3.1.1. Daily Bitcoin Price Data

Mean-centered log-prices *X*(*t*) and the log-returns for daily Bitcoin price are displayed in Figure 5. Two bull runs followed by two bear runs are observed in late 2013 and late 2017 represented by two prominent spikes. These two price surges are believed to be results of the increases in Bitcoin's popularity and media coverage. A price drop is spotted in early 2018 followed by the price falls of most cryptocurrencies. This period is known as 'the 2018 cryptocurrency crash' (Popken 2018). The Bitcoin price collapsed by 80% from January to September 2018, which is reported to be worse than the dot-com bubble's 78% collapse (Patterson 2018). Subsequently, a price hike on a smaller scale is observed between early 2019 and 3 September 2019.

A stylised feature check gives uncorrelated and stationary return increments but correlations in the absolute value of returns.

The estimation of *τ*(*q*) exhibits a relatively linear scaling function, as is shown in Figure 6. However, both concavity measures are less than 0, indicating the presence of concavity. A global simplex statistic of value −0.2222 and a localised concavity measure of value −0.6256 are obtained. The hypothesis tests based on these two measures indicate that the null hypothesis should be rejected in favour of the alternative; see Table 1.

<sup>6</sup> For example, if Hill's estimator takes value *h*Hill = 2, we generate simulated student *t*-distributed processes with 2 degrees of freedom, select those with tail indices in the interval [1.5, 2.5], and construct the null distribution using the empirical distribution of their concavity measures.

**Figure 5.** Daily mean-centered log Bitcoin open price process (**top**) and mean-centered log returns process (**bottom**).

**Figure 6. Left**—Autocorrelations of mean-centered log-returns (black) and absolute returns (grey) for daily Bitcoin open price (28 April 2013–3 April 2019). **Right**—Scaling function for daily Bitcoin open price (28 April 2013–3 September 2019). The dashed line is the *q*/2 reference line (same for all the scaling function figures).

**Table 1.** Hypothesis test results on daily Bitcoin open prices (28 April 2013–3 September 2019).


*Sample size* refers to the number of student *t*-distributed processes used in estimating the distribution of concavity measures under *H*0. In the case of Bitcoin open prices from 28/04/2013 to 03/09/2019, there are 1479 student *t*-distributed processes with tail indices ranging in the interval [2.5630, 3.5630] when constructing the distribution of localised concavity measures. 99 student *t*-distributed processes are used in the construction of the distributions of global simplex statistics.

Thus, despite appearances in Figure 6, we have preliminary evidence that Bitcoin may follow a multifractal process.

To observe whether different results appear when the sample changes, the log-price time series is broken down into smaller time periods. The same hypothesis tests are performed to examine multifractality. Some literature indicates that 2017 is a turning point for Bitcoin's price behaviour. Zhang et al. (2018) concluded an increase in market inefficiency from late 2016, which is believed to be a result of an increase in speculation. Consequently, we break the daily price process into two parts,


The corresponding scaling functions are shown in Figure 7 with hypothesis test results in Tables 2 and 3. The hypotheses tests for *<sup>X</sup>*1(*t*) give various results. This may be a result of a smaller data set or it could indicate a change to the way Bitcoin prices scale.

**Table 2.** Hypothesis test results on daily Bitcoin open prices (28 April 2013–16 July 2017).


**Table 3.** Hypothesis test results on daily Bitcoin open prices (17 July 2017–3 September 2019).


**Figure 7.** Scaling functions for Daily Bitcoin open price from 28 April 2013 to 16 July 2017 (**left**) vs. from 17 July 2017 to 3 September 2019 (**right**).
