*3.1. Results from the MFDFA*

The results from the MFDFA investigation revealed evidence of mean-reverting long memory process across BECI UB, LB, and average (see Table 4). Further, no pure short memory was detected (*d* = 0). The effect of long memory was found to be consistent across most sliding windows, barring one (window 10). The persistent pattern was observed for 17/17 (100%) cases in BECI UB and 16/17 (94%) cases in BECI LB. Most importantly, all these cases recorded a Hurst exponent significantly larger than 0.5, indicating a consistently higher intensity of long memory. Extremely strong long memory (H > 0.85) was found in more than 90% cases through this method. Multifractal or singularity spectrum was used to describe the fractal dimension (d) having the same HÖlder exponent. Here, all of the empirical values of 'd' ranged between (−0.07 to +0.50). Both Figures 2 and 3 (chosen randomly out of 51 Windows in consideration), exhibit similar multifractal spectrums, peak around Hq = 1.1 to 1.3. This is more consistent with FMH rather than EMH, which would suggest a peak around Hq = 0.5. The spectrum in each case ranges from 0.6 to 0.8 on the lower side and from 1.6 to 1.7 on the higher side. Small fluctuations are clearly persistent from these two figures. The lower limit of the spectrums in most cases (barring one) ended around Hq > 0.5, indicating the presence of long memory throughout. Another interesting observation is related to the width of the multifractal spectrum. In all 51 windows the width (multifractal) was around 1, exhibiting clear asymmetry (more for BECI LB). This indicates that the degree of complexity in BECI is quite high. This finding remains consistent with existing research [34]. However, future research may further investigate the differences between the left side (small fluctuations from individual cascades reaching white noise faster) and the right side (contracts far sooner to monofractal) [35].

**Table 4.** d and H values for BECI UB, LB and Average- MFDFA method.


Note: This table shows evidence of more observations having long memory using MFDFA (*q* = 5th order), 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.

**Figure 2.** BECI UB window 1 exhibiting the Multifractal Spectrum.

**Figure 3.** BECI LB window 14 exhibiting the Multifractal Spectrum.
