*2.2. Methodology*

Methodologically, we used a pluralistic approach combining two different methods to calibrate 'd' and 'H'. The FIGARCH is an extension of the famed GARCH family as described by Baillie (1996), which is consistent with EMH [20]. The MFDFA is an extension of detrended fluctuation analysis (DFA) as described by Kantelhardt (2008), which is consistent with FMH [21,22].

Firstly, we employed a FIGARCH model to uncover evidence of long memory in BECI indices. FIGARCH considers conditional heteroscedasticity and is comparable to ARCH, but allows for long memory in the conditional variance. It is preferred over autoregressive fractionally integrated moving average (ARFIMA) models because it can detect meanreverting long memory. Usually, financial time series have d = 1 (fractional integrating parameter), which is consistent for log closing prices of various tradeable securities. Furthermore, it is perfectly in harmony with the efficient market hypothesis (EMH), which concludes that closing levels are martingales, and log returns are martingale differences (usually first difference). Martingales are sequences of random variables with the future expectation equaling the present value. Squared returns typically carry a fractional value of *d*.

Consider a time series, such as the first level difference of each of the three BECI indices:

$$
\Delta \text{BECI}\_{l} = \mu + \varepsilon\_{l} \quad \text{with } \varepsilon\_{l} = \nu\_{l} \sigma\_{l}^{2} \tag{1}
$$

where *ν<sup>t</sup>* is a serially uncorrelated process with zero-mean and unit variance; *σ<sup>t</sup>* is a time-varying measurable function with respect to the information set available at time <sup>t</sup>−1 (ψ− (t−1)); and *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* is the time dependent conditional variance of ΔBECI*t*. The FI-GARCH model of Baillie and his co-researchers [20] is given by:

$$(1 - \beta\_1 L)\sigma\_t^2 = \omega\_0 + [1 - \beta\_1 L - \alpha\_1 L (1 - L)^d] \varepsilon\_t^2 \tag{2}$$

where, 0 ≤ *d* ≤ 1 is the fractional differential (long memory) parameter; *L* is the lag operator; β(L) is a finite order lag polynomial with the roots assumed to be situated outside the unit circle; and ∝*<sup>k</sup>* represents the autoregressive coefficient of an ARFIMA (1, *d*, 0) model. Unlike ordinary ARCH and GARCH, the FIGARCH model does not reach a constant level quickly. It is reduced to a standard GARCH when d = 0 and to an integrated GARCH (IGARCH) when *d* = 1.

Secondly, we used multifractal detrended fluctuation analysis (MF-DFA) to find h(q) value, where 'h' is the Hurst Exponent and 'q' is the order [23,24]. To this end, we relied on Espen Ihlen's algorithm in MATLAB 13 [25]. It involves a five-step process, as follows:

i. Determining the profile

$$Y(i) = \sum\_{k=1}^{i} \left[ \mathbf{x}\_k - \langle \mathbf{x} \rangle \right] \tag{3}$$

where, *xk* is the series, and mean subtraction occurs. Further, *i* = 1, 2, . . . ., *N*


$$F^2(S, v) \equiv 1/s \sum\_{i=1}^{s} \left\{ Y[(v-1)s+i] - y\_v \begin{pmatrix} i \end{pmatrix} \right\}^2 \tag{4}$$

where *yv* (*i*) is the curve fitting polynomial is segment *v*.

iv. Averaging across all segments to find *q*th order fluctuation function:

$$F\_q\left(\mathcal{S}\right) \equiv \{\frac{1}{2N\_s} \sum\_{v=1}^{2N\_b} \left[F^2(s,v)\right]^{q/2}\}^{1/q} \tag{5}$$

where *q* can be any real number, but not zero. It is interesting to note that *q* = 2 coincides with the standard DFA process. Research suggests that extremely large *q* values (−10 or +10) increase the error in the multifractal spectrum tails [26]; therefore, *q* = 5 was used to calibrate such series, which is recommended by another research work [27]. v. Determination of the scaling property of the fluctuation function:

$$F\_q \text{ (s) } \sim \text{ s}^{H(q)} \tag{6}$$

where *H*(*q*) represents the generalised Hurst exponent of the underlying series.

To understand this process intuitively, we referred to Mandelbrot's research, according to which, scaling exponents are unique in nature and depend upon time. Hence, monofractal is not a full proof. It depicts an incorrect narrative. Stochastic time series, such as Bitcoin energy consumption (BECI), have multiple dimensions that add further complexity. For this reason, multifractal is preferred over monofractal. Asset returns tend to deviate from the normal distribution. Moreover, they tend to obey Lévy stable condition. In other words, α ranges from 0 to 2, where α = 2 satisfies the condition for Gaussian distribution. Thus, [28] reformulate the Rescaled Range Analysis (R/S) approach proposed by Hurst in 1951. The Hurst exponent expresses H = 1/α; when α = 2, it becomes stochastic; i.e., it follows a Brownian motion. The legacy of fractals was investigated by a group of researchers [29] who constructed the mathematical formulae to measure the impact of multifractality in a noisy time series. Time series with consistent noises can be transformed into 'random walk' series by subtracting the mean value [25]. Ihlen (2012) integrates it further. According to his process, calculation for the root mean square variation (RMS) is crucial. RMS values are calculated for the localised areas with clear patterns or trends. Finally, all these RMS values are summarised. These RMS samples usually exhibit 'power law' characteristics. In technical terms, this process is known as detrended fluctuation analysis (DFA). The exponent for this relation is the Hurst exponent [29]. Kantelhardt (2002) formulated MFDFA formally for calibration. Ihlen (2012) extended this calculation to the qth-order, suggesting the multifractal detrended fluctuation analysis (MFDFA). Multifractal power law has more than one exponent. Further, Power law relationship and persistent pattern in most cases are two important facets of time series [30,31], such as BECI. The Hurst exponent and fractal dimensions do change from monofractal to multifractal, with the latter being more reliable [32].

Since our data points were 200 for each sliding panel, we altered the segments and scale in the MATLAB code proposed by Ihlen (2012). We took segment = 200 and scale = 4. In the first loop, samples 0–200 (Window 1) were taken. In the second loop, samples 100–300 (Window 2) were considered; the third loop considered 200–400 (Window 3), etc. A polynomial trend fit in each loop was conducted. Quadratic and cubic polynomials were used in this code. We obtained values for the fifth order (*q* = 5) Hurst exponent, and considered it for interpretations as suggested by Kantelhardt. It has recently been proved that a window size of 288, with four sub-windows having 72 observations each, works well through MFDFA [33]; therefore, our window size qualified for a robust calibration.
