**4. Results and Discussion**

Tables 3–5 show the results of the various test for the efficient market hypothesis on Ethereum, Bitcoin, and Litecoin, respectively. These tests were conducted over two fixed subperiods, during the bull market and during the bear market. In each case, corresponding *p*-values are shown. For the bull-market period, the majority of the *p*-values (with the exception of the Runs test for Ethereum and the AVR test for Litecoin) for all cryptocurrencies rejected the null hypotheses of no autocorrelation, independence, and random walk. Similarly, during the bear-market period, the majority of the *p*-values (with the exception of the Ljung–Box test and AVR test for Ethereum) for all cryptocurrencies rejected the null hypotheses of no autocorrelation, independence, and random walk. Overall, these tests indicated that high-frequency (hourly) cryptocurrency returns exhibited behaviour consistent with an inefficient market during both a bull and bear market. When compared to other financial markets, similar results can be seen, for example, Gil-Alana et al. (2018) noted that the Baltic stock market rejected the theory of market efficiency during the bull and bear markets; Jiang and Li (2019) investigated market efficiency for the Chinese, Japanese, and U.S. stock markets, and found market inefficiency in both the bull- and bear-market states, which could be explained by behavioural finance theory.

**Table 3.** Market-efficiency test for hourly returns of Ethereum during bull and bear market.



**Table 4.** Market-efficiency test for hourly returns of Bitcoin during bull and bear market.

**Table 5.** Market-efficiency test for hourly returns of Litecoin during bull and bear market.


The results of Amihud's illiquidity ratio are shown in Table 6. Results were multiplied by 10<sup>8</sup> for a simpler comparison, and this did not lead to loss of information. When comparing the three cryptocurrencies on the basis of the Amihud ratio, Bitcoin had the smallest value, followed by Ethereum and Litecoin. This illustrates that Bitcoin is the most liquid cryptocurrency. This result is consistent with our expectations, as Bitcoin holds the largest share of the cryptocurrency-market capitalisation, making it the most actively traded cryptocurrency. Other factors that led to Bitcoin being the most actively traded cryptocurrency include numerous trading platforms requiring users to hold Bitcoin before being able to trade other cryptocurrencies; the launch of Bitcoin futures, which allowed speculators to long and short Bitcoin and increased Bitcoin volatility; and the majority of exchanges providing other products, such as the trading of cryptocurrency pairs (e.g., BTC/ETH, BTC/LTC, BTC/XRP), with the majority of pairs involving Bitcoin.

**Table 6.** Amihud illiquidity ratio on hourly returns for Ethereum, Bitcoin, and Litecoin during bull and bear market.


When comparing bull- and bear-market liquidity and volatility, there was a strong relationship between liquidity and volatility for Ethereum and Litecoin. For Ethereum and Litecoin, market volatility decreases during a bear market, and the market becomes less liquid. However, results were different for Bitcoin, as market volatility decreases during a bear market, and Bitcoin becomes more liquid than in the bull market. This phenomenon could be explained by investors becoming irrational and worrying about the whole cryptocurrency market collapsing, leading to a majority of investors cashing out their cryptocurrency holdings. Most trading exchanges only allow cashing out cryptocurrencies through Bitcoin; therefore, this also causes Bitcoin to be traded by more active traders, which makes the market more liquid.

The long-range memory for all three cryptocurrencies during a bull and bear market was computed using the DFA method. The difference between this and previous methods is that this technique uses a rolling-window approach, as discussed in Section 2. A dotted black line in each plot was included to enable easier comparison between plots. Figure 2, illustrates the fluctuating behaviour of the Hurst exponent for the hourly returns of Ethereum, Bitcoin, and Litecoin during the different market periods. Ethereum and Bitcoin Hurst exponent values follow a similar pattern during the bull and bear market. In contrast, Litecoin results look slightly different. Throughout the bull period, Ethereum and Bitcoin generated a Hurst exponent of around 0.5, with Ethereum being slightly higher than Bitcoin. However, at around 2700 lags (bull period), the value significantly drops to below 0.4 before correcting and fluctuating back to around 0.5. In general, a Hurst exponent close to 0.5 indicates that the series is more random and resembles a random walk. This suggests that hourly Ethereum and Bitcoin returns are relatively efficient during a bull market. During the bear market, Ethereum and Bitcoin exhibit an increasing trend in the Hurst exponent as lag times increase. This indicates that the returns of both cryptocurrencies experience long-term positive autocorrelation. For Litecoin, the pattern of the Hurst exponent during a bull market is very close to 0.5 for the first 3000 lags, which illustrates that returns follow a random walk. However, after 3000 lags, the Hurst exponent suddenly increases to a value over 0.6, suggesting persistent behaviour. During a bear market, Litecoin exhibits a similar pattern to Bitcoin and Ethereum, suggesting that the market experiences long-term positive autocorrelation. Overall, we can conclude that, during a bull market, cryptocurrencies exhibit random-walk behaviour. However, when a bear market occurs, returns start to show persistent positive autocorrelation behaviour (market inefficiency). These results are also in line with those of Wang and Yang (2010), who identified intraday market inefficiency in heating-oil and natural-gas energy future markets during bull-market states, but not during bear markets.

**Figure 2.** Plots of Hurst exponent of high-frequency 1 h log returns of (**top left**, **top right**) Ethereum, (**middle left**, **middle right**) Bitcoin, and (**bottom left**, **bottom right**) Litecoin using a sliding window of 720 lagged data points. Bull and bear markets indicated by blue and red lines, respectively.

A contrast in results from both methods during the bull-market phase could be interpreted in the following way. Analysis involving the Hurst exponent utilises a rolling-window approach; during these individual rolling windows (subsamples of a specific size) within the bull market, the cryptocurrency market had actually steadily grown with investors gradually entering the market, thus leading to an efficient-market phenomenon. However, if we look at the overall picture of the bull-market phase, we see the market significantly rises; when considering the method that uses a fixed sample period, this appears to create an inefficient market. On the contrary, during the bear-market phase, the price consistently decreased throughout the whole period, which may have caused panic and irrational trading by investors, leading to a downward spiral in market prices, resulting in market inefficiency.

The actual methods and tests used here could also lead to result divergence due to the difference in tested samples in each case. For example, methods such as the Ljung–Box, Runs, and Bartels tests analyse all bull-/bear-market phase data as one, as opposed to the DFA method that analyses dynamic rolling data windows from the bull-/bear-market periods. In other words, the size of the samples analysed by the DFA method is smaller, but data points vary, while samples used in general tests are larger and data are fixed. Therefore, it is possible that the DFA method picks up variations within particular subsamples of return data (thus affecting results relating to market efficiency), which may be masked when considering bull-/bear-market samples as a whole. Furthermore, our results illustrate that, in a bear market, hourly Bitcoin returns become more liquid during this period of market inefficiency. In contrast, hourly Ethereum and Litecoin returns exhibit less liquidity in this period.

Here, we only tested the hypothesis of market efficiency through a range of different tests (including classical tests and the DFA method). We cannot claim that the DFA is better or more trustworthy, but results of DFA analysis sugges<sup>t</sup> that the level of market efficiency is different in bull and bear markets. Although the Hurst exponent (as a proxy for market efficiency) shows general trends in bull and bear markets, there are shorter-term changes that are also captured, likely due to the rolling-window approach. One could run the classical tests over different sample periods, but a problem that remains is that classical tests only generate a p-value. This p-value only gives us an indication of whether we can reject or fail to reject the null hypothesis of tests for properties such as independence, autocorrelation, and random walk. This is in contrast to DFA analysis, which not only gives us a numerical value indicating deviations from market inefficiency, but could provide further information, such as trend-reinforcing or antipersistence behaviour.
