**1. Introduction**

Recently, the market for cryptocurrencies has exhibited one of the most volatile periods in its history. While the total market capitalisation for cryptocurrencies reached a record high of over USD 800 billion in the last quarter of 2017, it was followed by a massive correction in the market leading to significantly reduced market capitalisation, which now stands at under USD 100 billion. This clearly suggests that the market has experienced a bull (cryptocurrency-price rising, precrisis) and bear (cryptocurrency-price falling, crisis) market throughout this period. Overspeculation, and interest from academics and those in the industry in this new financial technology are a few reasons behind this recent market phenomenon.

Over the past few years, the cryptocurrency literature has been rapidly expanding. The general literature on cryptocurrencies covers topics including (but not limited to) statistical analysis, modelling,

### *J. Risk Financial Manag.* **2020**, *13*, 8

and predicting the Bitcoin/USD exchange rate, measuring the volatility of the Bitcoin exchange rate against different financial assets and commodities, stylised facts of cryptocurrencies, and the market efficiency of cryptocurrencies. Chu et al. (2015) provided the first statistical and risk modelling analysis on Bitcoin returns. The generalized hyperbolic distribution provided the best fit; Glaser et al. (2014) investigated whether Bitcoin users see it as a currency or asset, and found that most uninformed users were not interested in Bitcoin as a transaction system but instead saw it as an alternative investment method. Kristoufek (2013) investigated the relationship between Bitcoin prices and search queries from Google and Wikipedia. They found that there was a significant positive correlation between prices and search queries, and that search queries had asymmetric effects on Bitcoin prices depending on whether prices were above or below the short-term trend. The significant volatility in Bitcoin prices and returns cannot simply be explained by economic or financial theory. Sapuric and Kokkinaki (2014) analysed the volatility of the exchange rate of Bitcoin during its early years and found that it was significantly greater than that of major exchange rates. However, when they accounted for transaction volume, volatility appeared to be more stable. Baur et al. (2018) analysed the statistical properties of Bitcoin and found that they were "uncorrelated with traditional asset classes such as stocks, bonds, and commodities, both in normal times and in periods of financial turmoil". In addition, the authors found that Bitcoin is primarily used as an investment asset and not as a currency. Briere et al. (2015) investigated Bitcoin from an investment perspective and found that it had significantly high average return and volatility, and little correlation with traditional financial assets. Results showed that, by including Bitcoin in well-diversified portfolios, the risk-return trade-off could be significantly improved.

The efficient market hypothesis (EMH) is a core and fundamental concept used in finance that was introduced by Malkiel and Fama (1970) through modelling financial data. There are three main forms of efficiency, with the most common being the weak form. The weak form states that investors cannot use historical-price information to make future-price predictions. The importance of understanding market efficiency can be beneficial to investors, academics, and financial practitioners, as historical-price pattern information can assist in the greater understanding or discovery of arbitrage returns. On the other hand, liquidity is a concept of how easily capital and assets can be traded without causing a dramatic change in an asset's price. In general, an illiquid asset would procure a higher bid ask spread and transaction cost, increasing the cost for speculators and investors to trade. Hence, if cryptocurrency markets are very illiquid, this results in market inefficiency, as the lack of market makers and traders causes a delay in market participants acting on new information.

Many attempts were made so far to study the market efficiency of various cryptocurrency markets, but the vast majority of the known work has been exclusively directed towards the Bitcoin market. For example, Bariviera (2017) studied the long-range memory of the Bitcoin market by analysing the Hurst exponent via the R/S and detrended-fluctuation-analysis (DFA) methods, and confirmed that daily volatility exhibits long-range memory; Alvarez-Ramirez et al. (2018) implemented the DFA method to estimate the long-range dependence of Bitcoin and found that the Bitcoin market exhibited periods of efficiency, alternating in different periods; Tiwari et al. (2018) reported that the Bitcoin market is informationally efficient, by using a battery of robust long-range dependence estimators; Khuntia and Pattanayak (2018) examined the efficiency of the Bitcoin market by using the Dominguez–Lobato consistent test and generalized spectral test, and concluded that dynamic efficiency in the Bitcoin market actually follows the proposition of adaptive market hypothesis (AMH); Jiang et al. (2018) employed the generalised Hurst exponent to investigate long-term memory in the Bitcoin market, and results suggested that the Bitcoin market was inefficient over the whole sample period; Zhang et al. (2018a) illustrated that the nine most popular cryptocurrency markets were inefficient by employing a battery of efficiency tests, and the MF-DFA and MF-DCCA approaches; Zhang et al. (2018b) analysed the stylised facts of cryptocurrencies in terms of long-range dependence by using the Hurst exponent with both the R/S and DFA methods for high-frequency-return data of the four most popular cryptocurrencies, while features of dependence between the different cryptocurrencies were also provided; Chu et al. (2019) analysed the efficiency of the high-frequency markets of the two largest cryptocurrencies, Bitcoin and Ethereum, versus the euro and US dollar, by investigating the existence of the AMH.

Our main motivation was to analyse and understand market-efficiency patterns and liquidity behaviour during a bull (precrisis) and bear (crisis) market for cryptocurrencies. These periods are very intriguing as they represent different market conditions. The main contributions of this paper are: (i) utilising algorithms for detecting turning points to identify bull and bear phases for the three largest cryptocurrencies of Bitcoin, Ethereum and Litecoin in high-frequency (hourly) markets; and (ii) analysing and understanding the characteristics of market efficiency and liquidity in high-frequency cryptocurrency returns during a bull or bear market. This is the first study of detecting bull and bear periods in high-frequency cryptocurrency markets, and analysing their market efficiency and liquidity during such periods.

For each cryptocurrency, we analysed data from 1 July 2017 to 19 September 2018. This time period was divided into two subperiods, corresponding to a bull market (precrisis period) from 1 July 2017 to 16 January 2018 (4789 observations), and a bear market (crisis period) from 17 January 2018 to 19 September 2018 (5888 observations). Sections 2 and 3 provide a detailed justification of how the bull and bear markets were identified.

For each cryptocurrency, we performed analysis by using two different methods. The first was to apply the DFA method to compute the Hurst exponent over a rolling window during a bull and bear market to analyse the behaviour of the high-frequency (hourly) returns of Ethereum, Bitcoin and Litecoin. The DFA method is most commonly implemented by using a rolling-window approach for analysing the Hurst exponent in financial time series (see, for example, Matos et al. 2008, Grech and Mazur 2004, and Carbone et al. 2004). The second was to use a series of tests, presented in Section 2, which examined the efficient market hypothesis within fixed periods (bull and bear markets). A rolling-window approach splits a dataset into subsamples of a specific size rather than analysing the whole data sample in one process. The initial subsample is analysed before the next most recent data are added to the subsample, and the earliest data in the subsample are removed. This process is then repeated until the subsample reaches the most recent data in the whole sample. A conventional fixed-period method analyses the whole data sample in one go.

The contents of the paper are organised as follows. The algorithms used in detecting bull and bear markets in cryptocurrencies and the methods used to measure the long-range memory, liquidity, and market efficiency of cryptocurrencies in a bull and bear market are discussed in Section 2. The three cryptocurrency datasets and their summary statistics are described in Section 3. Data analysis using a range of different methods, including analysis of the Hurst exponent, is presented in Section 4. Finally, conclusions are drawn in Section 5.
