**1. Introduction**

Market prices should incorporate and reflect all available information at any point in time, according to the Efficient Market Hypothesis (EMH) [1,2]. Yet, various studies [3–6] show that financial markets often become inefficient, and their behavior no longer follows that of a random walk. Stock markets can instead deviate from the rules of the EMH in the form of anomalies. Anomalies can be broadly categorized into calendar, fundamental and technical anomalies [7]. The most studied set of pricing anomalies is calendar or seasonal anomalies that represent systematic patterns of security returns around certain calendar points. Calendar anomalies include the day-of-the-week effect [8–11], turn-ofthe-month effect [12–15], turn-of-the-year effect [16–19] and holiday effect [20–23]. The day-of-the-week effect refers to the tendency of stocks to exhibit significantly higher returns on one particular day compared with other days of the week. Cross [24] first provided evidence of day-of-the-week effects on the Standard and Poor's index, reporting that price returns are significantly negative on Mondays. Since then, this phenomenon has been extensively studied and discovered in other financial markets such as specific equity markets [25–27], exchange rates [28,29], fixed-income securities [30], crude oil [31], gold [32] and cryptocurrencies [33]. For a detailed review of seasonal anomalies, please see [34,35].

**Citation:** 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

Academic Editor: Stanisław Drozd˙ z˙

Received: 6 January 2022 Accepted: 13 April 2022 Published: 17 April 2022

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Financial markets have attracted much attention from researchers in related fields such as econophysics, paving the road for new perspectives and understanding of financial markets by drawing concepts from statistical physics such as fractals and multifractals [36–39], information theory [40,41] and network structures [42–44] (see [45] and the references therein for a comprehensive review). While many well-known conclusions in the literature on an array of financial markets (including market indices, stocks, exchange rates and commodities) can be attributed to econophysics, there are still a number of important phenomena to be investigated from this perspective. To the best of our knowledge, one such phenomenon that remains to be unearthed is the calendar anomaly, and our study makes a contribution in this direction.

In this paper, we use multifractal analysis to evaluate if the temporal dynamics of market returns exhibit calendar anomalies such as day-of-the-week effects. We apply multifractal detrended fluctuation analysis (MF-DFA) [46] to the daily returns of market indices around the world for each day of the week (Monday returns, Tuesday returns and so on). We then compare the multifractal parameters, the position of maximum width and asymmetry of the multifractal spectrum, which quantify long-term correlations, the degree of multifractality and the dominance of large or small fluctuations in the return series for each day of the week. The economic literature states that market practitioners have been aware of the Monday effect as early as the 1920s [47]. For some markets, this effect disappears as the market becomes more efficient [48,49]. Other studies offer insight into the Monday effect being more prominent toward the end of the month [50] and during periods dominated by bad news [51]. To observe this behavior over time, we perform time-dependent multifractal analysis on the United States (GSPC) market by calculating the multifractal spectra of the return series in a sliding window. This computationally intensive and relatively novel approach, which has been implemented in only a few studies [52–54], permits us to analyze the temporal evolution of multifractal parameters which are related to different properties of market fluctuation, leading to better understanding of the underlying stochastic processes. The rest of this paper is organized as follows. Section 2 introduces the MF-DFA and the time-dependent methods. Section 3 describes the market data. Section 4 presents the results, and Section 5 draws the conclusion.
