**3. Results**

Looking at Table 1, the first differences in the three BECI indices are non-normally distributed, as evidenced by the Jarque–Bera statistics, with low kurtosis points to low volatility and high persistence, which can be confirmed by both the fractional integrating parameter and the Hurst exponent (H). The results of the augmented Dickey–Fuller (ADF) test show that a significant level of stationarity is achieved at first difference.

Table 2 gives the ranges of d and H and their interpretation, indicating the difference between intermediate memory tending towards short and long memory.


**Table 1.** Summary statistics of the first difference of BECI indices.

Notes: The sample period is 25th February 2017 to 25th January 2022. BECI upper bound (BECI UB), BECI lower bound (BECI LB). \* Indicates statistical significance at the 1% level.

**Table 2.** Ranges of d and H and their interpretations.


Notes: d is the fractional differential (long memory) parameter. H stands for Hurst exponent, which measures the extent of long memory in time series.

Table 3 shows that all the estimated d parameters are below the 0.5 level. This shows that there are no cases of non-stationarity (*d* = 1). At the same time, there is no purely stationary case (*d* = 0). All cases are mean-reverting (*d* < 1), but they have different degrees of decaying autocorrelations.

**Table 3.** d and H values for BECI UB, LB and Average- FIGARCH method.


Note: This table shows evidence of more observations having long memory using FIGARCH, but of various degrees. d is the fractional differential (long memory) parameter. H stands for Hurst exponent, which measures the extent of long memory in time series. BECI (Bitcoin Energy Consumption Index), BECI UB (BECI upper bound), BECI LB (BECI lower bound), and BECI Average. The sample period is 25 February 2017 to 25 January 2022.
