Persistence in the Cryptocurrency Market

This paper examines persistence in the cryptocurrency market. Two different long-memory methods (R/S analysis and fractional integration) are used to analyse it in the case of the four main cryptocurrencies (BitCoin, LiteCoin, Ripple, Dash) over the sample period 2013-2017. The findings indicate that this market exhibits persistence (there is a positive correlation between its past and future values), and that its degree changes over time. Such predictability represents evidence of market inefficiency: trend trading strategies can be used to generate abnormal profits in the cryptocurrency market.


Introduction
The exponential growth of BitCoin and other cryptocurrencies is a phenomenon that has attracted considerable attention in recent years. The cryptocurrency market is rather young (BitCoin was created in 2009, but active trade only started in 2013) and therefore still mostly unexplored (see Caporale and Plastun, 2017 for one of the very few existing studies, with a focus on calendar anomalies). One of the key issues yet to be analysed is whether the dynamic behaviour of cryptocurrencies is predictable, which would be inconsistent with the Efficient Market Hypothesis (EMH), according to which prices should follow a random walk (see Fama, 1970). Long-memory techniques can be applied for this purpose. Several studies have provided evidence of persistence in asset price dynamics (see Greene and Fielitz, 1977;Caporale et al., 2016), and also found that this changes over time (see Lo, 1991), but virtually none has focused on the cryptocurrency market. One of the few exceptions is due to Bouri et al. (2016), who find lon-memory properties in the volatility of Bitcoin.
The present study carries out a more comprehensive analysis by considering four main cryptocurrencies (the most liquid ones: BitCoin, LiteCoin, Ripple, Dash) and applying two different long-memory methods (R/S analysis and fractional integration) over the period 2013-2017 to investigate their stochastic properties. Moreover, it also examines the evolution of persistence over time (by looking at changes in the Hurst exponent). Any predictable patterns could of course be used as a basis for trading strategies aimed at making abnormal profits in the cryptocurrency market.
The layout of the paper is the following. Section 2 provides a brief review of the relevant literature. Section 3 describes the data and outlines the empirical methodology.
Section 4 presents the empirical results. Section 5 provides some concluding remarks.

Literature Review
As already mentioned above, the cryptocurrency market has only been in existence for a few years, and therefore only a handful of studies have been carried out. ElBahrawy et al. (2017) provide a comprehensive analysis of 1469 cryptocurrencies considering various issues such as market shares and turnover. Cheung et al. (2015), Dwyer (2014), Bouoiyour and Selmi (2015) and Carrick (2016) show that this market is much more volatile than others. Halaburda and Gandal (2014) analyse its degree of competitiveness Urquhart (2016) and Bartos (2015) focus on efficiency finding evidence for and against respectively. Anomalies in the cryptocurrency market are examined by Kurihara and Fukushima (2017) and Caporale and Plastun (2017). Bariviera et al. (2017) test the presence of long memory in the Bitcoin series from 2011 to 2017. They find that the Hurst exponent changes significantly during the first years of existence of Bitcoin before becoming more stable in recent times. Bariviera (2017) also use the Hurst exponent and detect long memory in the daily dynamics of BitCoin as well as its volatility; in addition, they find more evidence of informational efficiency since 2014. Bouri et al. (2016) examine persistence in the level and volatility of Bitcoin using both parametric and semiparametric techniques; they detect long memory in both measures of volatility considered (absolute and squared returns). Catania and Grassi (2017) provide further evidence of long memory in the cryptocurrency market, whilst Urquhart (2016) using the R/S Hurst exponent obtains strong evidence of anti-persistence, which indicates non-randomness of Bitcoin returns.

Data and Methodology
We focus on the four cryptocurrencies with the highest market capitalisation and longest span of data (see Table 1 below): BitCoin, LiteCoin, Ripple and Dash. The frequency is daily, and the data source is CoinMarketCap (https://coinmarketcap.com/coins/). The two approaches followed are R/S analysis and fractional integration respectively. The following algorithm is used for the R/S analysis (see Mynhardt et al., 2014 for additional details):

1.
A time series of length M is transformed into one of length N = M -1 using logs and converting prices into returns: (1) 5 2. This period is divided into contiguous A sub-periods with length n, such that A n = N, then each sub-period is identified as I a , given the fact that a = 1, 2, 3. . . , A.
Each element I a is represented as N k with k = 1, 2, 3. . . , N. For each I a with length n the average a e is defined as: ( 3. Accumulated deviations X k,a from the average a e for each sub-period I a are defined as: ( The range is defined as the maximum index X k,a minus the minimum X k,a , within each sub-period (I a ): The standard deviation Ia S is calculated for each sub-period I a :

5.
Each range R Ia is normalised by dividing by the corresponding S Ia .
Therefore, the re-normalised scale during each sub-period I a is R Ia /S Ia . In step 2 above, adjacent sub-periods of length n are obtained. Thus, the average R/S for length n is defined as: 6. The length n is increased to the next higher level, (M -1)/n, and must be an integer number. In this case, n-indexes that include the start and end points of the time series are used, and Steps 1 -6 are repeated until n = (M -1)/2.

7.
The least square method is used to estimate the equation log (R / S) = log (c) + H*log (n). The slope of the regression line is an estimate of the Hurst exponent H. (Hurst, 1951). In addition we also employ I(d) techniques for the log prices series, both parametric and semiparametric. Note that the differencing parameter d is related to the Hurst exponential described above through the relationship H = d + 0.5. Also, R/S analysis is used for the return series (the first differences of the log prices), while I(d) models are estimated for the log prices themselves, in which case the relationship becomes H = (d -1) + 0.5 = d -0.5. We consider processes of the form: where B is the backshift operator (Bx t = x t-1 ); u t is an I(0) process (which may incorporate weak autocorrelation of the AR(MA) form) and, x t are the errors of a regression model of the form: where y t stands for the log price of each of the cryptocurrencies. Note that under the Efficient Market Hypothesis the value of d in (7) should be equal to 1 and u t a white noise process. As mentioned before, we use both parametric and semiparametric methods, in each case assuming uncorrelated (white noise) and autocorrelated errors in turn. More specifically, we use first the Whittle estimator of d in the frequency domain (Dahlhaus, 1989;Robinson, 1994), and then the "local" Whittle estimator initially proposed by Robinson (1995) and then further developed by Velasco (1999)

Empirical Results
The results of the R/S analysis for the return series of the four cryptocurrencies are presented in Table 2. As can be seen, the series do not follow a random walk, and are persistent, which is inconsistent with market efficiency. The most efficient cryptocurrency is Bitcoin, which is the oldest and most commonly used, as well as the most liquid.
The dynamic R/S analysis shows the evolution over time of persistence in the cryptocurrency market (see Figure 1). In Table 3 we assume that u t in (1) is a white noise process, and consider the three cases of: i) no deterministic terms, ii) an intercept, and iii) an intercept with a linear time trend. This table shows that the estimates are around 1 in all cases, which implies non-stationary behaviour. A time trend is required in the cases of Bitcoin and Dash, but not for the other two series, Litecoin and Ripple.     Next we allow for autocorrelated disturbances, and for this purpose we use the exponential spectral model of Bloomfield (see Bloomfield, 1973 for details). Table 5 displays the estimates of d for the three cases of no regressors, an intercept and a linear time trend while Table 6 focuses on the selected models. As in the case of uncorrelated errors, a time trend is required for Bitcoin and Dash but not for the other two series. As before, the estimates of d are within the I(1) interval for all series except Ripple, for which the estimate of d is significantly higher than 1, which implies a high degree of persistence.  Given the differences between the results from the (non-parametric) R/S analysis and the parametric I(d) estimation, we also employ a semiparametric I(d) method, namely a "local" Whittle estimator in the frequency domain (see Robinson, 1995, etc.).
The estimates for selected bandwidth parameters are shown in Table 7. 1 In bold those cases where the estimates of d are significantly higher than 1 at the 5% level.