Covid-19 impact on cryptocurrencies: evidence from a wavelet-based Hurst exponent
M. Belén Arouxet, Aurelio F. Bariviera, Verónica E. Pastor, Victoria Vampa
CCovid-19 impact on cryptocurrencies: evidence from awavelet-based Hurst exponent
M. Bel´en Arouxet , Aurelio F. Bariviera ∗ , Ver´onica E. Pastor , and VictoriaVampa Universitat Rovira i Virgili, Department of Business, Av. Universitat 1, 43204 Reus, Spain Universidad de Buenos Aires, Facultad de Ingenier´ıa, Departamento de Matem´aticas Universidad Nacional de La Plata, Facultad de Ingenier´ıa, Departamento de Ciencias B´asicas, Argentina
September 15, 2020
Abstract
Cryptocurrency history begins in 2008 as a means of payment proposal. However, cryp-tocurrencies evolved into complex, high yield speculative assets. Contrary to traditional finan-cial instruments, they are not (mostly) traded in organized, law-abiding venues, but on onlineplatforms, where anonymity reigns. This paper examines the long term memory in return andvolatility, using high frequency time series of eleven important coins. Our study covers the pre-Covid-19 and the subsequent pandemia period. We use a recently developed method, based onthe wavelet transform, which provides more robust estimators of the Hurst exponent. We detectthat, during the peak of Covid-19 pandemic (around March 2020), the long memory of returnswas only mildly affected. However, volatility suffered a temporary impact in its long rangecorrelation structure. Our results could be of interest for both academics and practitioners.
Keywords: cryptocurrencies; Hurst exponent; wavelet transform; Covid-19
Cryptocurrencies have become one of the most traded financial assets in the last decade. In order toput their importance into perspective, two of the most important stock exchanges in the world, theNew York Stock Exchange and Nasdaq, report an average of $
30 billions and $
85 billions in dailyvolume, respectively. Over the last six months, daily transactions of cryptocurrencies varied between $ $
31 billions, depending on the day. As a consequence, cryptocurrencies have beenreceiving increasing interest from both academics and practitioners. As a new object of study, itposes several challenges. One of them is to examine the statistical properties of the price generatingprocess. According to the Efficient Market Hypothesis (EMH), the price of any speculative assetmust convey all available information [23]. In particular, the weak version of the EMH states thatthe current price includes all the information contained in the series of past prices. As a consequenceof the non-arbitrage possibility, price returns time series should follow a random process with nomemory. In particular, it is excluded the possibility of long-term memory, as it could allow forprofitable trading strategies.The EMH, specially regarding the presence of long-term memory has been subject to debatesince the work by Mandelbrot [35]. It has been extensively studied in developed and emerging stock ∗ Corresponding author. [email protected] a r X i v : . [ q -f i n . S T ] S e p arkets [13, 43, 16], in fixed income markets [18, 10, 11], interest rates [25, 45, 17], and exchangerates [41, 48, 51], among other financial time series.Studies related to cryptocurrencies are much more recent. Specially on these days it is relevantto investigate how markets react to such a global event as Covid-19 pandemic.Early papers studying the informational efficiency of Bitcoin time series [49, 42] find that thereturns (and some power transformations) had been informational inefficiency, but the also report atrend toward a more efficient behavior. Shortly after these papers, [8] confirms the diminishing mem-ory in daily returns, along with a highly persistent component in daily volatility. Such persistencejustifies the use of GARCH-type models in volatility, as proposed by [32, 33].One-time-only events such as a hughe price crash or a pandemic, could alter the stochasticprocess governing returns and volatilities. In this line, [15] finds that before the big price crash of2013, Bitcoin volatility was asymmetric in the opposite way of the traditional assets, whereas thisasymmetry is not found after the price crash. Similarly, a global event such as Covid-19 pandemic,could have non-trivial effects on volatility and returns. Consequently, it begins to be an area ofinterest for researchers. For example, [19] reports significant growth in both returns and volumestraded in large cryptocurrencies, and [26] affirms that levels of Covid-19 caused a rise in Bitcoinprices.According to the World Health Organization (WHO) [50], Covid-19 pandemic originates in thePeople’s Republic of China. On December 31st, WHO’s country office gathers information issued bythe Wuhan Municipal Health Commission reporting cases of ‘viral pneumonia’. At the beginning,the virus seemed to be circumscribed to that region. On January 24th. France informed of somecases from people who had been in the Wuhan region, constituting the first confirmed cases inEurope. However, infections and deaths associated to Covid-19 remained at relatively very lowlevels during January and February. As displayed in Figure 1, daily number of infected and deadpeople in Europe rocketed in March, and showed a diminishing trend by the end of that month.Europe has been the first region (after the inception in China) to suffer the pandemic, with its mostdeadly effect concentrated in a short time frame. Consequently, we would like to investigate if sucha sudden public health problem has had an impact on the cryptocurrency market.In this line, the aim of this paper is to study the long memory profiles of returns and volatilitiesof eleven cryptocurrencies, during a period spanning before and after the inception of the pandemicevent. We contribute to the literature in multiple ways: (i) we propose a new method that has notbeen applied before to compute the Hurst exponent in cryptocurrencies time series; (ii) we study aset of the eleven most important cryptoassets at high frequency; (iii) we discuss the effect of Covid-19pandemic on returns and volatilities.This remaining of the article is organized as follows: Section 2 presents the methodology usedin the paper; Section 3 describes the data used and discusses the empirical findings; and Section 4concludes. Harold Edwin Hurst was a British engineer, whose name is intrinsically connected to the study oflong range dependence in time series. His original method was presented in a series of papers inthe 1950s [28, 29, 30, 31]. Although it was originally formulated for the resolution of an specifichydrological problem, it turned out to have more universal applications in the field of time seriesanalysis. The Hurst exponent describes the persistent or anti-persistent character of a time series,arising from its long-range memory.The presence of long-range memory is compatible with the fractional Brownian motion modelpostulated in [37, 38]. It was precisely Benoit Mandelbrot [36] who proposed in the early 1970s the Assymetry in volatility means that the reaction of volatility to unexpected positive and negative changes is notthe same.
R/S method developed by Hurst, has some drawbacks, as it is bi-ased towards finding spurious long memory. This situation triggered the development of alternativemethods (e.g. aggregated variance approach [14], Higuchi method [27], Detrended Fluctuation Anal-ysis [44], etc.) to find better estimators for long-range dependence. The review conducted in [46]provides a comprehensive guide for the proper use of the different methods according to the signalcharacteristics.Previous works acknowledges that, for an arbitrary artificial time series, the wavelet method doesnot only find a more accurate H value than the R/S method, but also that it is not necessary toassess a priori the series stationarity [46]. Consequently, in this work we use a wavelet-based methodto estimate the Hurst exponent.The continuous wavelet transform allows to decompose the time series in the time-frequencydomain, and is defined as: W ψ f ( a, b ) = 1 √ a (cid:90) ∞−∞ f ( s ) ψ (cid:18) s − ba (cid:19) ds for a > a is the scale parameter, b is the shift parameter and ψ is the mother wavelet. Hereinafter,we will refer to this equation as W a,b .If the time series has H − self-affinity, i.e. if the time series satisfies a power law of the kind X ( ct ) ≈ c H X ( t ), the variance of the wavelet transform (Eq. (1)) will be asymptotically affected bya scale parameter: V ar ( a ) = E ( W a,b ) − ( E ( W a,b )) ≈ a β (2)where β ∈ [ − , H and β for self-affine series. Consequently, theHurst exponent is defined as: • H = β +12 , with β ∈ [ − , • H = β − , with β ∈ [1 , W a,b ;2. Compute, for a fixed scale, a , the average wavelet coefficients;3. Draw the log-log plot of coefficients vs. scale a .The key improvement presented in [4] consists in the use of more robust estimators for thecomputation of the coefficients required in step 2, compared to the estimators used in [47].Based on Eq. (1), it is proposed to estimate the coefficients of the variance of the time seriesusing two estimators. The first one is the well-known unbiased variance estimator (cid:100) V ar , which givena set of data x , . . . , x m is defined as: (cid:100) V ar ( x ) = 1 m − m (cid:88) i =1 ( x i − ¯ x ) (3)4he second one is the median absolute deviation (MAD), which is a more robust estimator of thevariance (see [40]): M AD ( W a,b ) = M ed ( | W a,b − M ed ( W a,b ) | ) (4)where M ed ( · ) is the median operator. This improved method will be referred, hereinafter, as AWC-MAD. In [4] AWC-MAD was benchmarked against the classical R/S method and competing alter-natives for averaging wavelet coefficients. In all cases AWC-MAD estimations of the Hurst exponentwere closer to the theoretical Hurst exponent in synthetic series with 32768 datapoints. Subsequently,[5, 6, 7] compare R/S and AWC-MAD methods for climate time series as well as for synthetic timeseries.Considering that the main goal of this paper is to study long range dependence in the cryptocur-rency market, with shorter time series, we benchmark theoretical and estimated Hurst exponent onfBm and fGn sinthetic series with 1435 datapoints. We generate rolling windows of 500 observations,moving forward one observation in each window. The algorithm to generate the artificial series wasproposed by [1], and implemented in MatLab with the function wfbm.m , using Daubechies waveletof order 10. Results displayed in Table 1. The first column is the theoretical Hurst exponent value,and the other columns display the mean and standard deviation of the estimations computed on allthe rolling windows with the AWC-MAD.Table 1: Estimation of the Hurst exponent for fBm and fGn sinthetic series, using the AWC-MADmethod. Estimated H (fBm series) Estimated H (fGn series)Theoretical H Mean Std. Dev. Mean Std. Dev.0.2 0.2649 0.0712 0.2602 0.01170.4 0.3513 0.0425 0.4097 0.02740.6 0.5697 0.0501 0.6044 0.02800.8 0.7552 0.0683 0.7898 0.0321 Cryptocurrencies are traded in online platforms that are not compelled to abide by national financialregulations. Thus, a careful selection of a reliable data source is crucial for the validity of results.Following [2] and [9], we downloaded data from [21]. We use high frequency (every two hour)price data of eleven important cryptocurrencies in terms of to traded volume. The period underexamination goes from 14/11/2019 to 08/06/2020, for a total of 2496 observations. We studiedtwo-hour returns and volatility. Considering that there are several ways of capturing volatility infinancial markets, we selected two widely used proxies [24, 3, 20]. Additionally, in order to take intoaccount the dynamic character of the market, we compute the Hurst exponents by means of rollingwindows. Each rolling window cointains 360 datapoints (30 trading days). Each window moves oneobservation forward, and deletes the first observation. Data covers the period that lies before andafter the onset of the Covid-19 pandemic. Namely, we compute the following measures from theprice time series: • Logarithmic or continuously compounded return: R t = log( P t ) − log( P t − ) (5)where P t and P t − are two consecutive closing prices every two hours.5 Max-Min volatility:
V ol
MinMaxt = log( P maxt ) − log( P mint ) (6)where P maxt and P mint the maximum and minimum prices observed in a two-hour period. • Absolute return volatility:
V ol
Abst = abs (log( P t ) − log( P t − )) (7)We obtained important results regarding long memory in returns and in volatility. With respectto returns, as can be appreciated in Figure 2, the effect of Covid-19 on the Hurst exponent is mild.The average Hurst exponent around 0.5, which is similar to previous findings for time series at lowand high frequency sampling [8, 12]. Table 2 displays the mean Hurst exponent of each return timeseries, for all windows, windows before 03/03/2020, windows during two weeks in March when therewas a peak in the pandemic, and windows after 18/03/2020. We observe that between March 3rd.and March 18th. the Hurst exponents are greater than in the preceding period, but they recoverprevious levels soon after this date.A different picture is obtained when we analyze volatility. There has been a striking effect ofCovid-19 on the long-term memory of volatility. This paper uses two proxies for volatility. Thefirst one, is the Max-Min volatility, and the dynamic evolution of its Hurst exponent is displayedin Figure 3. Considering that this way of measuring volatility uses the highest and lowest priceof a given cryptocurrency within a two-hour period, it is suitable to capture extreme events. InTable 3 we observe that the period that goes from November to the early March exhibits veryhighly persistence (Hurst exponent greater than 0.69). On the contrary, between March 3rd. andMarch 18th. the Hurst exponents drop suddendly, reflecting an antipersistent time series. Thissituation could reflect a moment of market panic, where large positive returns are followed by largenegative returns and vice versa. We should recall that during these days stocks markets around theworld were in free fall (for example, the German DAX index plunged by 26%). Cryptocurrencies,as alternative assets, were subject to speculative moves, and those large swings in volatilities couldreflect the uncertainty around the true value and feasibility of cryptocurrencies as a safe haven amidthe pandemic. After March 18th. the Hurst values reversed to values above 0.7, returning volatilityinto a highly persisting process.Absolute returns have been proposed to measure financial time series volatility. Similar to Max-Min volatility, absolute returns also exhibit a persistent behavior during the period before March.However, and at odds with the previous proxy, absolute returns increased in their persistence duringthe upheaval of the Covid-19 crisis (See Figure 4 and Table 4).These two apparently contradictory findings could be reconciled as follows. First, there is un-doubtedly a strong Covid-19 effect on volatility, as both proxies reflect strong changes in their Hurstexponents. Second, Max-Min and absolute returns measure different aspects of volatility. The for-mer is more sensible to extreme events, considers only the positive side of volatility, and measuresthe range of prices during a given hour. The latter, conversely, considers both positive and negativechanges as single absolute returns, being less sensitive to extreme events. Analyzing both resultsjointly, we can conclude that hours with large price swings were followed by hours with smaller priceswings during the worst days of the pandemic. However, two-hour returns were more persistentduring the same period. After March 18th. extreme events (as measured by Max-Min volatility)became again to be highly persistent, along with a persistence (albeit to a lesser extent) of absolutereturns. This paper’s contribution is twofold. On the one hand, it applies a new, improved, wavelet-basedmethod to compute the Hurst exponent. On the other hand, it provides an analysis of high frequency6igure 2: Hurst exponent of two-hour returns, using rolling windows.7igure 3: Hurst exponent of Max-Min two-hour volatility, using rolling windows.8igure 4: Hurst exponent of two-hour absolute returns, using rolling windows.9able 2: Mean Hurst exponent of return time series, for all windows, windows before 03/03/2020,windows between 03/03/2020 and 18/03/2020, and windows after 18/03/2020
Hurst exponentAll windows Before 03/03/2020 Between 03 and 18/03/2020 After 18/03/2020ADA 0.3869 0.3538 0.4727 0.4259BCH 0.4589 0.4727 0.5198 0.4092BTC 0.4747 0.4657 0.5529 0.4664DASH 0.4662 0.5014 0.5321 0.3706EOS 0.4655 0.4755 0.5430 0.4179ETC 0.4576 0.4711 0.4409 0.4353ETH 0.4670 0.4790 0.5203 0.4234IOTA 0.4448 0.4666 0.5089 0.3772LTC 0.4433 0.4461 0.4778 0.4257XMR 0.4085 0.3914 0.5061 0.4102XRP 0.4656 0.4855 0.4632 0.4253
Table 3: Mean Hurst exponent of Max-Min volatility time series, for all windows, windows before03/03/2020, windows between 03/03/2020 and 18/03/2020, and windows after 18/03/2020
Hurst exponentAll windows Before 03/03/2020 Between 03 and 18/03/2020 After 18/03/2020ADA 0.7278 0.7593 0.5529 0.7231BCH 0.6907 0.7005 0.4626 0.7493BTC 0.7050 0.6738 0.5894 0.8097DASH 0.6732 0.6625 0.4485 0.7731EOS 0.6814 0.7138 0.4833 0.6825ETC 0.6794 0.6994 0.2783 0.7766ETH 0.6597 0.6954 0.4468 0.6593IOTA 0.6895 0.7399 0.4887 0.6543LTC 0.6645 0.6911 0.4162 0.6952XMR 0.6883 0.6927 0.5452 0.7284XRP 0.7114 0.7214 0.5960 0.7306
Hurst exponentAll windows Before 03/03/2020 Between 03 and 18/03/2020 After 18/03/2020ADA 0.5920 0.5420 0.8421 0.6093BCH 0.6874 0.7078 0.8087 0.6033BTC 0.6651 0.6367 0.7892 0.6811DASH 0.7258 0.7493 0.8206 0.6443EOS 0.6882 0.7084 0.8499 0.5906ETC 0.6825 0.6937 0.8012 0.6184ETH 0.6518 0.6825 0.8446 0.5216IOTA 0.6578 0.6824 0.8012 0.5572LTC 0.6536 0.6588 0.8549 0.5730XMR 0.6233 0.6049 0.8319 0.5896XRP 0.7182 0.7271 0.8460 0.6555 return and volatility of eleven cryptocurrencies during Covid-19 pandemic. We find that, even thoughthe ongoing pandemic has produced only a mild effect on the long-range memory of cryptocurrencyreturns, it imprinted a strong transitory effect on volatility. We use two alternative measures ofreturn volatility: Max-Min and absolute returns. Both proxies reflect a significant change in theirlong memory profile. Regarding Max-Min, we observe a momentary swap, from a highly persistentprocess to an antipersistent one. Differently, absolute returns time series suffer a reinforcementin their persistent behavior during the upheaval of the pandemic. The divergent reaction of bothvolatility measures could be explained by the sensibility of the Max-Min measure to more extremeevents, as it only measures positive changes (
V ol
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