*2.1. Data*

We used daily data from the Bitcoin Energy Consumption Index (BECI) over the period 25 February 2017 to 25 January 2022, according to the data availability from Digiconomist. BECI data have been recently used in academia [12,17]. They consist of daily data covering three series, BECI upper bound (BECI UB), BECI lower bound (BECI LB), and BECI average. Accordingly, in this work, 1825 daily observations were used for each of the three series. The plots of the three series are shown in Figure 1, in which an increase in the three series is observed from around the second quarter of 2021, which coincides with the spike in the price of Bitcoin.

**Figure 1.** Plots of BECI UB, LB, and average during the study period (25 February 2017 to 25 January 2022).

Considering both BECI upper bound (BECI UB) and BECI lower bound (BECI LB) for electricity consumptions, we argue the following. The BECI UB is defined as the breakeven point of mining revenues and electricity costs; therefore, it is more sensitive to the economic parameters. In contrast, the BECI LB is a state where all miners use the most efficient hardware, which makes it more stable and reliable for our current study [13,14] examining the long memory traits of the energy consumption series. Therefore, our analysis emphasises the lower bound results.

Our analysis involved the application of 17 windows to the three series, with the length of each sliding window being 200 days. Given that each series consists of 3400 (17\*200) daily observations, a total of 51(17\*3) windows (with 10,200 total observations, 3400\*3) were considered in our calculation for the sample period 25 February 2017 to 25 January 2022. The choice of sliding window-based estimation procedure is backed by the academic literature [18], which points to the suitability of the application of an increasing window size in a dynamic model for estimating long memory. A sliding window approach to modelling is suitable and an increasing estimation window leads to an increase in the estimation accuracy when calibrating the long memory [19].

Each series observation in the BECI indices is expressed in terawatt-hours (TWh), the standard unit of electricity consumption. The three BECI series are significantly meanreverting at first difference. Therefore, all the calculations and analyses were conducted at first difference (Δ). Long-range dependence or long memory was characterised by a slow, power law decay of the autocorrelation function (ACF).
