Detrended fluctuation analysis on spot and futures markets of West Texas Intermediate crude oil

In this paper, we study the auto-correlations and cross-correlations of West Texas Intermediate (WTI) crude oil spot and futures return series employing detrended fluctuation analysis (DFA) and detrended cross-correlation analysis (DCCA). Scaling analysis shows that, for time scales smaller than a month, the auto-correlations and cross-correlations are persistent. For time scales larger than a month but smaller than a year, the correlations are anti-persistent, while, for time scales larger than a year, the series are neither auto-correlated nor cross-correlated, indicating the efficient operation of the crude oil markets. Moreover, for small time scales, the degree of short-term cross-correlations is higher than that of auto-correlations. Using the multifractal extension of DFA and DCCA, we find that, for small time scales, the correlations are strongly multifractal, while, for large time scales, the correlations are nearly monofractal. Analyzing the multifractality of shuffled and surrogated series, we find that both long-range correlations and fat-tail distributions make important contributions to the multifractality. Our results have important implications for market efficiency and asset pricing models.     

Multifractal analysis on international crude oil markets based on the multifractal detrended fluctuation analysis

The multifractal nature of WTI and Brent crude oil markets is studied employing the multifractal detrended fluctuation analysis. We find that two crude oil markets become more and more efficient for long-term and two Gulf Wars cannot change time scale behavior of crude oil return series. Considering long-term influence caused by Gulf Wars, we find such “turning windows” in generalized Hurst exponents obtained from three periods divided by two Gulf Wars so that WTI and Brent crude oil returns possess different properties above and below the windows respectively. Comparing with the results obtained from three periods we conclude that, before the First Gulf War, international crude oil markets possessed the highest multifractality degree, small-scope fluctuations presented the strongest persistence and large-scope fluctuations presented the strongest anti-persistence. We find that, for two Gulf Wars, the first one made a greater impact on international oil markets; for two markets, Brent was more influenced by Gulf Wars. In addition, we also verified that the multifractal structures of two markets’ indices are not only mainly attributed to the broad fat-tail distributions and persistence, but also affected by some other factors.     

Asymmetric multifractal scaling behavior in the Chinese stock market: Based on asymmetric MF-DFA

We utilized asymmetric multifractal detrended fluctuation analysis in this study to examine the asymmetric multifractal scaling behavior of Chinese stock markets with uptrends or downtrends. Results show that the multifractality degree of Chinese stock markets with uptrends is stronger than that of Chinese stock markets with downtrends. Correlation asymmetries are more evident in large fluctuations than in small fluctuations. By discussing the source of asymmetric multifractality, we find that multifractality is related to long-range correlations when the market is going up, whereas it is related to fat-tailed distribution when the market is going down. The main source of asymmetric scaling behavior in the Shanghai stock market are long-range correlations, whereas that in the Shenzhen stock market is fat-tailed distribution. An analysis of the time-varying feature of scaling asymmetries shows that the evolution trends of these scaling asymmetries are similar in the two Chinese stock markets. Major financial and economical events may enhance scaling asymmetries.     

Auto-correlated behavior of WTI crude oil volatilities: A multiscale perspective

In this paper, we investigate the long-range auto-correlated behavior of WTI crude oil volatility series employing multifractal detrended fluctuation analysis. Our findings show that the for small time scales, the auto-correlations of volatilities were multifractal while for large time scales, the auto-correlations were nearly monofractal. Based on multiscale analysis, we also investigate the dynamics of auto-correlations for different intervals of time scales and find that several shocks could make significant effects on the auto-correlated behaviors for small time scales. Analyzing the dynamics of multifractality degrees of auto-correlations for small time scales, we find that the stronger auto-correlations were always related to the lower degrees of multifractality. At last, we have discussions on the determination factors of price behavior, the predictive implications of scaling behavior in volatilities for oil markets and the reasons why long-range auto-correlations of volatility were always strong for both small time scales and large time scales. Our results are very important theoretically and practically.     

Multifractal analysis of stock exchange crashes

We analyze the complexity of rare events of the DJIA Index. We reveal that the returns of the time series exhibit strong multifractal properties meaning that temporal correlations play a substantial role. The effect of major stock market crashes can be best illustrated by the comparison of the multifractal spectra of the time series before and after the crash. Aftershock periods compared to foreshock periods exhibit richer and more complex dynamics. Compared to an average crash, calculated by taking into account the larger 5 crashes of the DJIA Index, the 1929 event exhibits significantly more increase in multifractality than the 1987 crisis.     

Multifractal detrended fluctuation analysis of the Chinese stock index futures market

Based on the multifractal detrended fluctuation analysis (MF-DFA) and multifractal spectrum analysis, this paper empirically studies the multifractal properties of the Chinese stock index futures market. Using a total of 2942 ten-minute closing prices, we find that the Chinese stock index futures returns exhibit long-range correlations and multifractality, making the single-scale index insufficient to describe the futures price fluctuations. Further, by comparing the original time series with the transformed time series through shuffling procedure and phase randomization procedure, we show the existence of two different sources of the multifractality for the Chinese stock index futures market. Our results suggest that the multifractality is mainly due to long-range correlations, although the fat-tailed probability distributions also contribute to such multifractal behaviour.     

Are developed and emerging agricultural futures markets multifractal? A comparative perspective

Although there are many reports on the empirical evidence of the existence of multifractality in various financial or commodity markets in current literature, few can be found to compare the multifractal properties of emerging and developed economies, especially for agricultural futures markets in those countries (regions). We therefore chose China as the representative of the transition and emerging economies, and USA as the representative of developed ones. We attempt to find the answers to the following questions: (1) Are all those different markets multifractal? (2) What are the dynamical causes for multifractality in those markets (if any)? (3) Are the multifractality strengths in those markets of the transition and emerging economies weaker (or stronger) than those of the developed ones? To answer these questions, Multifractal Detrended Fluctuation Analysis (MF-DFA) are applied to study some of the representative agricultural futures markets in China and USA, namely, wheat, soy meal, soybean and corn. Our results suggest that all the markets of China and USA exhibit multifractal properties except US soybean market, which is much closer to mono-fractal comparing with China’s soybean market. To investigate the sources of multifractality, shuffling and phase randomization procedures are applied to destroy the temporal correlations and non-Gaussian distributions respectively. We found that multifractality can be mainly attributed to the non-Gaussian probability distribution and secondarily to the nonlinear temporal correlation mechanism for all the markets, except US soybean and soy meal, which derives from some other unknown factors. Furthermore, the average of τ(q) are applied to obtain the multifractal spectra of the two markets as a whole. The results show that the width of the multifractal spectrum of US agricultural futures markets is significantly narrower than that of China’s. Based on our findings, we proposed a hypothesis that the strength of multifractality in developed economies may be weaker than that in emerging and transition ones.     

Are crude oil markets multifractal? Evidence from MF-DFA and MF-SSA perspectives

In this article, we investigated the multifractality and its underlying formation mechanisms in international crude oil markets, namely, Brent and WTI, which are the most important oil pricing benchmarks globally. We attempt to find the answers to the following questions: (1) Are those different markets multifractal? (2) What are the dynamical causes for multifractality in those markets (if any)? To answer these questions, we applied both multifractal detrended fluctuation analysis (MF-DFA) and multifractal singular spectrum analysis (MF-SSA) based on the partition function, two widely used multifractality detecting methods. We found that both markets exhibit multifractal properties by means of these methods. Furthermore, in order to identify the underlying formation mechanisms of multifractal features, we destroyed the underlying nonlinear temporal correlation by shuffling the original time series; thus, we identified that the causes of the multifractality are influenced mainly by a nonlinear temporal correlation mechanism instead of a non-Gaussian distribution. At last, by tracking the evolution of left- and right-half multifractal spectra, we found that the dynamics of the large price fluctuations is significantly different from that of the small ones. Our main contribution is that we not only provided empirical evidence of the existence of multifractality in the markets, but also the sources of multifractality and plausible explanations to current literature; furthermore, we investigated the different dynamical price behaviors influenced by large and small price fluctuations.     

A multifractal approach for stock market inefficiency

In this paper, the multifractality degree in a collection of developed and emerging stock market indices is evaluated. Empirical results suggest that the multifractality degree can be used as a quantifier to characterize the stage of market development of world stock indices. We develop a model to test the relationship between the stage of market development and the multifractality degree and find robust evidence that the relationship is negative, i.e., higher multifractality is associated with a less developed market. Thus, an inefficiency ranking can be derived from multifractal analysis. Finally, a link with previous volatility time series results is established.     

Coupled Criticality Analysis of Inflation and Unemployment

In this paper, we focus on the critical periods in the economy that are characterized by unusual and large fluctuations in macroeconomic indicators, like those measuring inflation and unemployment. We analyze U.S. data for 70 years from 1948 until 2018. To capture their fluctuation essence, we concentrate on the non-Gaussianity of their distributions. We investigate how the non-Gaussianity of these variables affects the coupling structure of them. We distinguish “regular” from “rare” events, in calculating the correlation coefficient, emphasizing that both cases might lead to a different response of the economy. Through the “multifractal random wall” model, one can see that the non-Gaussianity depends on time scales. The non-Gaussianity of unemployment is noticeable only for periods shorter than one year; for longer periods, the fluctuation distribution tends to a Gaussian behavior. In contrast, the non-Gaussianities of inflation fluctuations persist for all time scales. We observe through the “bivariate multifractal random walk” that despite the inflation features, the non-Gaussianity of the coupled structure is finite for scales less than one year, drops for periods larger than one year, and becomes small for scales greater than two years. This means that the footprint of the monetary policies intentionally influencing the inflation and unemployment couple is observed only for time horizons smaller than two years. Finally, to improve some understanding of the effect of rare events, we calculate high moments of the variables’ increments for various q orders and various time scales. The results show that coupling with high moments sharply increases during crises.     

Multifractal detrended fluctuation analysis of derivative and spot markets

We investigate the multifractal properties of price increments in the cases of derivative and spot markets. Through the multifractal detrended fluctuation analysis, we estimate the generalized Hurst and the Renyi exponents for price fluctuations. By deriving the singularity spectrum from the above exponents, we quantify the multifractality of a financial time series and compare the multifractal properties of two different markets. The different behavior of each agent-group in transactions is also discussed. In order to identify the nature of the underlying multifractality, we apply the method of surrogate data to both sets of financial data. It is shown that multifractality due to a fat-tailed distribution is significant.     

Research on the relationship between the multifractality and long memory of realized volatility in the SSECI

We examine the multifractal properties of the realized volatility (RV) and realized bipower variation (RBV) series in the Shanghai Stock Exchange Composite Index (SSECI) by using the multifractal detrended fluctuation analysis (MF-DFA) method. We find that there exist distinct multifractal characteristics in the volatility series. The contributions of two different types of source of multifractality, namely, fat-tailed probability distributions and nonlinear temporal correlations, are studied. By using the unit root test, we also find the strength of the multifractality of the volatility time series is insensitive to the sampling frequency but that the long memory of these series is sensitive.     

Multifractal properties of the Indian financial market

We investigate the multifractal properties of the logarithmic returns of the Indian financial indices (BSE & NSE) by applying the multifractal detrended fluctuation analysis. The results are compared with that of the US S&P 500 index. Numerically we find that qth-order generalized Hurst exponents h(q) and τ(q) change with the moments q. The nonlinear dependence of these scaling exponents and the singularity spectrum f(α) show that the returns possess multifractality. By comparing the MF-DFA results of the original series to those for the shuffled series, we find that the multifractality is due to the contributions of long-range correlations as well as the broad probability density function. The financial markets studied here are compared with the Binomial Multifractal Model (BMFM) and have a smaller multifractal strength than the BMFM.     

Forecasting volatility in Shanghai and Shenzhen markets based on multifractal analysis

This paper analyzes the multifractality in Shanghai and Shenzhen stock markets using multifractal spectrum analysis and multifractal detrended fluctuation analysis. We find that the main source of multifractality is long-range correlations of large and small fluctuations. Then, we introduce a multifractal volatility measure (MV) and find that by taking MV as daily conditional volatility, the simulated series displayed similar “stylized facts” to the original daily return series. By capturing the dynamics of MV using the ARFIMA model, we find that the out-of-sample forecasting performance of the ARFIMA-MV model is better than some GARCH-class models and the ARFIMA-RV model under some criteria of loss function.     

Size selected growth of nanodots: analytical prediction for the selected size

We analyze the multifractal spectra of daily foreign exchange rates for Japan, Hong-Kong, Korea, and Thailand with respect to the United States in the period from 1991 until 2005. We find that the return time series show multifractal spectrum features for all four cases. To observe the effect of the Asian currency crisis, we also estimate the multifractal spectra of limited series before and after the crisis. We find that the Korean and Thai foreign exchange markets experienced a significant increase in multifractality compared to Hong-Kong and Japan. We also show that the multifractality is stronger related to the presence of high values of returns in the series.     

Multifractal detrending moving average analysis on the US Dollar exchange rates

In this paper, we investigate the multifractal behavior of the US dollar (USD) exchange rates. The results from the multifractal detrending moving average algorithm show that twelve exchange rate series were multifractal. The major source of multifractality are long-range correlations of small and large fluctuations. Fat-tail distributions have important effects on the multifractality of USD/AUR, USD/EUR and CNY/USD exchange rates. We also find evidence that extreme events play an important role in the contributions to multifractality for the USD/EUR exchange rate.     

Multifractal detrended cross-correlations between the Chinese exchange market and stock market

Based on the daily price data of the Chinese Yuan (RMB)/US dollar exchange rate and the Shanghai Stock Composite Index, we conducted an empirical analysis of the cross-correlations between the Chinese exchange market and stock market using the multifractal cross-correlation analysis method. The results demonstrate the overall significance of the cross-correlation based on the analysis of a statistic. Multifractality exists in cross-correlations, and the cross-correlated behavior of small fluctuations is more persistent than that of large fluctuations. Moreover, using the rolling windows method, we find that the cross-correlations between the Chinese exchange market and stock market vary with time and are especially sensitive to the reform of the RMB exchange rate regime. The previous reduction in the flexibility of the RMB exchange rate in July 2008 strengthened the persistence of cross-correlations and decreased the degree of multifractality, whereas the enhancement of the flexibility of the RMB exchange rate in June 2010 weakened the persistence of cross-correlations and increased the multifractality. Finally, several relevant discussions are provided to verify the robustness of our empirical analysis.     

Statistical properties of the yuan exchange rate index

We choice the yuan exchange rate index based on a basket of currencies as the effective exchange rate of the yuan and investigate the statistical properties of the yuan exchange rate index after China’s exchange rate system reform on the 21st July 2005. After dividing the time series into two parts according to the change in the yuan exchange rate regime in July 2008, we compare the statistical properties of the yuan exchange rate index during these two periods. We find that the distribution of the two return series has the exponential form. We also perform the detrending moving average analysis (DMA) and the multifractal detrending moving average analysis (MFDMA). The two periods possess different degrees of long-range correlations, and the multifractal nature is also unveiled in these two time series. Significant difference is found in the scaling exponents τ(q)$\tau \left(q\right)$ and singularity spectra f(α)$f\left(\alpha \right)$ of the two periods obtained from the MFDMA analysis. Besides, in order to detect the sources of multifractality, shuffling and phase randomization procedures are applied to destroy the long-range temporal correlation and fat-tailed distribution of the yuan exchange rate index respectively. We find that the fat-tailedness plays a critical role in the sources of multifractality in the first period, while the long memory is the major cause in the second period. The results suggest that the change in China’s exchange rate regime in July 2008 gives rise to the different multifractal properties of the yuan exchange rate index in these two periods, and thus has an effect on the effective exchange rate of the yuan after the exchange rate reform on the 21st July 2005.     

Dynamics of bid–ask spread return and volatility of the Chinese stock market

The bid–ask spread is taken as an important measure of the financial market liquidity. In this article, we study the dynamics of the spread return and the spread volatility of four liquid stocks in the Chinese stock market, including the memory effect and the multifractal nature. By investigating the autocorrelation function and the Detrended Fluctuation Analysis (DFA), we find that the spread return is the lack of long-range memory, while the spread volatility is long-range time correlated. Besides, the spread volatilities of different stocks present long-range cross-correlations. Moreover, by applying the Multifractal Detrended Fluctuation Analysis (MF-DFA), the spread return is observed to possess a strong multifractality, which is similar to the dynamics of a variety of financial quantities. Different from the spread return, the spread volatility exhibits a weak multifractal nature.     

Multifractal detrended cross-correlation analysis between the Chinese stock market and surrounding stock markets

In this paper, we investigate the cross-correlations between the stock market in China and markets in Japan, South Korea and Hong Kong. We use not only the qualitative analysis of the cross-correlation test, but also the quantitative analysis of the MF-X-DFA. Our findings confirm the existence of cross-correlations between the stock market in China and markets in Japan, South Korea and Hong Kong, which have strongly multifractal features. We find that the cross-correlations display the characteristic of multifractality in the short term. Moreover, the cross-correlations of small fluctuations are persistent and those of large fluctuations are anti-persistent in the short term, while the cross-correlations of all kinds of fluctuations are persistent in the long term. Furthermore, based on the multifractal spectrum, we also find that the multifractality of cross-correlation between stock markets in China and Japan are stronger than those between China and South Korea, as well as between China and Hong Kong.