoin, Etherium, Ripple, amongst other cryptocurrencies are an innovation that simplifiespayment without the need for a third party. Bitcoin is believed to have been first mintedon January 4th in 2009, its first payment occurred on January 11th and the software waspublicly released as open source on the 15th of the same month and from there onwards,anyone with required technical skills could participate.With a market capitalization that reached 314 billion dollars in December 2017, bitcoin isan independent currency that has become popular among investors, consumers and retailers.The growing popularity of the use and acceptance of cryptocurrencies suggest that they canbecome an alternative currency in the future. European Central Bank (2012) pointed thatan increase of electronic commerce particularly digital goods, growing access to and use ofinternet, higher degree of anonymity and lower transactions will precipitate the growth ofdigital currencies in the future. Cryptocurrencies are a form of digital currencies that aredifferent from deposits. Dwyer (2014) clarifies this difference in that deposits are representedby a bank account balance at an institution while digital currency is viewed as storage ofvalue which can be transferred without the intermediation of a financial institution. Inthe absence of this intermediation, digital currencies must not allow users to spend theirbalances more than once. To avoid double spending, Bitcoin uses peer-to-peer networksand open source software which generate computational proof of the chronological order oftransactions, secured by a system of verifiable nodes, (Nakamoto 2008). Bariviera, Basgall,Hasperue and Naioufu (2017) note that a distributed ledger which came with the invention2f bitcoin is a key innovation in decentralizing and democratizing the currency. Distributedledger is a consensus of replicated and synchronized digital data shared across the world bya group of peers who share responsibility for maintaining the ledger, (Deloitte, 2016).Researchers have expressed diverging views on whether bitcoin is a currency or not. Yer-mack (2013) posits that with an exchange rate volatility that is higher than commonly usedcurrencies, bitcoin exhibit zero correlation with other currencies and concluded that bitcoindoes not behave as a currency. Paul Krugerman has been a longtime critic of ccryptocur-rencies especially bitcoin, he argues that the currency is not a reasonably stable store ofvalue. The rise in price of bitcoin from 4.951 cents on the first day of trading in July 2010(Yermack, 2013) to its highest peak of 18 737.60 US dollars (coinmarketcap.com) in Decem-ber 2017 shows that the criticism does not seem to thwart the demand for bitcoin. Thetotal number of cryptocurrencies has exceeded 5000 in over 20 000 markets, commanding atotal market capitalization of 270 billion dollars of which over 60% is dominated by bitcoin(coinmarketcap.com, 2020).Since inception, cryptocurrencies have become widely acceptable as a medium of ex-change across the world. Hankin, (2017) records the first use of bitcoin to have been a pizzabought for 10 000 bitcoins in 2010, and afterwards, internet reports propagate the use ofbitcoin to purchasing of illegal drugs, raising concern over its anonymity. Despite its pop-ularity, the road to fame for bitcoin has been marred by challenges stemming from moneylaundering, drug dealing, fraud and security concerns. With approximately 1 billion dollarsworth of bitcoin in circulation in 2013, the U.S Senate Committee set up a hearing to lookinto bitcoin, fearing that the system was a vehicle for money laundering and drug dealing.The hearing occurred after FBI shut down Silk Road, a website which sold illegal goods anddrugs in bitcoin, (Dwyer, 2014). BBC (2013) reports that the currency trebled after newsthat the committee was told cryptocurrencies were legitimate financial services comparableto other online payment systems. Another major blow hit bitcoin in February 2014 when theTokyo based Mt. Gox, the first and largest bitcoin exchange trading platform which han-dled over 70% of all bitcoin transactions worldwide immediately suspended all transactions.According to Popper and Abrams (2014), Mt. Gox filed for bankruptcy citing "a weaknessin our system" referring to what Hern (2014) called a loophole in bitcoin system that wasexploited by hackers to get over 800 000 free bitcoins which accounted for 6% of the totalbitcoins at the time.In academic literature, cryptocurrencies have not been fully examined. Dwyer (2014)focused on the price and returns of bitcoin and concluded that bitcoin is 10 times morevolatile than stocks. Cheah et al (2015) find that bitcoin exhibits speculative bubbles andfurther postulate that the fundamental value of bitcoin is zero. Blau (2017) argues thatspeculative trading does not explain bitcoinâĂŸs price volatility. Bouri et al (2017) usesa dynamic conditional correlation model to examine whether Bitcoin acts as a safe havenor hedge for stocks, bonds, oil and gold; and the empirical results conclude that Bitcoinis a poor hedge and can be used to eliminate idiosyncratic risk only. Dyhrberg (2015)demonstrate that bitcoin behaves like a currency and it has many similarities with goldand the dollar, one extreme being the pure store of value and the other extreme beingpure medium of exchange. This gives bitcoin more advantages over other currencies asit can be used as an asset for portfolio management in the financial market. Urquhart(2017) examines the efficiency of bitcoin and finds that bitcoin market is still inefficient butmoving towards an efficient market. Bradvold et al (2015) submits that bitcoin exchangeshave significant contributions to bitcoinâĂŸs price discovery due to information sharing.Ali, Barrdear, Clews and Southgate (2014) posit that digital currencies such as bitcoindo not pose a material risk in the United Kingdom because they are only exploited bya few people. Trimborn, Li and HÃďrdle (2017) show that cryptocurrencies add valueto a portfolio and using Markowitz optimization framework, they demonstrate that theirapproach can increase return of a portfolio while lowering volatility. Briere, Oosterlinck andSzafarz (2015) include bitcoin to a portfolio of traditional assets and arrived at the sameconclusion. Our work focuses on modelling the returns of crytpocurrencies. In this paperwe model the top three cryptocurrencies that account for over 70% of the cryptocurrencymarket capitalisation return data using stable distribution. Understanding the distribution3f returns is critical in evaluating risk and managing a portfolio of assets. To calculate returns, we used the first difference of cryptocurrency’s log close price from1 Dec 2011 to 31 Dec 2017, 7 August 2015 to 20 April 2018, 5 August 2013 to 20 April 2018for bitcoin, ethereum and ripple, respectively. We use daily data from Yahoo Finance andit was stationary at first difference of log close price. Table 1 shows summary statistics ofbitcoin return data and we note that, for instance, ripple data has a kurtosis of 22.325 whichis by far more than the kurtosis that is fit for a normal distribution.Table 1: Summary StatisticsCrypto Mean Std.dev Skewness Kurtosis Min Max J-B test ObsBTC 3.655E-03 0.0642 4.888 160.27 -0.849 1.474 2.261E6 2 187ETH 0.489E-02 0.083 -1.214 20.901 -0.916 0.3383 5.926 971XRP 0.286E-02 0.121 0.747 22.325 -0.997 1.028 5.939 1165Figure 1 shows the bitcoin closing price and the first difference of closing price from 31December 2011 to 31 December 2017. We note that, for instance, return for ripple has aFigure 1: Bitcoin Closing Price and Returnskurtosis of 22.325 which is by far more than the kurtosis that is fit for a normal distribution.As a result, a non-Gaussian approach is necessary in order to understand the distribution ofcryptocurrenciesâĂŹ returns. Jarque-Bera test rejects the null hypothesis that the returnsare normally distributed at 5% significance level. The purpose of this paper is to fit stabledistribution to bitcoin returns and compare the goodness of fit with other heavy taileddistributions such as student-t distribution. Furthermore, we observe frequent jumps in thereturn data of bitcoin and consequently the assumption of finite variance in this case maynot hold. Stable distribution which assumes infinite variance can explain the data betterthan the normal distribution in this case. α -Stable Distribution Stable distributions are a rich class of distributions that includes the Gaussian, Levy andCauchy distributions in a family that allows for skewness and heavy tails, (Nolan, 1999b).Price changes are a result of new information into the market and of the re-evaluation ofexisting information, thus, changes in price represent the effect of many different bits ofinformation (Fama,1965). Consequently, these bits of information may combine in additivefashion to produce stable distributions for daily, weekly or monthly periods. Stable dis-tribution has been popular in statistical analysis of financial data since they are the onlypossible limiting distributions for sums of independent, identically distributed (i.i.d) randomvariables (Lux, 1996). Researchers have shown that changes in stock prices exhibit highvolatility and statistical techniques such as the normal distribution which depend on theasymptotic theory of finite variance distributions are inadequte. An α -stable is a levy pro-cess whose departure from the Brownian motion is controlled by the tail index α , which liesin the range < α < . The additive property of the stable distribution can be expressedas follows: 4f X, X , X · · · X n are random variables, then for very positive integer n , there existconstants a n > , B n such that X + X + ... + X n ≈ a n X + B n thus LHS converges in distribution to the RHS. The most common parameterization for stable distribution is defined by Samorodnitskyand Taqqu (1994): A random variable X is S ( α, β, γ, δ ) if it has characteristic function. E ( exp i tX ) = exp (cid:18) − γ α | t | α (cid:20) − iβ (tan πα sign t ) (cid:21) + iδt (cid:19) if α (cid:54) = exp (cid:18) − γ | t | (cid:20) iβ π ( sign t ) ln | t | (cid:21) + iδt (cid:19) if α = 1The parameter α is the index of stability andsign t = if t >0 if t =0 − if t<0A stable class has four parameters α, β, γ, δ , where α describes the tail of the distribu-tion, β is the skewness of the distribution, δ is the location parameter, and γ is a scaleparameter. As α increases, the effect of β decreases. Figure 2 shows the shapes of α -stabledistribution for different values of α and β .When compared with other models used to capture leptokurtic features such as affinejump diffusion models and generalised hyperbolic models, α -stable distribution is not onlyparsimonious with its four free parameters but also a creative model that is close to reality.Furthermore, setting α below two effects a pure jump process with fat tails in the returndistribution of cryptocurrencies and with such an infinite number of jumps, α -stable distri-bution incorporates extreme market movements traditionally handled by diffusion processes.The major drawback of α -stable distribution is that the density and distribution functionsdo not have closed form solutions except for a few members of the stable family.The distri-bution functions of stable distribution are known analytically under rare situations, that is,Cauchy distribution where α =1 and Gaussian distributions where α = 2 and the stable lawof characteristic component α = 1 / .However, empirical efforts have been made to alleviate this challenge. For instance, esti-mators for scale parameter and characteristic component were suggested by Fama and Roll(1971) who further provided probability tables of symmetric members of stable class withfinite mean. They further suggested estimators for scale parameter and characteristic com-ponent; and examined goodness of fit test and stable test as a robustness checks for dataanalysis. Koutrouvelis (1980) used a regression-type method of estimating the four param-eters of a stable distribution and found that the estimators were consistent and unbiasedwhen analyzing large sample sizes. Paulson, Holcomb and Leitch (1975) improved the workof Fama and Roll (1971) by relaxing the hypothesis that stable distribution is symmetric, β = 0 . When they allowed β to vary, the maximum absolute difference between the empiri-cal and fitted distribution decreased significantly by 50% when compared to Fama and Roll(1971) procedure which is applicable only to symmetric distributions.Since the closed form probability density function for stable distribution is unknown exceptfor a few members of the stable family, most of the conventional methods in mathematicalstatistics could not be used. The probability densities of α -stable random variables exist andare continuous but, with a few exceptions, they are not known in closed form, (Zolotarev1986b). These exceptions are: For proofs and derivation of the stable distribution properties, see Samorodnitsky and Taqqu, 1994 < α (cid:53) , | β | (cid:53) min ( α, − α ), γ > - ∞ < δ < + ∞ . When α = 2 , the resulting distribution is a normaldistribution with mean δ and variance 2 γ . When β = 0 , the distribution is symmetric, and if β is greater than one the distribution is skewed tothe right and if beta is less than one the distribution is skewed to the left. δ is equal to the mean of the distribution if α equals one. δ shifts the distribution either to the left orto the right γ compresses or extends the distribution about δ in proportion to γ
5. The Gaussian distribution S ( σ, , µ )=N ( µ, σ ). A Gaussian distribution is a specialcase of stable distribution with α =2, such that N ( µ, σ )= S (2 , , σ √ , µ ), where µ isthe mean of the nomal distribution and σ is the standard deviation of the normaldistribution. As noted earlier, when β = 2 there is no effect on stable distributionas the resulting distribution will be a normal distribution. The probability densityfunction is given by σ √ π exp − ( x − µ ) / σ
2. The Cauchy distribution. The Cauchy distribution is also another form of stable distri-bution with α =1 and β =0, such that Cauchy ( δ, γ )= S (1 , , γ, δ ), where γ is the scaleparameter and δ is the location parameter of the Cauchy distribution. The probabilitydensity function is given by γπ (( x − δ ) + γ ) , −∞ < x < ∞ If X ∼ S ( γ, , ), then for x > , its can be shown that P(X ≤ x )= + π arctan( xγ )3. The Levy distribution is also special case of stable distribution where α =0.5 and β =1.In other words, Levy ( δ, γ )= S / (0 . , , γ, δ ). The probability density function is givenby (cid:114) γ π x − δ ) / exp (cid:20) − γ x − δ ) (cid:21) , δ < x < ∞ The PDF is concentrated on δ, ∞ . If X ∼ S / (0 . , , γ , δ ), then for x > P ( X ≤ x ) = 2 (cid:18) − φ (cid:18)(cid:114) γx (cid:19)(cid:19) where φ denotes cumulative distribution function of the N (0 , distribution. DuMouchel (1971), (1973) was the first to propose the method of Maximum Likelihood(ML) principle to bracketed data in order to estimate parameters of stable distribution; andto further provided a table of the asymptotic standard deviations and correlations of the MLestimators. McCulloch (1986) introduced the quantile-based method to estimate the fourparameters of a stable distribution using five predetermined sample quantiles with the aid ofaccompanying tables. ML method has received wide acceptance and use in approximatingstable parameters for financial data. Mittnik et al (1999) implemented FFT-based MonteCarlo procedure to compare ML method with quantile-based method of McCulloch (1986)and found that the ML method is not only fast but also performs accurately compared toPDF calculations based on direct numerical integration. They also concluded that unlikeML estimator which can be easily modified to accommodate complicated extensions, thequantile-based method cannot be extended to complex estimation problems such as regres-sions that contain stable paretian disturbances, ARMA and GARCH models that are drivenby stable paretian innovations.Furthermore, Nolan and Ojeda (2013) showed that ML outperforms OLS regressionmethod and the performance of ML increases as the error distribution deviates from nor-mality. Nolan (1998), (1999a) implemented the parameterization used by ZolotarevâĂŸs(1986), Samorodnitsky and Taqqu (1994), McCulloch (1985) and DuMouchel (1971) (1973)into STABLE programme that can be used to give reliable computations of stable densities.Nolan (2001a) warned that stable distribution should be used to summarise the shape ofthe distribution and not to make statements about tail probabilities. In this paper we useSTABLE programme to estimate the parameters and densities of the stable distribution. α -stable distribution for different values of α and β ( X , ...., X T )be a vector of T i.i.d stable Paretian random variables, and also x ∼ S α ( α, β, γ, δ ). Defining θ = ( α, β, γ, δ ), Mittnik, Rachev Doganoglu and Chenayo (1999) developed a ML algorithmand showed that the estimate of θ can be obtained by maximising the log-likelihood function (cid:96) ( θ, x ) = (cid:88) Ti =1 log f ( x i , θ ) with respect to the unknown parameter vector θ . DuMouchel (1973) applied ML estimationto stable distribution inference and defined the likelihood function by L ( θ ) = n (cid:89) k =1 S α,β (cid:18) X k − δγ (cid:19)(cid:30) γ where θ = ( α, β, γ, δ ) based on x = ( x , ...x n ) for a sample size n .Another technique to estimate the parameters of a stable distribution is the quantilebased approach introduced by McCulloch (1986). Using bitcoin as an example, we have 2186 independent drawings X i , from stable distribution S α , ( α, β, γ, δ ). We let X p be the p − th population quantile such that S α ( X p , α, β, γ, δ ) = p . Given the above, we let (cid:99) X p be the corresponding sample quantile with continuity correction . Thus, (cid:99) X p is therefore aconsistent estimator of X p . McCulloch defined the following υ α = X . − X . X . − X . ; υ β = X . + X . − X . X . − X . where υ α and υ β are independent of γ and δ . By letting (cid:99) υ α and (cid:99) υ β be corresponding valuesof υ α and υ β , respectively, and given that υ α and υ β are functions of α and β , the followingrelationship can be established: υ α = φ ( α, β ); υ β = φ ( α, β ) The above relationship can further be inverted to yield the following α = ψ ( υ α , β ); β = ψ ( υ α , β The parameters of α and β may now be consistently estimated by ˆ α = ψ ( (cid:99) υ α , (cid:99) υ β ); ˆ β = ψ ( (cid:99) υ α , (cid:99) υ β ); McCulloch (1986) showed the results of the relationship between υ α and υ β in table I-Vof his paper. We also used quantile-based method to estimate the parameters of the stabledistribution. A detailed approach of the sample characteristic method is found in Koutrou-velis (1980), Kogon and Williams (1998) and further clarified by Kateregga, Mataramvuraand Zhang (2017). In literature, ML Method has been found to yield consistent and accu-rate parameters of the stable distribution. The following table shows results of parameterestimates using Maximum Likelihood Method.Table 2: Estimates of α -stable distribution for bitcoinEstimator α β γ δ ML Estimator 1.186 ± ± ± ± α -stable distribution for EtheriumEstimator α β γ δ ML Estimator 1.186 ± ± ± ± α -stable distribution for RippleEstimator α β γ δ ML Estimator 1.1750 ± ± ± ± Results and Goodness of fit test
There are different ways that we can explore to establish whether the data is from astable distribution. Nolan (1999), (2001) underscored that many heavy tailed distributionsare not stable. We therefore need to test NolanâĂŸs (1999) proposition by comparing esti-mates from three different estimation techniques namely ML by DuMouchel (1971), quantile-based method by McCulloch (1986) and lastly the sample characteristic method proposedby Koutrovelis (1980), and Kogon and Wiliams (1998). For more details on the parameteri-zations of the characteristic function, refer to Zolotarev (1986). The argument is that thesedifferent methods are consistent estimators of parameters of a stable distribution. If theestimates are close then the hypothesis that the data is drawn from an α -stable distributionis supported. However, Nolan (1999) does not state a boundary of how close the estimatesshould be relative to each other.We use the non-parametric Kolgomorov-Smirnov test (K-S test) to compare the goodnessof fit for the three subclasses of stable distributions and the student-t distribution for eachcryptocurrency under study. We further examine the estimation technique that yields thebest fit for the cryptocurrencies under study. Most heavy tailed continuous distributionsused in financial econometrics such as Log-logistic, Weibull and Log-normal only assume apositive vector of returns, hence, we could not use them. Although Generalised Gammaand Generalised Extreme Value distributions, among others, can be used with a vector ofnegative and positive values, they were insignificant at all levels and we included only Levydistribution as an example of that case.Table 5: Results of the K-S test for BitcoinSig. level α - Stable Cauchy Student-t Levy20% 0.0229 0.0229 0.0229 0.022910% 0.0261 0.0261 0.0261 0.02615% 0.0290*** 0.0290 0.0290*** 0.02901% 0.0347*** 0.0347*** 0.0347*** 0.0347test stat 0.0261 0.0292 0.0270 0.2923p-value 0.0989 0.0472 0.0805 3.0913E-163Table 6: Results of the K-S test for EtheriumSig. level α - Stable Cauchy Student-t Levy20% 0.0343*** 0.0343 0.0343 0.034310% 0.0392*** 0.0392 0.0392*** 0.03925% 0.0435*** 0.0435 0.0435*** 0.04351% 0.0521*** 0.0521*** 0.0521*** 0.0521test stat 0.0337 0.0517 0.0371 0.2955p-value 0.2176 0.0110 0.1350 2.7866E-74Table 7: Results of the K-S test for RippleSig. level α - Stable Cauchy Student-t Levy20% 0.0313*** 0.0313 0.0313 0.031310% 0.0357*** 0.0357 0.0357*** 0.03575% 0.0396*** 0.0396*** 0.0396*** 0.03961% 0.0475*** 0.0475*** 0.0475*** 0.0475test stat 0.0291 0.0394 0.0322 0.2864p-value 0.2725 0.0524 0.1746 6.5926E-84* means that the p-value is higher than the corresponding significance level, hence we acceptthe null hypothesis that the data comes from the stated distributionWe note that alpha stable, Student-t and Cauchy distributions are significant at diferentlevels for the currencies under study, however, when comparing the p-values, we find moreevidence in support of the alpha stable distribution than Student-t and Cauchy distributionssince the p-value of alpha stable is higher than that of other distributions. We also find thatStudent-t distribution outperforms the Cauchy distribution when considering the p-values.Levy distribution was not significant even at 1% level and this suggests that the distribution9annot be used to model cryptocurrencies and other highly speculative assets with similarcharacteristics. In this paper we have applied α -stable distribution to model cryptocurrency return dataand compared the goodness of fit with other heavy tailed distributions used in financialeconometrics. The empirical study shows that α -stable distribution with parameters esti-mated by ML method is better fitted to model highly speculative cryptocurrenciesâĂŹ returndata particularly bitcoin, ethereum and ripple. The leptokurtic features that exist in bitcoindue to high volatility can be captured by an α -stable distribution. For cryptocurrency data,we found that student-t distribution outperforms Cauchy distribution. However, the tailbehavior of the data deviates from that of Stable Paretian distribution, a phenomenon thatcould be associated with a generalized Pareto or simple Pareto tail-index estimate above 2which has been frequently cited as evidence against infinite-variance stable distribution. Ina critique, McCulloch (1995) argued that the inference is invalid since a tail index above 2can result from a stable distribution with α as low as 1.65.10 eferences [1] Ali, R.,Barrdear, J., Clews, R., and Southgate, J. 2014. The economics of digital curren-cies Bank of England Quarterly Bulletin . Q3, 276-286.[2] Bariviera, A.F.,Basgall, M.J, Hasperue, W., Naioufu, M. 2017. Some stylized facts of thebitcoin.
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