This paper represents a pioneering effort to investigate multifractal dynamics that exclusively encompass the return time series of USA Education Group Stocks concerning two non-overlapping periods (before, during, and after COVID-19). Given this, we employ the Multifractal Detrended Fluctuations Analysis (MF-DFA). In this sense, we investigate the generalized Hurst exponent ℎ(𝑞) and the Rényi exponent 𝜏 (𝑞) for each asset and quantify their statistical properties, which allowed us to observe separately the contributing small scale (primarily via the negative moments 𝑞) and the large scale (via the positive moments 𝑞). We perform a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process. Also, we shall apply the inefficiency multifractal index to assess the COVID-19 shock for both periods. Our findings show that for both periods, the majority of these assets are marked by multifractal dynamics associated with persistent behaviour (𝛼 0 > 0.5), a higher degree of multifractality and the dominance of large fluctuations. At the same time, most of these assets show asymmetry parameter (𝑅 > 1) for both periods, indicating that large fluctuations contributed more to multifractality in the time series of returns.

 This paper promotes an exchange of scientific information considering two cornerstones for modern economic-financial theory: the Efficient Market Hypothesis in weak form (EMH), Asymmetric Information and Multifrac-tal Theory. We employ the Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) to check the effects caused by COVID-19 in one of the most relevant economic sectors to the Economic Growth theory, the Education sector. Therefore, our analysis considers two non-overlapping periods (before, during and after COVID-19) to investigate the intricate multifractal dynamics between volatility time series pairs of these assets fixed the EDU asset. For both periods, our findings categorically rejected the Efficient Market Hypothesis in Fama's sense and ratified the Fractal Market Hypothesis. Also, analyzing the asymmetry multifractal parameter validates the Asymmetric Information in Stiglitz's sense for both periods. Our empirical evidence clarifies the complex dynamics of peculiar USA education group stocks and provides a better understanding of the most diverse stakeholders.

This paper sheds light on the changes suffered in cryptocurrencies due to the COVID-19 shock through a non-linear cross-correlations and similarity perspective. We have collected daily price and volume data for the seven largest cryptocurrencies considering trade volume and market capitalization. For both attributes (price and volume), we calculate their volatility and compute the Multifractal Detrended Cross-Correlations (MF-DCCA) to estimate the complexity parameters that describe the degree of multifractality of the underlying process. We detect (before and during COVID-19) a standard multifractal behaviour for these volatility time series pairs and an overall persistent long-term correlation. However, multifractality for price volatility time series pairs displays more persistent behaviour than the volume volatility time series pairs. From a financial perspective, it reveals that the volatility time series pairs for the price are marked by an increase in the non-linear cross-correlations excluding the pair Bitcoin vs Dogecoin (í µí»¼ í µí±¥í µí±¦ (0) = −1.14%). At the same time, all volatility time series pairs considering the volume attribute are marked by a decrease in the non-linear cross-correlations. The K-means technique indicates that these volatility time series for the price attribute were resilient to the shock of COVID-19. While for these volatility time series for the volume attribute, we find that the COVID-19 shock drove changes in cryptocurrency groups.

This paper disrupts mistaken applications of multifractal approaches in financial time series. Specifically, we have examined the non-linear cross-correlation between the São Paulo time series of the weekly price of ethanol and the other 14 Brazilian capitals' time series of the weekly price of the same biofuel using the Multifractal Detrended Cross-Correlation Analysis (MF-DCCA). Given the statistical peculiars of stationary and non-stationary financial time series, we suggest two possibilities for employing multifractal approaches to these time series. Our findings shed light and promote alignment between basic time series analysis techniques and multifractal dynamics. Also, we discover that the use of MF-DCCA is highly impacted by choice of time series (stationary or non-stationary).

This letter has investigated the exogenous shocks (climate change and the Russia-Ukraine war) in the top ten digital currencies' price dynamics. Therefore, we use the information theory quantifiers. We discover that the exogenous shocks have an ambiguous effect on the informational efficiency of these digital currencies. Our results shed light on the potential of altcoins to support exogenous shocks and their capability to use with portfolio selection, risk diversification and herding behaviour.

We analyze multifractality for green bonds, stock sector indices, and US economic sector bonds. Green bonds and US bonds show non-linear cross-correlations. We perform Mul-tifractal Detrended Cross-Correlations Analysis (MF-DCCA) to analyze multifractal cross-correlations and the weak version of the Efficient Market Hypothesis (EMH). Our findings are relevant to academics, financial professionals and the general public. Although green bonds are bonds used exclusively to finance sustainable investments, they are still inefficient assets. We find that bond indices for consumer staples and equity indices for information technology and the real state sector can be used to hedge investments in green bonds.

We examine the price disorder and market efficiency of five cryptocurrencies (Bitcoin, BNB, Cardano, Ethereum, and XRP) before and during COVID-19 pandemic period. Using permutation entropy and Fisher information measure (FIM), we construct the Shannon-Fisher causality plane (SFCP) to map these cryptocurrencies and their respective locations in a two-dimensional plane and then apply sliding time window approach to study the temporal evolution of efficiency. All cryptocurrencies exhibit high but slightly varying informational efficiency during both periods. Cardano is the most efficient. These results might point to the increasing maturity and lower potential for price predictability, which matter to cryp-tocurrencies usage for liquidity risk diversification strategy.

This paper provides an overview of the commodities market, considering three relevant attributes: predictability, similarity, and efficiency. We examine the monthly spot and futures prices time series for 22 commodities from January 1984 until January 2022 with 457 observations. We estimate the permutation entropy (í µí°¸íµí°¸í µí±) and Fisher information measure í µí°¹ í µí± using the Bandt & Pompe method (BPM). We employ the value of these two complexity measures to construct the Shannon-Fisher Causality Plane (SFCP), which allows us to evaluate the disorder and assess the randomness present in the monthly spot and futures prices time series for these commodities. Moreover, we apply í µí°¸íµí°¸í µí± and í µí°¹ í µí± to classify the commodities using complexity hierarchy. We find that the commodities that are located farther from the random ideal position (í µí°¸íµí°¸í µí± = 1, í µí°¹ í µí± = 0) in the SFCP, such as Natural gas, Europe; Iron ore, cfr spot, and Potassium chloride are marked by lower entropy, higher predictability and lower efficiency. In contrast, the commodities that are located near the random ideal position (í µí°¸íµí°¸í µí± = 1, í µí°¹ í µí± = 0) in the SFCP, such as Crude oil-Brent; Crude oil-average, and Silver are characterized by higher entropy, lower predictability and higher efficiency. The K-means algorithm and the hierarchical cluster grouped commodities into only three distinct groups, which is a strong indication that commodity prices have very similar behaviour.

This research explores the multifractal dynamics of the daily number of vaccinated time series for COVID-19, considering six European countries (Belgium, Denmark, France, Ger-many, Greece, and Italy) using the Multifractal Detrended Fluctuations Analysis (MF-DFA). We calculate the multifractal spectrum í µí±“ (í µí»¼) and apply a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process. We find that the multifractal dynamics of all these countries are characterized by strongly anti-persistent behaviour or anti-persistent long-term correlations (í µí»¼ 0 < 0.5) a lower degree of multifractality, and small fluctuations are dominant in the mul-tifractal spectrum. Our findings shed light on the period of immunization of the population that adhered to the vaccination campaigns is short and that the application of new doses of vaccines must comply with this phenomenology to keep the people safe.