Changes to the extreme and erratic behaviour of cryptocurrencies during COVID-19
CChanges to the extreme and erratic behaviour ofcryptocurrencies during COVID-19
Nick James a , Max Menzies b , Jennifer Chan a a School of Mathematics and Statistics, University of Sydney, NSW, Australia b Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
Abstract
This paper introduces new methods for analysing the extreme and erratic be-haviour of time series to evaluate the impact of COVID-19 on cryptocurrencymarket dynamics. Across 51 cryptocurrencies, we examine extreme behaviourthrough a study of distribution extremities, and erratic behaviour through struc-tural breaks. First, we analyse the structure of the market as a whole and observea reduction in self-similarity as a result of COVID-19, particularly with respectto structural breaks in variance. Second, we compare and contrast these twobehaviours, and identify individual anomalous cryptocurrencies. Tether (USDT)and TrueUSD (TUSD) are consistent outliers with respect to their returns, whileHolo (HOT), NEXO (NEXO), Maker (MKR) and NEM (XEM) are frequentlyobserved as anomalous with respect to both behaviours and time. Even among amarket known as consistently volatile, this identifies individual cryptocurrenciesthat behave most irregularly in their extreme and erratic behaviour and showsthese were more affected during the COVID-19 market crisis.
Keywords:
COVID-19, cryptocurrencies, nonlinear dynamics, time series,anomaly detection
1. Introduction
The COVID-19 pandemic has had significant impacts on society, and prompteda substantial amount of attention and research. In epidemiology, studies havefocused on the spread of COVID-19 and potential measures of containment[1, 2, 3, 4, 5, 6], while clinical research has explored potential treatments for thedisease [7, 8, 9, 10, 11, 12], including a vaccine [13]. Researchers in nonlineardynamics have used a wide range of new techniques to analyse and predict thespread of COVID-19 cases and deaths [14, 15, 16, 17, 18, 19].By contrast, most research studying the impact of COVID-19 on financialmarkets has used traditional statistical methods such as parametric models[20, 21, 22, 23]. For example, [24, 25] use dynamic conditional correlations
Email address: [email protected] (Max Menzies) a r X i v : . [ q -f i n . M F ] N ov etween stock returns and hedging costs to study financial contagion and safeasset classes. Similarly, [26] studies financial contagion via cross-correlationanalysis, [27] uses a GARCH(1,1) model, while [28] is a qualitative overview offinancial contagion. [29, 30] use a value at risk measure to study safe havensin the cryptocurrency market. [31] uses existing metrics and tests (such asapproximate entropy, Lyapunov Exponents, t-tests and F-tests) to study thepredictability of price fluctuations. All these methods are well-studied in theexisting finance literature. Notably, [32] proposes new methods to detect assettail risk and determine if such risk can be reduced with safe asset classes.Since their inception, cryptocurrencies have been of great interest to re-searchers in dynamical systems and econophysics. There has been substantialresearch on individual cryptocurrencies such as Bitcoin [33, 34, 35, 36, 37] and thedisorder and fractal behaviour within cryptocurrencies in general [38, 39, 40, 41].Notably, Drozdz and coauthors have analysed several quantitative aspects of thecryptocurrency market closely, including cross-correlations, auto-correlations andscaling effects. Across three papers [42, 43, 44] they have proposed significantevidence that the cryptocurrency market has become an independent regularmarket decoupled from and resembling foreign exchange. Structural breaks ofcryptocurrencies have been analysed by [45, 46], who imply, but do not explicitlystate, that these points in time where statistical properties change herald erraticand unpredictable behaviour.The goal of this paper is to extend the study of the nonlinear dynamics ofcryptocurrencies and introduce new methods to analyse their extreme and erraticbehaviour, before and during the COVID-19 pandemic. Extreme behaviour isanalysed via restricted distributions that capture the extreme values of the logreturns and Parkinson variance time series; erratic behaviour is analysed viastructural breaks. We build on the existing literature in several ways. The theoryof extreme events has been studied in [47] and [48, 49] where machine learningmethods have been used to predict extreme events and detect outliers, respectively.While [29] and [23] study the impact of COVID-19 on 3 and 5 cryptocurrencies,respectively, and [45] studies the structural breaks of 7 cryptocurrencies, weanalyse the impact of COVID-19 on the extreme behaviour and structural breaksof 51 cryptocurrencies. While [24, 25, 26, 27, 29, 30, 31] have studied thefinancial impact of the COVID-19 pandemic with existing techniques, all ourtechniques are new. Finally, we develop a more general framework than thatproposed by [32]. We determine changes due to market dynamics in general,and identify several specific cryptocurrencies that are anomalous with respect toreturns or variance. Further, we develop new inconsistency matrices to comparethe relationship between extreme and erratic behaviours before and after theemergence of COVID-19.In Section 2, we describe our methodology to analyse extreme and erraticbehaviour of an arbitrary collection of time series. The methods therein may beapplied more generally than in the instance of this paper. In Section 3, we applyour methods to the log returns and Parkinson variance [50] time series for 51cryptocurrencies. We conclude in Section 4.2 . Methodology In this paper, the most general object of study is a collection of real-valuedtime series X ( i ) t , i = 1 , ..., n over a time interval t = 1 , ..., T . We analyse foursuch collections: the log returns and Parkinson variance of 51 cryptocurrenciesbefore and during the COVID-19 pandemic. Let P t , H t , L t be the closing price,the daily high, and the daily low, respectively, of a financial instrument at time t .Let R t and σ t be the log returns and Parkinson variance time series, respectively,defined as follows: R t = log (cid:18) P t P t − (cid:19) , (1) σ t = (log H t − log L t ) . (2)The Parkinson variance σ t defined above is a measure of the instantaneousprice variance of a financial asset based on its intra-day volatility [50]. Thisgives a time-dependent variance rather than a scalar value through the standarddeviation computation. The time series R t takes both positive and negativevalues, while σ t is non-negative everywhere, a distinction that is necessary inSection 2.1. The precise methodology that we describe below is not exhaustive.Below we outline the detailed implementation to model the erratic and extremebehaviour of cryptocurrencies. Modifications may be made for other contexts. In this section, we describe how we extract and measure the distance betweenthe extreme values of various time series. Let µ be a probability distributionthat records the values of a time series X t . For full generality, suppose µ is acontinuous probability measure of the form µ = f ( x ) dx , where dx is Lebesguemeasure, and f ( x ) a probability density function. As such, f ( x ) is non-negativeeverywhere with integral .For the log return time series R t , we extract the points of density % and %, respectively, by the equations (cid:90) s −∞ f ( x ) dx = 0 . (3) (cid:90) ∞ t f ( x ) dx = 0 . (4)The range x ≤ s gives the left extremal 5% of the distribution, while the range x ≥ t gives the right extremal 5%. Next, we define the restricted function by g ( x ) = f ( x ) { x ≤ s }∪{ x ≥ t } = f ( x ) , x ≤ s , s < x < tf ( x ) , x ≥ t. (5)3bove, denotes an indicator function of a set; this construction essentially trun-cates f only in its tail range. Next, we form the associated measure ν = g ( x ) dx ,where dx is Lebesgue measure. This construction is common in probabilitytheory [51].For the Parkinson variance time series σ t , the associated function f issupported on (0 , ∞ ) . In this case, we extract the point of density 90% by (cid:90) ∞ l f ( x ) dx = 0 . . (6)In this instance, we truncate f only in its extremal positive range by defining h ( x ) = f ( x ) { x ≥ l } = (cid:40) f ( x ) , x ≥ l , x < l. Again, we form the associated measure η = h ( x ) dx where dx is Lebesgue measure.In both cases, this procedure works even more simply for a discrete distributiongiven by a finite data set. For the log returns, we form the empirical distributionfunction, then remove the middle % of the values by order; for the Parkinsonvariance, we remove the bottom 90% of the values.Now suppose we are given n time series X ( i ) t , i = 1 , ..., n . We form n associatedprobability measures µ , ..., µ n and then the restricted measures ν , ..., ν n for thelog returns and η , ..., η n for the Parkinson variance. All these restricted measureshave total measure equal to 0.1 so we may compute the L -Wasserstein metric d [52] between them. Also known as the Earth-mover’s distance, this is a commonmetric between two measures with the same total mass. The constructionof the “associated measure” of ν and η is a technical step from probabilitytheory [51] that means the Wasserstein metric can compute distance betweentwo distribution functions by transforming them into measures. We concludeby forming a matrix between the distribution extremities of the time series.Let D ERij = d ( ν i , ν j ) be the matrix between the log return distributions, and D EVij = d ( η i , η j ) be the matrix between the Parkinson variance distributions.In Section 3, we will consider the means E ( ν i ) of the restricted distributionsassociated to the log returns. Individually, E ( ν i ) > if the correspondingcryptocurrency has greater extreme positive than negative returns, on average.Collectively, they will offer us insight into the market as a whole.Altogether, studying the log returns, Parkinson variance time series andtheir tail ends gives us insight into not only the mean and standard deviation ofthe cryptocurrencies but also higher moments from studying the tail behaviour.In particular, applying our method to the time-dependent Parkinson varianceallows us to study extremities in variance in detail. Let X t be a time series, t = 1 , ..., T . We apply the two-phase change pointdetection algorithm described by [53] to obtain a set of structural breaks, also4nown as change points. These are points in time where the statistical propertiesof the time series change, as detected by a particular algorithm. Applied to acollection X ( i ) t , i = 1 , ..., n, t = 1 , ..., T of time series, this produces a collectionof finite sets S , ..., S n , each a subset of { , ..., T } . Further details are given inAppendix B. This describes a specific change point detection algorithm. Ourmethodology is flexible and may build off any such algorithm.Next, we measure appropriate distances between the sets S i . Traditionalmetrics such as the Hausdorff distance are unsuitable, being too sensitive tooutliers [54] so we adopt and modify the semi-metrics developed in [46] betweenthe sets of structural breaks S i . We define a normalised distance as follows: D ( S i , S j ) = 12 T (cid:32) (cid:80) b ∈ S j d ( b, S i ) | S j | + (cid:80) a ∈ S i d ( a, S j ) | S i | (cid:33) , (7)where d ( b, S i ) is the minimal distance from b ∈ S j to the set S i . The expressionabove is a L norm average of all minimal distances from elements of S to S and vice versa, normalised by both the size of the sets and the length of the timeseries. As in Section 2.1, we form a matrix between the sets of structural breaks, D Bij = D ( S i , S j ) . Let D BR and D BV be the matrices between sets of structuralbreaks for the log return and Parkinson variance time series, respectively. In this section, we describe how we analyse the dynamics of returns andvariance across the entire cryptocurrency market, before and after COVID-19.Let the Frobenius norm of a vector v ∈ R n and an n × n matrix D be defined as || v || = (cid:0)(cid:80) ni =1 | v i | (cid:1) and || D || = (cid:16)(cid:80) ni,j =1 | d ij | (cid:17) , respectively. Let R t and Σ t bethe length n vectors of all log returns ( R ( i ) t ) and Parkinson variances (( σ t ) ( i ) ) attime t , respectively. The values || R t || and || Σ t || give the magnitude of the first twodistribution moments across the market, and represent its volatility as a whole.The Frobenius norm of an n × n distance matrix D represents the total size of alldistances within a collection of n elements. A greater Frobenius norm indicatesless self-similarity in the collection. Having studied overall market dynamics, wenext seek to understand the changing relationships between cryptocurrencies.For this purpose, we study four different time series consisting of the log returnsand Parkinson variance over two different periods, and compare the distancematrices we have defined, pertaining to return extremes, variance extremes,return breaks and variance breaks. We compute the respective matrix normsbefore and after the emergence of COVID-19 to understand the pandemic’simpact on the similarity and dynamics of the cryptocurrency market. Ourperiods of analysis are 30-06-2018 to 31-12-2019 as “pre-COVID,” which wedenote with the subscript “pre” and 1-1-2020 to 24-06-2020 as “post-COVID,”which we denote with the subscript “post.” Our methods may appropriatelycompare periods of different length as all matrices are appropriately normalised.Thus, we have eight different distance matrices:5 For the log returns pre-COVID time series, we have D ER pre and D BR pre ;• For the Parkinson variance pre-COVID time series, have D EV pre and D BV pre ;• For the log returns post-COVID time series, have D ER post and D BR post ;• For the Parkinson variance post-COVID time series, have D EV post and D BV post .We compute the Frobenius norms for these eight matrices in Section 3. In this section, we describe how we measure the consistency between theextreme and erratic behaviours of cryptocurrencies. To do so, we introduce anew method of comparing distance matrices and apply this to compare D B and D E for both the log return and Parkinson variance time series.Given an n × n distance matrix A , the affinity matrix is defined as A ij = 1 − D ij max { D } . (8)All elements of these affinity matrices lie in [0 , , so it is appropriate to comparethem directly by taking their difference. Let the affinity matrices associated to D ER , D EV , defined in Section 2.1, and D BR , D BV , defined in Section 2.2, be A ER , A EV , A BR and A BV respectively. We define the behaviour inconsistencymatrix between extreme and erratic behaviour be defined as follows, for logreturns and Parkinson variance, respectively:INC R,EB = A ER − A BR (9)INC V,EB = A EV − A BV . (10)As described in Section 2.3, we analyse the log returns and Parkinson variancesover two distinct periods (pre- and post-COVID), so we have four differentbehaviour inconsistency matrices INC pre ,R bhvr , INC pre ,V bhvr , INC post ,R bhvr and INC post ,V bhvr . Finally, we can also define time inconsistency matrices . By comparing corre-sponding distances matrices, extreme or erratic, over time, we can continue thegoal of Section 2.3 and identify individual cryptocurrencies that have significantlychanged with respect to their similarity with others. We define four inconsistencymatrices with respect to time as follows:INC ER time = A ER pre − A ER post (11)INC EV time = A EV pre − A EV post (12)INC BR time = A BR pre − A BR post (13)INC BV time = A BV pre − A BV post . (14)Thus, we have defined eight different inconsistency matrices above. Four,INC pre ,R bhvr , INC pre ,V bhvr , INC post ,R bhvr and INC post ,V bhvr , we refer to as behaviour incon-sistency matrices ; the other four, INC ER time , INC EV time , INC BR time and INC BR time , we6efer to as time inconsistency matrices . The behaviour inconsistency matrices reveal cryptocurrencies that are irregular with respect to the rest of the marketin the comparison between extreme and erratic behaviour. This may pertainto the returns or variance, pre- or post-COVID, depending on which of thefour behaviour inconsistency matrices we examine. As a shorthand, we willsay such cryptocurrencies are inconsistent with respect to behaviour. The timeinconsistency matrices reveal cryptocurrencies that have been anomalously im-pacted by the emergence of COVID-19. This may pertain to its extreme orerratic behaviour of its log returns or variance, depending on which of the fourtime inconsistency matrices we examine. We will say such cryptocurrencies are inconsistent with respect to time .We reveal inconsistent cryptocurrencies as follows. Assume the cryptocur-rencies are given some ordering j = 1 , ..., n . Given any inconsistency matrixINC of the eight above, define the anomaly score of the j -th cryptocurrencyas a j = (cid:80) nj =1 | INC ij | . Larger values indicate cryptocurrencies that are moreanomalous between the two affinity matrices under consideration (recall thatevery inconsistency matrix is the difference of two affinity matrices). In Section3, we highlight the top 3 cryptocurrencies across a range of these inconsistencymatrices to determine the most prominent anomalies. We have chosen to list thetop 3 for ease of interpreting the results, and to point out that examining themost inconsistent cryptocurrencies reveals a considerable number of repetitions -that is, certain cryptocurrencies that are inconsistent in one regard frequentlyare inconsistent in other regards.
3. Experimental results and discussion
We draw data from Coinmarketcap. Of the cryptocurrencies with pricehistories that go as far back as 30-06-2018, we analyse the 51 largest by marketcapitalisation. These are all detailed in Table A.3, ordered by decreasing marketcapitalisation. For each cryptocurrency, we draw closing price, daily high, anddaily low, and first calculate the log returns and Parkinson variance as definedin (1) and (2). In the proceeding sections, we report a broad range of findings onthe contrasting impact COVID-19 has had on the extreme and erratic behaviourof cryptocurrency returns and variance, and the identification of inconsistenciesin various cryptocurrency behaviours. We refer to the period from 30-06-2018to 31-12-2019 as “pre-COVID” and the period from 1-1-2020 to 24-06-2020 as“post-COVID.”We have chosen these periods deliberately to study the impact of the COVID-19 epidemiological crisis on financial assets, specifically on the cryptocurrencymarket. The post-COVID period can be characterised as a systemic crisisacross all asset classes, while the pre-COVID period was relatively stable, withconsistently positive returns across equity markets in 2019. Market crises referto market crashes that impact all asset classes rather than individual drops inassets, even if they are precipitous.This is not to say that there was not some drawdown within the cryptocur-rency market during 2019, nor that all volatility during the post-COVID period7an be explained by the pandemic. Indeed, the crash of the BitMEX exchangemarket crash in March 2020 caused profound losses in Bitcoin [55]. However, thepre-COVID and post-COVID periods are distinct in nature - the latter is char-acterised by the emergence of COVID-19 around the world and with subsequentcrises in various equities and economic measures. Generally speaking, we areinterested in studying the impact of broader financial crises on cryptocurrencies.The periods are of different length, however this is appropriate given thatall constructions in Section 2 are normalised with respect to time. Moreover,such a difference in lengths between crisis and non-crisis periods is inevitable.Historically, financial markets have exhibited a tendency for collective asset pricesto rise slowly during “bull” markets and drop more quickly during “bear” markets.For example, The Dow Jones Industrial Average exhibited losses of -22.61% on19-10-1987 (Black Monday), -7.87% on 15-10-2008 (the Global Financial Crisis),and -9.99% on 12-03-2020 (Black Thursday) [56]. Accordingly, we have chosento study a shorter period of time as our crisis period to more accurately reflectthe market dynamics during the crash.
In this first section, we study the cryptocurrency market dynamics overtime as a whole, without reference to the internal similarity between distinctcryptocurrencies. We use the same 51 cryptocurrencies as elsewhere in the paper,listed in Table A.3. In Figures 1a and 1b, we depict the Frobenius norms || R t || and || Σ t || , respectively, defined in Section 2.3. Larger magnitudes indicate moreunstable market dynamics, characterised by greater total returns and variance,regardless of direction. These figures reveal several insights: first, during 2018and 2019, the Parkinson variance exhibits less regular peaks than the log returns,but relative to the baseline these peaks are more significant. These peaks alsoappear to occur more periodically than that of returns, with our measure possiblyhighlighting some latent volatility clustering. Second, the sharpest peak of eachfigure is coincident: anomalously large changes in returns are accompanied byproportionally large changes in variance at the same time. The day of greatestchanges in returns and volatility is 12 March, 2020. Known as Black Thursday[56], this was the worst crash of the Dow Jones since 1987, and caused substantialdrops across all sectors. On the exact same day, the cryptocurrency exchangeBitMEX crashed, causing a substantial drop in the price of Bitcoin. After thisgreatest spike, both measures drop to levels below their 30-month average. Thatis, after this point, the volatility of the market is in fact reduced relative to theprevious year. In this section, we analyse the distance matrices between distribution extrem-ities and structural breaks for the four time series under consideration, that is,log returns and Parkinson variance before and after the emergence of COVID-19.Specifically, we examine the Frobenius norms, defined in Section 2.3, of theeight distance matrices defined in the same section. Greater Frobenius norms8 a) The Frobenius norm || R t || as a function of time(b) The Frobenius norm || Σ t || as a function of time Figure 1: The changing dynamics of the Frobenius norm for (a) log returns and (b) Parkinsonvariance over the whole market are plotted with time. A coincident sharp peak in March 2020can be seen, showing the extreme but brief impact of COVID-19. || D E || for returns and variance, pre- and post-COVIDPeriod Log returns Parkinson variancePre-COVID 1.30 0.64Post-COVID 1.51 0.73 Table 1: Frobenius norms for distribution extremity distance matrices pre- and post-COVID.
Matrix norms || D B || for returns and variance, pre- and post-COVIDPeriod Log returns Parkinson variancePre-COVID 5.70 0.84Post-COVID 5.22 2.44 Table 2: Frobenius norms for structural breaks distance matrices pre- and post-COVID. indicate greater overall distances and hence less self-similarity among a collectionof cryptocurrencies. This reveals several insights:First, both before and during COVID-19, the behaviour of cryptocurrencies ismore self-similar (measured by Frobenius matrix norms) with respect to variancethan returns. This is true for both distribution extremities and structural breaks,heralding extreme and erratic behaviour, respectively. This can be seen in theconsistently smaller values in Tables 1 and 2, where Frobenius norms, definedin Section 2.3, are computed. All distances in question, and hence the matrixnorms, are normalised, so this comparison is appropriate when comparing eitherdistribution extremities or structural breaks among themselves.Next, the post-COVID period generally exhibits less similarity (measuredby the Frobenius norms) in the cryptocurrency market than before COVID-19.Comparing distribution extremities and structural breaks with respect to thelog returns and variance time series, three of these distance matrices exhibitgreater Frobenius norm for the post-COVID period. Only the structural breaksin returns observe a slight decrease in Frobenius norm, implying greater self-similarity with respect to structural breaks of returns, in the post-COVID period.By contrast, the Frobenius norm for structural breaks in variance has increasedalmost threefold. Indeed, this is reflected in Figure 2. As we will discuss furtherin Section 3.3, Figure 2a shows essentially one cluster, broad self-similarity andno outlier elements between structural breaks with respect to variance, whileFigure 2b has less self-similarity - rather, several anomalous elements are visiblein the dendrogram.On the other hand, for the Frobenius norms of distribution extremities, amoderate and similar increase was observed for both returns and variance. Thatis, the total similarity (measured by Frobenius norms) in return and varianceextremes decreased by a similar amount due to COVID-19. The slight increasein Frobenius norm for distribution extremities of variance contrasts with thelarge increase in the case of structural breaks; while the increase in norms forlog returns contrasts with the decrease in the case of breaks.10 .3. Structural breaks with respect to Parkinson variance
In this section, we take a closer look at the structure of the cryptocurrencymarket with respect to structural breaks in Parkinson variance. In Figure 2, wedepict the results of hierarchical clustering of the corresponding affinity matrices A BV pre and A BV post . This supports our analysis in the preceding section. There aretwo types of hierarchical clustering: agglomerative (bottom-up) and divisive(top-down) clustering. The former starts with each object defined as a singleelement cluster and sequentially combines the most similar clusters into a largercluster, until there is only one cluster. The latter works by sequentially splittingthe most heterogeneous cluster in two, until each cluster contains only a singleelement. Our experiments apply agglomerative hierarchical clustering. UnlikeK-means clustering, hierarchical clustering does not require a specific numberof clusters k to be set, and the procedure generates an image of the clusterstructure. Tree colors are determined by the dendrogram, which computes theoptimal trade-off between the number of clusters and total error. An ideal fitwould have fewer clusters and smaller error. More details are provided in [57, 58].Figure 2a exhibits essentially one amorphous cluster lacking any visiblestructure, with no notable outliers and a high degree of self-similarity. Thecomplete lack of subclusters represents the significant risk for investors in thecryptocurrency market even during normal times. Indeed, there is really no wayto diversify against erratic behaviour in volatility, as all cryptocurrencies havehighly similar structural breaks, and this exposes investors to simultaneous draw-down risk among any held collection of cryptocurrencies. After the emergence ofCOVID-19, we see a considerable change in market structure, displayed in Figure2b. There, two primary clusters of cryptocurrencies are observed. Within thesmaller cluster, a subcluster of high similarity emerges containing cryptocurren-cies such as Chainlink (LINK) and Tezos (XTZ). Several slight outliers emerge:Maker (MKR), Tether (USDT) and Zcash (ZEC). All cryptocurrency tickersare indexed in Table A.3. That is, the emergence of COVID-19 has significantlyaltered the market structure with respect to structural breaks in variance, fromone highly homogeneous cluster with no outliers, to two clusters with somesubcluster structure and slight outliers. In this section, we analyse the impact of COVID-19 on the distributionextremities of the log returns. Analysis of the corresponding affinity matrices A ER pre and A ER post reveals more pronounced structure after the emergence of COVID-19. Hierarchical clustering of these matrices is displayed in Figures 3a and 3b.These dendrograms each highlight two outliers, Tether (USDT) and TrueUSD(TUSD), consistent before and during COVID-19. Both these cryptocurrenciesare highly anomalous with respect to their log returns distribution extremities,which is likely due to their low volume of trading and their being pegged to theUS dollar in a one-to-one ratio.Indeed, in Figure 4, we depict the distribution extremities of their log returnsalongside two cryptocurrencies in the majority cluster, Bitcoin (BTC) and11 a) A BV pre dendrogram(b) A BV post dendrogram Figure 2: Hierarchical clustering of affinity matrix between structural breaks with respect toParkinson variance for (a) pre-COVID and (b) post-COVID. a) A ER pre dendrogram outliers(b) A ER post dendrogram outliers Figure 3: Hierarchical clustering of affinity matrix between distribution extremities with respectto log returns for (a) pre-COVID, (b) post-COVID. a) USDT (b) TUSD(c) BTC (d) ETH(e) USDT (f) TUSD(g) BTC (h) ETH Figure 4: Extremities of log returns distributions, together with kernel density estimation plots.Figures (a)-(d) represent pre-COVID distributions, (e)-(h) represent post-COVID distributions. a) A ER pre dendrogram no outliers(b) A ER post dendrogram no outliers Figure 5: Hierarchical clustering of affinity matrix between distribution extremities with respectto log returns for (a) pre-COVID and (b) post-COVID, excluding outliers USDT and TUSD. E ( ν i ) all lie in the second cluster, and the average of the means E i E ( ν i ) isgreater than the mean of the first cluster under a simple statistical test,with p < . . Such measures of statistical significance are not necessarilydispositive of a meaningful relationship [59], but ordering the means of16he restricted distributions shows that the latter cluster contains morerestricted distributions with positive mean, including the greatest four.2. During COVID-19, excluding outliers USDT and TUSD, the two anomalouscryptocurrencies LEND and DGB have the two greatest means E ( ν i ) ofthe entire post-COVID collection. That is, the mean E ( ν i ) is once againsignificantly related to the cluster structure.3. Before COVID-19, 88% of cryptocurrencies had an negative value of E ( ν i ) .After the emergence of COVID-19, just 37% of cryptocurrencies had anegative mean. That is, a majority of post-COVID return extremes hada positive mean. This is an unexpected result and demonstrates thatalthough there was a sharp initial drop in returns due to COVID-19, thisbehaviour was anomalous relative to the rest of the post-COVID period. In this section, we focus on identifying two types of anomalies within thecryptocurrency market - cryptocurrencies that are inconsistent between extremeand erratic behaviour, or inconsistent with respect to time. As defined in Sec-tion 2.4, we use the shorthand inconsistent with respect to behaviour to meanthat a cryptocurrency is irregular with respect to the rest of the market in thecomparison between extreme and erratic behaviour, and detect this through our behaviour inconsistency matrices , also defined in Section 2.4. Similarly, we havethe shorthand inconsistent with respect to time , which describes cryptocurrencieswhose behaviour changed significantly in the period of COVID-19. Each incon-sistency matrix assigns anomaly scores, which can rank the most inconsistentcryptocurrencies in either regard. We list the top 3 most inconsistent cryptocur-rencies pertaining to a range of inconsistency matrices. We have chosen to listthe top 3 for ease of interpreting the results, and to point out that examining themost inconsistent cryptocurrencies reveals a considerable number of repetitions -that is, certain cryptocurrencies that are inconsistent in one regard frequentlyare inconsistent in other regards.First, we study the behaviour inconsistency matrices INC pre ,R bhvr , INC pre ,V bhvr , INC post ,R bhvr , INC post ,V bhvr . We implement hierarchical clustering in Figures 6 and7 to highlight clusters in returns and variance, and compute anomaly scores.The behaviour inconsistency dendrogram for pre-COVID returns is displayed inFigure 6a and reveals a separate cluster of Holo (HOT), 0x (ZRX) and Aave(LEND). Computing anomaly scores reveals that the top 3 most inconsistentcryptocurrencies with respect to differing behaviour are HOT, ZRX and Maker(MKR) - LEND is fourth. The anomaly scores may give slightly differentresults than the hierarchical clustering because the anomaly scores featureabsolute values to calculate absolute differences with other elements, while thedendrograms cluster based on the positive or negative values of the inconsistencymatrix. Rows or columns with negative entries may indicate cryptocurrencies ofsimilar behaviour that should be clustered together, but might not necessarilydistinguish the most absolutely inconsistent entries of the collection. The17ehaviour inconsistency matrix with respect to post-COVID returns is clusteredin Figure 6b. NEXO (NEXO), USDT and TUSD form their own cluster; thetop 3 anomaly scores are of NEXO, THETA (THETA) and MKR.When analysing inconsistency matrices between extreme and erratic behaviourof variance, we observe the repetition of inconsistent cryptocurrencies - the top3 anomaly scores for the pre-COVID variance behaviour inconsistency matrixare of NEM (XEM), NEXO and MKR, while MonaCoin (MONA), Ravencoin(RVN) and Bitcoin Diamond (BCD) form a separate cluster. Observing thedendrograms in Figures 7a and 7b side by side, slightly more structure is observedin the post-COVID period, with a growth in the total number of clusters. USDTand TUSD emerge as inconsistent in their behaviour after the emergence ofCOVID-19. Indeed, before COVID-19, both their distribution extremities andstructural breaks are outliers, leading to relative consistency between the twobehaviours, while during COVID-19, their structural breaks in variance becomesimilar to the rest of the market.Time inconsistency matrices are clustered in Figures 8 and 9 and anomalyscores computed again. Dendrograms for returns and variance extremes, dis-played in Figures 8a and 8b, are dominated by one cluster - with each inconsis-tency matrix producing one clearly anomalous cryptocurrency, DigiByte (DGBPand BCD respectively. The top 3 anomaly scorers for these inconsistency matri-ces are DGB, RVN and HOT, and BCD, MKR and DGB, respectively. We seethe repetition of inconsistent cryptocurrencies from prior experiments. This isalso the case for structural breaks: the top 3 anomalies for structural breaks inreturns are USDT, TUSD and HOT, while NEXO and XEM are the second andthird most inconsistent cryptocurrencies across time with respect to structuralbreaks in variance. That is, we see consistent repetition in the cryptocurrenciesthat are inconsistent between extreme and erratic behaviours and those whichwere most affected across time by COVID-19.
4. Conclusion
In this paper, we have developed new methods for the study of extreme anderratic behaviours of time series, both individually and in collections. Our workextends the new distance measures between finite sets proposed in [46]. Weintroduce inconsistency matrices and anomaly scores for identification of elementsthat are inconsistent with respect to behaviour and time. Such methodologyto study extreme behaviour complements the work of [47, 48, 49], who havedeveloped machine learning methods in other applications.Applied to the cryptocurrency market, we have uncovered several insightsand dissimilarities when studying the behaviour patterns of returns and variance.In general, cryptocurrency behaviour is more self-similar in variance than returns,both before and during the pandemic. Before COVID-19, the cryptocurrencymarket exhibited considerable homogeneity with respect to the structural breaksin variance. This was disrupted by the pandemic, with a reduction in self-similarity, reflected in the comparison of Frobenius norms and the emergence ofoutliers in hierarchical clustering. COVID-19 also had an impact on the return18 a) INC pre ,R bhvr (b) INC post ,R bhvr Figure 6: Hierarchical clustering of inconsistency matrices between extreme and erraticbehaviour for (a) log returns pre-COVID (b) log returns post-COVID a) INC pre ,V bhvr (b) INC post ,V bhvr Figure 7: Hierarchical clustering of inconsistency matrices between extreme and erraticbehaviour for (a) Parkinson variance pre-COVID (b) Parkinson variance post-COVID. a) INC ER time (b) INC EV time Figure 8: Hierarchical clustering of time inconsistency matrices for (a) log returns extremes(b) Parkinson variance extremes a) INC BR time (b) INC BV time Figure 9: Hierarchical clustering of time inconsistency matrices for (a) log returns structuralbreaks (b) Parkinson variance breaks.
Acknowledgements
Many thanks to Kerry Chen for helpful comments and edits.
Appendix A. Glossary
We include Tables A.3 and A.4, glossaries of cryptocurrency tickers andmathematical objects introduced in this paper, respectively. Table A.3 orderscryptocurrency by market capitalisation.23 ryptocurrency tickers and names
Ticker Coin Name Ticker Coin NameBTC Bitcoin DGB DigiByteETH Ethereum ZRX 0xUSDT Tether KNC Kyber NetworkXRP XRP (Ripple) OMG OMG NetworkBCH Bitcoin Cash THETA THETALTC Litecoin REP AugurBNB Binance Coin ZIL ZiliqaEOS EOS BTG Bitcoin GoldADA Cardano DCR DecredXTZ Tezos ICX ICONLINK Chainlink QTUM QtumXLM Stellar LEND AaveXMR Monero TUSD TrueUSDTRX Tron BCD Bitcoin DiamondHT Huobi Token ENJ Enjin CoinNEO NEO LSK LiskETC Ethereum Classic REN RenDASH Dash NANO NanoMIOTA IOTA RVN RavencoinZEC Zcash SC SyscoinMKR Maker WAVES WavesONT Ontology MONA MonaCoinBAT Basic Attention Token NEXO NexoXEM NEM HOT HoloDOGE Dogecoin IOST IOSTSNT Status
Table A.3: Cryptocurrency tickers and names athematical objects glossary Object Description D ER Distance matrix between return extremes D EV Distance matrix between variance extremes D BR Distance matrix between return structural breaks D BV Distance matrix between variance structural breaks A ER , A EV , etc Corresponding affinity matrices || R t || Time varying log return Frobenius vector norm || Σ t || Time varying variance Frobenius vector norm || D BR || Frobenius matrix normINC pre ,R bhvr Pre-COVID returns behaviour inconsistency matrixINC post ,R bhvr Post-COVID returns behaviour inconsistency matrixINC pre ,V bhvr Pre-COVID variance behaviour inconsistency matrixINC post ,V bhvr Post-COVID variance behaviour inconsistency matrixINC ER time Return extremes time inconsistency matrixINC EV time Variance extremes time inconsistency matrixINC BR time Return structural breaks time inconsistency matrixINC BV time Variance structural breaks time inconsistency matrix
Table A.4: Mathematical objects and definitions
Appendix B. Change point detection algorithm
In this section, we provide an outline of change point detection algorithms, anddescribe the specific algorithm that we implement. Many statistical modellingproblems require the identification of change points in sequential data. Bydefinition, these are points in time at which the statistical properties of a timeseries change. The general setup for this problem is the following: a sequenceof observations x , x , ..., x n are drawn from random variables X , X , ..., X n and undergo an unknown number of changes in distribution at points τ , ..., τ m .One assumes observations are independent and identically distributed betweenchange points, that is, between each change points a random sampling of thedistribution is occurring. Following Ross [53], we notate this as follows: X i ∼ F if i ≤ τ F if τ < i ≤ τ F if τ < i ≤ τ ,. . . While this requirement of independence may appear restrictive, dependence cangenerally be accounted for by modelling the underlying dynamics or drift process,then applying a change point algorithm to the model residuals or one-step-aheadprediction errors, as described by Gustafsson [60]. The change point modelsapplied in this paper follow Ross [53]. 25 ppendix B.1. Batch change detection (Phase I)
This phase of change point detection is retrospective. We are given a fixedlength sequence of observations x , . . . , x n from random variables X , . . . , X n .For simplicity, assume at most one change point exists. If a change point existsat time k , observations have a distribution of F prior to the change point, anda distribution of F proceeding the change point, where F (cid:54) = F . That is, onemust test between the following two hypotheses for each k : H : X i ∼ F , i = 1 , ..., nH : X i ∼ (cid:40) F i = 1 , , ..., kF , i = k + 1 , k + 2 , ..., n and end with the choice of the most suitable k .One proceeds with a two-sample hypothesis test, where the choice of testis dependent on the assumptions about the underlying distributions. To avoiddistributional assumptions, non-parametric tests can be used. Then one ap-propriately chooses a two-sample test statistic D k,n and a threshold h k,n . If D k,n > h k,n then the null hypothesis is rejected and we provisionally assume thata change point has occurred after x k . These test statistics D k,n are normalisedto have mean and variance and evaluated at all values < k < n , and thelargest value is assumed to be coincident with the existence of our sole changepoint. That is, the test statistic is then D n = max k =2 ,...,n − D k,n = max k =2 ,...,n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ D k,n − µ ˜ D k,n σ ˜ D k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where ˜ D k,n were our unnormalised statistics. This test statistic is known as the Mann-Whitney test [53].The null hypothesis of no change is then rejected if D n > h n for someappropriately chosen threshold h n . In this circumstance, we conclude that a(unique) change point has occurred and its location is the value of k whichmaximises D k,n . That is, ˆ τ = argmax k D k,n . This threshold h n is chosen to bound the Type 1 error rate as is standard instatistical hypothesis testing. First, one specifies an acceptable level α for theproportion of false positives, that is, the probability of falsely declaring that achange has occurred if in fact no change has occurred. Then, h n should be chosenas the upper α quantile of the distribution of D n under the null hypothesis. Forthe details of computation of this distribution, see [53]. Computation can oftenbe made easier by taking appropriate choice and storage of the D k,n .26 ppendix B.2. Sequential change detection (Phase II) In this case, the sequence ( x t ) t ≥ does not have a fixed length. New ob-servations are received over time, and multiple change points may be present.Assuming no change point exists so far, this approach treats x , ..., x t as a fixedlength sequence and computes D t as in phase I. A change is then flagged if D t > h t for some appropriately chosen threshold. 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