**2. Method**

### *2.1. Detecting Bull and Bear Markets*

In the finance literature, there is no generally accepted formal definition of a bull or bear market. Therefore, in this paper, the considered time period was split into bull and bear phases in the cryptocurrency markets on the basis of two well-known algorithms: the algorithm of Lunde and Timmermann (2004) (filtering method) and the algorithm of Bry and Boschan (1971) (dating method). Both of these methods were designed to capture financial and business cycles. Here, we give a brief explanation of these two methods.

The Lunde and Timmermann (2004) algorithm is based on imposing a minimum on the price change since the last peak or trough. Let *λ*1 be a scalar defining the threshold for a transition from a bear to bull market, and *λ*2 be a threshold for a transition from a bull to bear market. Suppose *Xt* denotes the hourly price of a cryptocurrency at time *t*, and a trough in *X* has been detected at time *t*0 ≤ *t*. A bull phase begins in the algorithm at time *t*0 + 1. The algorithm first detects the maximum value in *X* at time [*t*0, *t*]:

$$X\_{t\_0, t}^{\max} = \max \left\{ X\_{t\_0 \prime} X\_{t\_0 + 1 \prime} \dots \mathcal{X}\_t \right\} \dots$$

Then, the relative change in *X* is computed as

$$
\delta\_t = \frac{X\_{t\_0, t}^{\max} - X\_{\rm f}}{X\_{t\_0, t}^{\max}}.
$$

If *δt* > *λ*2, this point is denoted as a new peak (maximum) occurring at *tpeak* in interval [*<sup>t</sup>*0, *t*]. Then, [*t*0 + 1, *tpeak*] is labelled as a bull state period. By contrast, a Bear state period begins from *tpeak* + 1 and if a peak has been identified in *X* at time *t*0 ≤ *t*, then the algorithm finds the minimum value of *X* on the time interval [*<sup>t</sup>*0, *t*],

$$X\_{t\_0, t}^{\min} = \min\left\{ X\_{t\_0}, X\_{t\_0 + 1}, \dots, X\_{t} \right\}$$

and then the relative change in *X* is computed as

$$
\delta\_t = \frac{X\_{t\_0, t}^{\min} - X\_t}{X\_{t\_0, t}^{\min}}.
$$

If *δt* > *λ*1, this point is denoted as a new trough (minimum) occurring at *ttrough* in the interval [*<sup>t</sup>*0, *t*]. Then, [*t*0 + 1, *ttrough*] is labelled as a bear period. A bull period begins from *ttrough* + 1. For more details on this method, see Lunde and Timmermann (2004).

The main objective of the Bry and Boschan (1971) algorithm is to detect turning points in a financial cycle. This method consists of two main steps: identifying the initial turning points in *X*, followed by guided censoring operations. First, one identifies a window of length *τwindow* months on either side of the date and defines a peak (trough) in *X* as a point higher (lower) than other points within the window. Next, censoring requires eliminating peaks and troughs in the first and last *τcensor* months; eliminating phases that last less than *<sup>τ</sup>phase* months; and eliminating cycles that last less than *<sup>τ</sup>cycles* months. We repeated the procedure of the censoring operation many times, until the sequence of turning points satisfied all constraints. For more details on this method, see Bry and Boschan (1971).

A major drawback of the Bry and Boschan (1971) method is that it is mostly applied to monthly frequency data, and it is very sensitive to data frequency. On the other hand, one can just edit the parameters to account for the data frequency. Compared with the Bry and Boschan (1971) method, the Lunde and Timmermann (2004) method is not that sensitive when applied to either daily or hourly frequency data because parameters in the algorithm are computed as two relative changes in the cryptocurrency prices in the algorithm. Implementation of the two algorithms to our selected data is described in Section 3.

One may question if the use of these methods is adequate for analysing cryptocurrency markets or whether it can only be applied to traditional business cycles. To answer this question, it is best not to look specifically at the duration of business cycles and say whether cryptocurrency cycles are similar or not, but rather to look back at the algorithm itself (see Section 3). The algorithm determines bull and bear markets in any financial markets through the setting of a threshold relating to a level of price change that, if exceeded, represents a change in the market state. Hence, these methods are robust to application in any financial market, as discussed by Bry and Boschan (1971) and Lunde and Timmermann (2004).

### *2.2. Detrended Fluctuation Analysis (DFA) Method*

We tested for long-range memory in the bull and bear markets of Ethereum, Bitcoin, and Litecoin by computing the Hurst exponent via the DFA method. The DFA method examines dependence in these markets, and was an indicator of random and nonrandom behaviour in our time series. Other methods for detecting long-range memory include R/S analysis, which is one of the most popular extended methods that can be used to estimate long-term memory in time-series data; however, it is not that stable. For instance, when a process under investigation has short memory, the R/S statistic may wrongly indicate the presence of long-term memory. The DFA method has been shown to be more suitable in dealing with nonstationary time-series data. In addition, as highlighted by Grau-Carles (2000), the DFA method avoids the spurious detection of long-range dependence. Hence, this is the main reason why chose to use the method. The computation of the Hurst exponent was conducted using the R statistical software package (R Development Core Team 2019). We followed the method presented in Section 2.1 of Zhang et al. (2018b), to compute Hurst exponent values, using the default parameters given in the procedure, and a rolling window of 720 (approximately one month) lagged data points. Further details on these methods applied to cryptocurrency data can be found in Zhang et al. (2018b).

The values that the Hurst exponent (*α*) could take range from 0 to 1. A value of *α* = 0.5 indicates that the time series follows a random walk and does not exhibit a long memory. However, if *α* = 0.5, this indicates that the considered time series exhibits evidence of long-term correlations. If 0.5 < *α* < 1, the series indicates trend-reinforcing behaviour; if 0 < *α* < 0.5, the series exhibits antipersistence behaviour. The stochastic behaviour of the Hurst exponent computed using the DFA method for hourly returns of Ethereum, Bitcoin, and Litecoin in bull and bear markets is illustrated in Section 4.

### *2.3. Efficiency Market Hypothesis Tests*

Other methods used to test the efficiency market hypothesis include the Ljung–Box test (Ljung and Box 1978) that examines the null hypothesis of no autocorrelation; and the Wald–Wolfowitz Runs Test (Wald and Wolfowitz 1940) and the Bartels Rank Test (Bartels, 1982), both testing the null hypothesis of independence of the returns; the Wild Bootstrapping of Automatic Variance Ratio Test (Kim 2009) and the Spectral shape tests (Durlauf 1991), testing the null hypothesis that returns follow a random walk; and the Automatic Portmanteau Test (Escanciano and Lobato 2009), testing a null hypothesis of serial correlation.
