*2.1. Asymmetric Multifractal Detrended Fluctuation Analysis (A-MF-DFA)*

The A-MF-DFA extends the MF-DFA method by considering positive and negative market trends [44,45]. First, the profile time series of each return time series *xj* : *j* = 1, . . . , *N* are calculated as *X*(*t*) = ∑*<sup>t</sup> j*=1 *xj* − *x* for *t* = 1, ... , *N*, where *x* is the average of the entire return time series. Then, the profile time series and the return time series are both divided into *Nn* = *N*/*n* non-overlapping segments of length *n*. In case *N* is not a multiple of *n*, we repeat the division initially from the other end of the time series to take into account all the available data, making a total of 2*Nn* segments for both the profile and the return time series.

Next, the local trend of the profile series *Xv*(*i*), *i* = 1, ... , *n* is calculated for each segment *v* = 1, ... , 2*Nn*, by fitting a least-square polynomial of degree 2 in order to detrend the corresponding profile *Xv*(*i*), *i* = 1, ... , *n*. For the return time series, the local linear trend for each segment is also calculated to determine whether the return time series show an uptrend or downtrend. The different trends depend on the sign of each local slope *bn*,*<sup>v</sup>* = 0, where *bn*,*<sup>v</sup>* represents the coefficient of the linear trend for segment *v* at scale *n* [27]. If *bn*,*<sup>v</sup>* > 0 (*bn*,*<sup>v</sup>* < 0), the return time series have an upward (downward) trend within the *v*th segment.

Then, we define the residual variance as follows:

$$F^2(n, \upsilon) = \frac{1}{n} \sum\_{i=1}^{n} \left( X\_{\upsilon}(i) - \overline{X}\_{\upsilon}(i) \right)^2. \tag{1}$$

By taking the average over corresponding segments, we can obtain the asymmetric *q*th order average fluctuation functions, which are then calculated by taking the average over the corresponding segments:

$$F\_q^+\left(n\right) = \left\{\frac{1}{M^+} \sum\_{v=1}^{2N\_n} \frac{1 + \text{sgn}\left(b\_{n,v}\right)}{2} \left[F^2\left(n, \left.v\right)\right]^{\frac{q}{2}}\right\}^{\frac{1}{q}}\right.\tag{2}$$

$$F\_q^-\left(n\right) = \left\{ \frac{1}{M^-} \sum\_{v=1}^{2N\_n} \frac{1 - \text{sgn}\left(b\_{n,v}\right)}{2} \left[ F^2\left(n, \upsilon\right) \right]^{\frac{q}{2}} \right\}^{\frac{1}{q}},\tag{3}$$

where *M*<sup>+</sup> = ∑2*Nn <sup>v</sup>*=1(<sup>1</sup> <sup>+</sup> sgn(*bn*, *<sup>v</sup>*))/2 and *<sup>M</sup>*<sup>−</sup> <sup>=</sup> <sup>∑</sup>2*Nn <sup>v</sup>*=1(1 − sgn(*bn*, *<sup>v</sup>*))/2 are the number of total segments with directional trends. Note that for all *v* = 1, ... , 2*Nn*, *M*<sup>+</sup> + *M*<sup>−</sup> = 2*Nn* holds. Therefore, the *q*th order average fluctuation functions for the overall trend is written as:

$$F\_{\eta}(n) = \left\{ \frac{1}{2N\_n} \sum\_{v=1}^{2N\_n} \left[ F^2(n, \ v) \right]^{\frac{q}{2}} \right\}^{1/q}. \tag{4}$$

The calculation is repeated to find the fluctuation function for all box sizes *n*. If longrange power-law correlations are present, the function will increase with *n* as a power-law *Fq*(*n*) <sup>∼</sup> *<sup>n</sup>h*(*q*). The scaling exponent *<sup>h</sup>*(*q*), namely, the generalized Hurst exponent, is calculated by estimating the slope of the linear regression of log *Fq*(*n*) versus log(*n*). The asymmetric generalized exponents *h*+(*q*) and *h*−(*q*) are calculated in a similar way from the relationship *F*<sup>+</sup> *<sup>q</sup>* (*n*) <sup>∼</sup> *<sup>n</sup>h*+(*q*) and *<sup>F</sup>*<sup>−</sup> *<sup>q</sup>* (*n*) <sup>∼</sup> *<sup>n</sup>h*−(*q*). In this study, we consider *<sup>n</sup>* ranging from 8 to *N*/4 for the log-log linear regression to estimate the asymmetric generalized Hurst exponents.
