**3. Data**

We analyzed the time series of 19 major stock market indices that appear on the website https://finance.yahoo.com/world-indices/ (accessed on 2 January 2022), which are listed in Table 1. The period under study spanned the earliest recorded date for each index up to the end of 2018. For each of the market indices with consecutive workday closing price values *<sup>S</sup>*(*t*), *t* = 1, . . . , *N*, we calculated the daily logarithmic returns:

$$R\_t \equiv \ln \frac{S(t)}{S(t-1)} \quad t = 2, \dots, N,\tag{7}$$

where the returns for Monday were calculated using the closing price of the previous Friday, while for other days of the week, two consecutive workday closing price values were used. Next, we constructed time series from the returns *Rt* for each day of the week (Monday returns, Tuesday returns and so on):

$$\mathcal{R}^{\bar{i}} = \{ \mathcal{R}\_{t\_{i}}, \mathcal{R}\_{t\_{i} + \dots, \mathcal{I}}, \mathcal{R}\_{t\_{i} + \dots \mid \frac{N}{2}}] \}, \tag{8}$$

where *i* = 1, ... , 5 denotes the index of the weekday, *Rti* corresponds to the first occurrence of day *i* in the returns series *Rt*, *t* = 2, ... , *N* and the operator [.] denotes taking the integer part of the argument. Figure 1 reveals that the fluctuations in the returns varied between different days. While Monday exhibited the most pronounced negative returns, the fluctuations for other days dominated at specific time intervals. This is a well-known day-of-the-week effect which was found for the US market [8,25].

The MF-DFA method was applied to the day-resolved returns *Ri* of major stock market indices, where local trends were fitted with a second-degree polynomial *m* = 2. Next, we performed a fourth-order polynomial regression on the singularity spectra *f*(*α*) to determine the position of the maximum *α*0 and the zeros of the polynomial *α*max and *α*min. From the polynomial fits, we calculated three measures of complexity: the position of the maximum *α*0, the width of the spectrum *W* = *α*max − *α*min and the skew parameter *r* = (*<sup>α</sup>*max − *<sup>α</sup>*0)/(*<sup>α</sup>*0 − *<sup>α</sup>*min). These parameters were then used to determine the multifractal behavior of the day-resolved price returns.


**Table 1.** Information on analyzed time series for major market indices.

**Figure 1.** Time series for (**a**) Monday, (**b**) Tuesday, (**c**) Wednesday, (**d**) Thursday and (**e**) Friday day-resolved price returns *Ri* of the United States (GSPC) market index.

### **4. Results**

#### *4.1. Day-of-the-Week Effect*

Complexity measures derived from the singularity spectra were used to study the multifractal behavior of the price returns for every day of the week. We first considered multifractality in the day-resolved price returns from four distinct markets: the United States (GSPC), South Korea (KS11), Chile (IPSA) and France (FCHI). The multifractal spectra for each day using the four markets are illustrated in Figure 2. We observed that the day-of-the-week effects led to significant differences in multifractal behavior: (1) the positions of the maxima *α*0 were shifted to the right (*<sup>α</sup>*0 > 0.5) for the Monday returns, and (2) the spectrum widths *W* were wider on Monday than those for returns from other days. There seemed to be no consistent differences in the skew parameter *r*, which implies that both large and small fluctuations are present for different days of the week (e.g., see Table 2). These results indicate that the Monday returns exhibited more persistent behavior and richer multifractal structures, which led to more complex time series than other day's returns. Our findings are consistent with results obtained from [25], which indicated that Monday had the largest anomalies (day-of-the-week effect) because of the weekend gap in trading hours. Other days of the week did not exhibit any visible patterns in multifractal behavior for either the position or width of the spectrum.

**Figure 2.** Multifractal spectrum *f*(*α*) for day-resolved price returns *Ri* of (**a**) the United States (GSPC), (**b**) South Korea (KS11), (**c**) Chile (IPSA) and (**d**) France (FCHI) market indices.

**Table 2.** Multifractal parameters *α*0, *W* and *r* for day-resolved price returns *Ri* of major market indices.


We expanded our investigation to other markets listed in Table 1. Figure 3 reveals that the multifractal spectra of the Monday returns were dominantly right-shifted (*<sup>α</sup>*0 > 0.5) compared with other days for most analyzed markets. Notable exceptions included the United States (DJI), Australia (AORD, AXJO), where the Tuesday returns were more persistent, and Japan (N225), where the Thursday returns exhibited stronger persistency. The width of the multifractal spectrum displayed similar tendencies to its position, where the Monday returns possessed broader multifractal widths. Yet, we found that more markets tended to have other days with richer multifractal structures; the multifractal spectra were the widest for the Friday returns in Taiwan (TWII) and the Tuesday returns in Japan (N225) and Australia (AORD), as opposed to the markets with dominant Monday returns considered so far. It has been noted that the day-of-the-week effect occurs on different distinct days of the week for different markets [25]. Considering both parameters *α*0 and *W*, we observed that the North American, European and some Asian (South Korea, Indonesia and Hong Kong) and Latin American markets (Chile and Mexico) tended to show both stronger persistency and stronger multifractality for the Monday returns, while for Australia, Indonesia and Taiwan, this tendency was found for the Tuesday returns. This is also in agreemen<sup>t</sup> with the literature, where it was found that some Asian markets displayed a Tuesday anomaly, which is one day out of phase with North American markets due to different time zones [64]. Patterns in the skew of multifractal spectra for a given day of the week are again hard to discern across distinct markets, where both small and large fluctuations exist. Values of the multifractal complexity parameter are listed in Table 2. Our results indicate that while most markets exhibit more complex behavior for Monday returns, some markets have other days with largest anomalies (day-of-the-week effect) such as Tuesday, Thursday and Friday returns. This is expected from literature where it was found that different day-of-the-week effects exist for different markets [25].

**Figure 3.** Complexity parameters (**a**) position of maximum *α*0, (**b**) spectrum width *W*, and (**c**) skew parameter *r*, for day-resolved price returns of the market indices listed in Table 1, sorted from largest to smallest.

#### *4.2. Comparison to Bulk Behavior*

The day-resolved multifractal spectra could also be compared to those for the whole time series. The motivation for such a comparison is to provide more insight on the relation between multifractality and the day-of-the-week effect. From Figure 3, we found that many markets (IPSA, KS11, GSPTSE and MMX) exhibited distinct multifractal properties for a particular day (e.g., Monday returns), while the whole series displayed similar multifractal behavior to the bulk, or the remaining days of the week. For other markets (DJI, AXJI and N225), the overall multifractality of the series differed widely from the multifractal spectrum for each day of the week. This suggests that the day-of-the-week effects resulted in different multifractalities for these markets. We could further classify the markets into one of two multifractal behaviors: (1) bulk multifractality, which only differs for one

particular day of the week, and (2) day-of-the-week multifractality, which is unique to every day and differs from the bulk behavior.

#### *4.3. Source of Multifractality*

We shuffled the time series of the day-resolved returns for the four markets and then applied MF-DFA to determine the source of multifractality. The shuffling procedure performed 1000 × *N* transpositions on each series and was repeated 100 times with different random number generator seeds in order to obtain statistics such as the mean and standard deviation. Figure 4 reveals that for the United States (GSPC), the right-hand side of the spectrum (effect of small fluctuations) was mildly affected by shuffling on Mondays and Fridays, while the left side of the spectrum (effect of large fluctuations) was affected primarily on Thursdays (and less so on Wednesdays), and the position remained the same for all of the day-resolved returns. This indicates that multifractality arose primarily from a broad probability density function [65], and the long-term correlations had only a minor impact on some days of the week.

**Figure 4.** Original and shuffled multifractal spectra *f*(*α*) for (**a**) Monday, (**b**) Tuesday, (**c**) Wednesday, (**d**) Thursday and (**e**) Friday day-resolved price returns of the United States (GSPC) market.

While it may be argued that destroying correlations by shuffling leads to strictly monofractal behavior and leaving only finite size effects, as shown for the qGaussian distributions using MFDFA [66] and market volatility data using partition function formalism [67], in the current case, shuffling left the spectrum width only slightly narrowed down, in agreemen<sup>t</sup> with previous MFDFA studies of market returns [65]. Even if upon shuffling only a finite size effect remained, different effects on different days of the week on small and large fluctuations provided novel insight into the market behavior.

Table 3 lists the changes in spectra position (Δ*α*0) and width (Δ*W*) after shuffling the day-resolved returns for GSPC, KS11, IPSA and FCHI. We found that the Monday returns tended to exhibit the strongest effect from shuffling, where aside from the probability density function, long-term correlations also contributed to multifractality.

**Table 3.** Differences in multifractal parameters between original and shuffled day-resolved price returns.


#### *4.4. Time Evolution*

For intuition on how the multifractal day-of-the-week effects change over time, we could analyze the time evolutions of the multifractal spectra. We considered the United States (GSPC) market, since the day-of-the-week effects over time here are well known [48]. For each day-of-the-week return, we constructed a sliding window of a size *w* = 730 days with a sliding step Δ = 5 days, meaning that we applied the MF-DFA method over a 14-year period in monthly intervals. Figure 5 illustrates the time evolutions of the multifractal spectra for different day-resolved returns. We observed that the spectrum evolved differently for each day of the week. For the Monday returns, the spectrum shifted to the left, which means that the fluctuations became less persistent over time. Other dayof-the-week returns either exhibited small movements in the multifractal spectra or moved back to the same position after some time. For a more quantitative analysis, we calculated the differences over time in the complexity parameters, namely Δ*α*0 and Δ*W*, between Monday and other day-resolved returns. Figure 6a reveals that the spectra position of the Monday returns differed considerably from *α*0 of the other day returns in the first 15 years of the recorded period, but their differences dropped to zero in the subsequent years. This indicates the presence of strong day-of-the-week effects between 1950 and 1980 (Δ*α*0 → 0 after 1965, where 1980 is already included because of the 14-year long sliding window), which is consistent with the literature, where it was found that the day-of-the-week effects diminished around 1980 [48].

Fluctuations around Δ*α*0 = 0 after 1980 can be attributed to large financial crises that affected the entire market, such as Black Monday in 1987 and the global financial crisis in 2008. Figure 6b illustrates the time evolutions of the differences in the spectra width Δ*W* between Monday and other day-resolved returns. We observed that the Monday returns exhibited much wider multifractal spectra than other day's returns during either of the two financial crises in 1987 and 2008. The Monday returns were characterized by more complex structures and had significant day-of-the-week effects during the financial crises even after 1980, when the effects from the calendar anomalies should have vanished. A possible explanation for this phenomena is the weekend gap in trading hours, which leads to even more speculative behavior from investors during a crisis.

**Figure 5.** Time evolution of the multifractal spectrum *f*(*α*) for (**a**) Monday, (**b**) Tuesday, (**c**) Wednesday, (**d**) Thursday and (**e**) Friday day-resolved price returns of the United States (GSPC) market. A sliding window of 14 years and monthly intervals were used for the period spanning from 1950 to 2019.

**Figure 6.** Time evolution of differences in complexity parameters (**a**) *α*0 and (**b**) *W* derived from the multifractal spectra *f*(*α*) between Monday and other day-resolved price returns for the United States (GSPC) market. A sliding window of 14 years and monthly intervals were used for the period spanning from 1950 to 2019.
