# A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect

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## Abstract

**:**

## 1. Introduction

## 2. Methods

- i
- The first step is the integration of the original series $x\left(i\right),i=1,\dots ,N$ to produce$$\begin{array}{c}\hfill X\left(k\right)=\sum _{i=1}^{k}[x\left(i\right)-\langle x\rangle ],\phantom{\rule{1.em}{0ex}}k=1,\dots ,N,\end{array}$$
- ii
- Next, the integrated series $X\left(k\right)$ is divided into ${N}_{n}=int(N/n)$ non-overlapping segments of a length n, and in each segment $\nu =1,\dots ,{N}_{n}$, the local trend ${X}_{n,\nu}\left(k\right)$ is estimated as a linear or higher order polynomial least square fit and subtracted from $X\left(k\right)$.
- iii
- The detrended variance$$\begin{array}{c}\hfill {F}^{2}(n,\nu )=\frac{1}{n}\sum _{k=(\nu -1)n+1}^{\nu n}{\left[X\left(k\right)-{X}_{n,\nu}\left(k\right)\right]}^{2}\end{array}$$$$\begin{array}{c}\hfill {F}_{q}\left(n\right)={\left\{\frac{1}{{N}_{n}}\sum _{\nu =1}^{{N}_{n}}{\left[{F}^{2}(n,\nu )\right]}^{q/2}\right\}}^{1/q},\end{array}$$
- iv
- Repeating this calculation for all box sizes provides the relationship between the fluctuation function ${F}_{q}\left(n\right)$ and box size n. ${F}_{q}\left(n\right)$ increases with n according to a power law ${F}_{q}\left(n\right)\sim {n}^{h\left(q\right)}$ if long-term correlations are present. The scaling exponent $h\left(q\right)$ is obtained as the slope of the linear regression of $log{F}_{q}\left(n\right)$ versus $logn$.

## 3. Data

## 4. Results

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

#### 4.2. Comparison to Bulk Behavior

#### 4.3. Source of Multifractality

#### 4.4. Time Evolution

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Time series for (

**a**) Monday, (

**b**) Tuesday, (

**c**) Wednesday, (

**d**) Thursday and (

**e**) Friday day-resolved price returns ${R}^{i}$ of the United States (GSPC) market index.

**Figure 2.**Multifractal spectrum $f\left(\alpha \right)$ for day-resolved price returns ${R}^{i}$ of (

**a**) the United States (GSPC), (

**b**) South Korea (KS11), (

**c**) Chile (IPSA) and (

**d**) France (FCHI) market indices.

**Figure 3.**Complexity parameters (

**a**) position of maximum ${\alpha}_{0}$, (

**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.

**Figure 4.**Original and shuffled multifractal spectra $f\left(\alpha \right)$ for (

**a**) Monday, (

**b**) Tuesday, (

**c**) Wednesday, (

**d**) Thursday and (

**e**) Friday day-resolved price returns of the United States (GSPC) market.

**Figure 5.**Time evolution of the multifractal spectrum $f\left(\alpha \right)$ 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**) ${\alpha}_{0}$ and (

**b**) W derived from the multifractal spectra $f\left(\alpha \right)$ 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.

Market | Country | Index | Period |
---|---|---|---|

All Ordinares | Australia | AORD | 3 August 1984–26 December 2018 |

S&P500/ASX 200 | Australia | AXJO | 22 November 1992–26 December 2018 |

BEL 20 | Belgium | BFX | 9 April 1991–24 December 2018 |

IBOVESPA | Brazil | BVSP | 27 April 1993–21 December 2018 |

Dow30 | United States | DJI | 29 January 1985–26 December 2018 |

CAC 40 | France | FCHI | 1 March 1990–24 December 2018 |

DAX Performance | Germany | GDAXI | 30 December 1987–27 December 2018 |

S&P500 | United States | GSPC | 3 January 1950–24 December 2018 |

S&P/TSX Composite | Canada | GSPTSE | 29 June 1979–24 December 2018 |

Hang Seng Index | Hong Kong | HIS | 31 December 1986–27 December 2018 |

IPSA Santiago de Chile | Chile | IPSA | 2 January 2002–26 December 2018 |

Nasdaq | United States | IXIC | 5 February 1971–26 December 2018 |

Jakarta Composite | Indonesia | JKSE | 1 July 1997–27 December 2018 |

KOSPI Composite | South Korea | KS11 | 1 July 1997–26 December 2018 |

Merval | Argentina | MERV | 8 October 1996–26 December 2018 |

IPC Mexico | Mexico | MXX | 8 November 1991–26 December 2018 |

Nikkei 225 | Japan | N225 | 5 January 1965–27 December 2018 |

NYSE Composite | United States | NYA | 31 December 1965–26 December 2018 |

TSEC Weighted | Taiwan | TWII | 2 July 1997–27 December 2018 |

**Table 2.**Multifractal parameters ${\alpha}_{0}$, W and r for day-resolved price returns ${R}^{i}$ of major market indices.

Market | Monday | Tuesday | Wednesday | Thursday | Friday | All | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathbf{0}}$ | $\mathit{W}$ | $\mathit{r}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | $\mathit{W}$ | $\mathit{r}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | $\mathit{W}$ | $\mathit{r}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | $\mathit{W}$ | $\mathit{r}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | $\mathit{W}$ | $\mathit{r}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | $\mathit{W}$ | $\mathit{r}$ | |

AORD | 0.547 | 0.570 | 0.837 | 0.585 | 0.684 | 0.590 | 0.547 | 0.628 | 0.815 | 0.549 | 0.549 | 0.963 | 0.574 | 0.603 | 0.771 | 0.583 | 0.579 | 0.942 |

AXJO | 0.537 | 0.529 | 0.990 | 0.583 | 0.514 | 1.201 | 0.561 | 0.558 | 0.897 | 0.557 | 0.544 | 1.180 | 0.562 | 0.550 | 0.866 | 0.533 | 0.748 | 0.913 |

BFX | 0.619 | 0.633 | 0.730 | 0.561 | 0.662 | 1.383 | 0.571 | 0.541 | 0.754 | 0.574 | 0.553 | 0.940 | 0.553 | 0.556 | 0.715 | 0.534 | 0.676 | 1.188 |

BVSP | 0.616 | 0.581 | 1.562 | 0.601 | 0.472 | 0.932 | 0.587 | 0.455 | 1.492 | 0.615 | 0.465 | 0.939 | 0.592 | 0.666 | 0.943 | 0.550 | 0.643 | 0.917 |

DJI | 0.572 | 0.826 | 0.579 | 0.598 | 0.643 | 0.883 | 0.576 | 0.586 | 0.970 | 0.560 | 0.661 | 0.887 | 0.581 | 0.669 | 1.230 | 0.520 | 0.690 | 0.720 |

FCHI | 0.617 | 0.656 | 0.969 | 0.526 | 0.621 | 1.257 | 0.579 | 0.613 | 1.087 | 0.620 | 0.579 | 1.090 | 0.553 | 0.535 | 0.894 | 0.506 | 0.633 | 1.174 |

GDAXI | 0.606 | 0.612 | 0.682 | 0.556 | 0.619 | 0.886 | 0.574 | 0.530 | 0.986 | 0.616 | 0.538 | 1.034 | 0.555 | 0.485 | 1.397 | 0.534 | 0.648 | 1.176 |

GSPC | 0.590 | 0.787 | 0.709 | 0.565 | 0.539 | 0.856 | 0.557 | 0.635 | 0.760 | 0.528 | 0.573 | 0.718 | 0.551 | 0.627 | 1.416 | 0.528 | 0.605 | 0.782 |

GSPTSE | 0.611 | 0.632 | 0.683 | 0.587 | 0.647 | 0.841 | 0.581 | 0.618 | 0.956 | 0.587 | 0.552 | 0.733 | 0.554 | 0.681 | 0.775 | 0.585 | 0.613 | 0.928 |

HIS | 0.582 | 0.823 | 0.828 | 0.562 | 0.669 | 0.639 | 0.514 | 0.730 | 0.864 | 0.592 | 0.509 | 1.083 | 0.576 | 0.749 | 0.878 | 0.557 | 0.609 | 0.805 |

IPSA | 0.654 | 0.969 | 0.832 | 0.584 | 0.747 | 0.984 | 0.580 | 0.519 | 1.250 | 0.582 | 0.705 | 0.938 | 0.611 | 0.677 | 1.174 | 0.601 | 0.825 | 0.801 |

IXIC | 0.641 | 0.707 | 0.764 | 0.585 | 0.644 | 0.941 | 0.615 | 0.702 | 0.781 | 0.563 | 0.671 | 1.425 | 0.587 | 0.680 | 1.134 | 0.591 | 0.624 | 0.901 |

JKSE | 0.598 | 0.848 | 1.352 | 0.539 | 0.674 | 1.335 | 0.582 | 0.725 | 0.877 | 0.560 | 0.566 | 1.802 | 0.500 | 0.907 | 0.881 | 0.570 | 0.518 | 0.769 |

KS11 | 0.607 | 0.707 | 1.190 | 0.539 | 0.421 | 1.140 | 0.540 | 0.637 | 1.195 | 0.590 | 0.535 | 1.026 | 0.526 | 0.616 | 2.180 | 0.530 | 0.633 | 0.945 |

MERV | 0.651 | 0.520 | 0.927 | 0.537 | 0.625 | 1.265 | 0.537 | 0.681 | 1.163 | 0.611 | 0.647 | 0.652 | 0.540 | 0.602 | 1.135 | 0.574 | 0.534 | 0.985 |

MXX | 0.580 | 0.805 | 0.890 | 0.542 | 0.666 | 1.088 | 0.548 | 0.577 | 1.039 | 0.606 | 0.690 | 1.150 | 0.552 | 0.557 | 0.967 | 0.548 | 0.617 | 0.951 |

N225 | 0.584 | 0.472 | 1.041 | 0.573 | 0.745 | 0.714 | 0.550 | 0.639 | 0.901 | 0.614 | 0.505 | 0.732 | 0.553 | 0.530 | 0.804 | 0.539 | 0.406 | 0.559 |

NYA | 0.593 | 0.685 | 0.466 | 0.579 | 0.648 | 0.790 | 0.550 | 0.588 | 0.615 | 0.559 | 0.691 | 0.827 | 0.526 | 0.573 | 0.954 | 0.522 | 0.583 | 0.772 |

TWII | 0.659 | 0.474 | 1.661 | 0.594 | 0.564 | 1.453 | 0.519 | 0.453 | 1.069 | 0.540 | 0.494 | 1.584 | 0.503 | 0.764 | 1.303 | 0.539 | 0.491 | 1.053 |

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

Market | Monday | Tuesday | Wednesday | Thursday | Friday | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\Delta}{\mathit{\alpha}}_{\mathbf{0}}$ | $\mathbf{\Delta}\mathit{W}$ | $\mathbf{\Delta}{\mathit{\alpha}}_{\mathbf{0}}$ | $\mathbf{\Delta}\mathit{W}$ | $\mathbf{\Delta}{\mathit{\alpha}}_{\mathbf{0}}$ | $\mathbf{\Delta}\mathit{W}$ | $\mathbf{\Delta}{\mathit{\alpha}}_{\mathbf{0}}$ | $\mathbf{\Delta}\mathit{W}$ | $\mathbf{\Delta}{\mathit{\alpha}}_{\mathbf{0}}$ | $\mathbf{\Delta}\mathit{W}$ | |

GSPC | 0.049 | 0.115 | 0.030 | 0.019 | 0.021 | 0.094 | 0.008 | 0.058 | 0.014 | 0.126 |

KS11 | 0.033 | 0.010 | 0.034 | 0.219 | 0.028 | 0.052 | 0.012 | 0.138 | 0.044 | 0.052 |

IPSA | 0.073 | 0.200 | 0.022 | 0.119 | 0.028 | 0.069 | 0.023 | 0.064 | 0.051 | 0.098 |

FCHI | 0.064 | 0.035 | 0.023 | 0.078 | 0.031 | 0.054 | 0.066 | 0.033 | 0.002 | 0.025 |

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**MDPI and ACS Style**

Stosic, D.; Stosic, D.; Vodenska, I.; Stanley, H.E.; Stosic, T.
A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect. *Entropy* **2022**, *24*, 562.
https://doi.org/10.3390/e24040562

**AMA Style**

Stosic D, Stosic D, Vodenska I, Stanley HE, Stosic T.
A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect. *Entropy*. 2022; 24(4):562.
https://doi.org/10.3390/e24040562

**Chicago/Turabian Style**

Stosic, Darko, Dusan Stosic, Irena Vodenska, H. Eugene Stanley, and Tatijana Stosic.
2022. "A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect" *Entropy* 24, no. 4: 562.
https://doi.org/10.3390/e24040562