Download scientific diagram | Hurst exponent H calculated over 1-month windows for the BTC price from July to March Error bars reflect the standard.
Table of contents
For these missing data, we treat them as the price is unchanged. We eliminate the data that are larger than 40, i. This manipulation keeps the results almost unchanged except kurtosis.
Demystifying the Hurst exponent
In our data set, we find four outliers. Fig 2 d shows the volatility series s t defined by , where r t and stand for the return at t and the average of r t , respectively. The volatility series introduced in [ 58 , 59 ] can be utilized to identify the volatility clustering. Namely, the increasing decreasing trend of the volatility series implies the existence of the high low volatility clustering. Such trends indicating high and low volatility clusterings are seen in Fig 2 d. Table 1 provides descriptive statistics for the whole sample of returns, and we find a positive average, high kurtosis, and negative skewness.
We also explore the time variation in these using the rolling window method.
Some comments on Bitcoin market (in)efficiency
Fig 3 illustrates the average, standard deviation SD , kurtosis, and skewness calculated with a day rolling window. Interestingly, they vary considerably over time. Whilst the kurtosis before is very high i. Recently, it has taken a value of around 6, which is still higher than the Gaussian kurtosis. The origin of high kurtosis could be a fat-tailed return distribution that means higher price variations are observed more often.
The inverse square law observed in the Bitcoin market, however, is not permanent. The recent Bitcoin data up to show that the tail index comes close to 3, which suggests that the Bitcoin market is becoming more mature [ 47 ].
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Further studies [ 66 , 67 ] also indicate the change of the tail index to 3. It is also worth noting that the recent COVID pandemic considerably affects the cryptocurrency market and as a result the market experiences a volatile period in which the tail index varies [ 13 ]. SD stands for standard deviation. These are calculated using the rolling window method with a window size of days.
Bars in the data points represent one sigma error, estimated by the Jackknife method. Although the skewness is mostly negative over the whole period, the magnitude of the negative skewness decreases gradually with time, and the skewness seems to disappear in recent returns. The disappearance of skewness possibly means the efficiency improvement of the Bitcoin market, and agrees with the observation in efficiency-related measures such as the Hurst exponent and the multifractal degree that indicate that the market efficiency of the Bitcoin market varies over time and the recent market efficiency is being improved.
We describe the results on the market efficiency in more detail later. Overall, these findings in descriptive statistics also suggest that properties of the Bitcoin market vary over time. Thus, it is important to consider time variation in analysis. Its magnitude, however, is not large, which is consistent with the result of [ 31 ]. It is evident that the parameters vary over time.
We also observe a strong inverted asymmetry between and For the robustness check on the IID distribution in Eq 2 , we perform the parameter estimation with the normal and generalized error distributions, and find that the similar asymmetric volatility patterns to that from the Student t distribution are obtained. Therefore, the choice of distributions in the IID process is irrelevant. The error bars show the standard errors. The AR 1 parameter c 1 , which captures serial correlation, also varies considerably. It is argued that non-zero serial correlation implies that uninformed investors dominate in trading and that price changes due to uninformed investors will increase volatility more than price changes caused by informed investors [ 68 ].
In line with [ 68 ], it is claimed that non-zero AR 1 coefficients are found for cryptocurrencies; and the inverted asymmetry due to uniformed investors is consistent with phenomena such as fear of missing out, pump and dump schemes, and the disposition effect [ 31 ]. Our results of c 1 indicate that there are periods in which c 1 is consistent with zero, which suggests that the Bitcoin market is not always dominated by uninformed investors.
We find both strong inverted asymmetry and non-zero c 1 from to Thus, the period from to is considered to be dominated by uninformed investors, thereby affecting the volatility asymmetry. As seen in Fig 2 , in this period, the Bitcoin price increases considerably and recorded the highest value on December Consequently, Bitcoin price movement in this period is associated with the inverted asymmetry induced dominantly by uninformed investors. We also estimate the model parameters for high-frequency returns 6h and 12h.
We find that high-frequency returns exhibit similar variation with daily returns except that the 6h returns for which no significant inverted asymmetry is seen before These results imply that whilst the asymmetric volatility pattern remains for higher frequency returns, the detail of the asymmetry pattern depends on the frequency of returns.
As representatives, Fig 8 a and 8 b shows fluctuation functions F q s calculated using the first window data and h q , respectively. This finding is consistent with the results of kurtosis and skewness in Fig 3 that shows low kurtosis and insignificant skewness for the recent Bitcoin market, meaning that the return distribution becomes more Gaussian shaped than before.
Two measures of the multifractal degree calculated here show the same time-varying pattern. Broadly speaking, the multifractal degree decreases with time, which suggests that the Bitcoin market gradually approaches the efficient market. Using the Amihud illiquidity measure [ 69 ], it is argued that the market inefficiency in the cryptocurrency market is caused by illiquidity and the illiquidity is related with anti-persistency [ 4 ]. For the Bitcoin market, liquidity in terms of the Amihud illiquidity measure turns out to be improving [ 44 ], which agrees with the improved efficiency of the Bitcoin market in recent years.
These results are obtained by the rolling window method with a day window and a step of 1 day. Therefore, the results of Fig 10 a and 10 b imply that in the more efficient market, the kurtosis takes smaller values close to the Gaussian one. Fig 11 c indicates that for a region near the Gaussian kurtosis e.
This is consistent with the consequence of the efficient market that any predictable patterns such as the asymmetric volatility should not exist. Correct understanding of the market state is of great importance for investors who change trading strategy depending on the state of the market.
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This study could contribute to offer such information. For example, according to the efficient market hypothesis [ 46 ], the technical analysis is not supported on the efficient market. Our results imply that the recent Bitcoin market is being efficient and it might be difficult to make high profits by the technical analysis. By monitoring the market efficiency of Bitcoin, if the Bitcoin market becomes inefficient substantially again, one could use the technical analysis to gain profits.
Another suggestion from the efficient market hypothesis is that on the efficient market, the most efficient portfolio is a market portfolio consisting of every asset weighted in proportion to its market capitalization. Of course, it is difficult to form a completely diversified portfolio in practice [ 70 ]. However, it may be possible to make an approximate diversified portfolio including the efficient Bitcoin. Furthermore, although not all cryptocurrencies are efficient, when the cryptocurrency markets become more mature and more efficient, an index or portfolio consisting of cryptocurrencies could be a proxy of a fully diversified portfolio on the cryptocurrency markets.
We use the rolling window method to investigate time-varying properties of Bitcoin. We find that various measurements, such as volatility asymmetry, kurtosis, skewness, serial correlation, and multifractality, are time varying. Thus, the Bitcoin market may have inherently variable properties. Although the inverted asymmetry is observed in Bitcoin and the strong inverted asymmetry is found around , the recent volatility asymmetry is weak. The magnitude of the volatility asymmetry may relate with the market state, especially the market efficiency. To investigate a relationship between the volatility asymmetry and the market efficiency, we examine efficiency-related measures: the Hurst exponent, multifractal degree, and kurtosis.
Paper methodology
We find that when these efficiency-related measures indicate that the market is more efficient, the volatility asymmetry is more likely to weaken. The efficiency-related measures indicate that the recent Bitcoin market has become more efficient. It might be interesting to investigate whether these models lead to the similar results on the volatility asymmetry.
In addition to the inverted asymmetry, other remarkable properties are observed in the Bitcoin market. This difference is important to construct a correct volatility model with monofractal or multifractal behavior.