Narrow Results

This paper aims to analyze whether the financial crises of the past 20 years have reduced efficiency, in its weak form, in 19 stock markets belonging to the 20 most developed economies (G-20). The sample period comprises the period from 2 January 2000 to 5 February 2021 with the respective financial crises, namely, Dot-com, Argentina, Subprime, Sovereign debt, China stock market crash (2015–2016), UK’s withdrawal from the European Union and the global pandemic of 2020. The results highlight that most markets show signs of (in)efficiency in each sliding window (1000 days), that is, they show asymmetries and non-Gaussian distributions, and ${\alpha }_{\mathrm{\text{DFA}}}\ne 0.5$. These findings suggest that the random walk hypothesis is rejected in certain markets, which has implications for investors, since some returns may be expected, creating arbitrage and abnormal profit opportunities, contrary to the random walk and informational efficiency hypotheses. The results found also open room for market regulators to take steps to ensure better informational data across international financial markets.

It was conjectured by Černý in 1964, that a synchronizing DFA on $n$ states always has a synchronizing word of length at most ${(n-1)}^2$, and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for $n\le 5$, and with bounds on the number of symbols for $n\le 12$. Here we give the full analysis for $n\le 7$, without bounds on the number of symbols.

For PFAs (partial automata) on $\le 7$ states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding ${(n-1)}^2$ for $n\ge 4$. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on $\le 10$ states and two symbols we investigate all occurring synchronizing word lengths.

We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of $n$. For $n=6,7,8,9$, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than ${(n-1)}^2$, although still quadratic.

Based on string rewriting, for arbitrary size we construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation. Both PFAs are transitive.

Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.

Based on the daily return and volatility series of the Chinese yuan (RMB)/US dollar (USD) exchange rate and the Shanghai Stock Composite Index, the time-varying long memories of the Chinese currency and stock markets are investigated by comprehensively using the rescaled range (R/S), the modified R/S, and the detrended fluctuation analysis methods. According to the results drawn: (1) the efficiency of the Chinese currency market has not improved significantly, whereas the efficiency of the Chinese stock market has improved steadily, (2) volatility series presents longer memory than return series either in the Chinese currency or stock market and (3) the time-varying Hurst exponent of the Chinese currency market is sensitive to the reform that enhances the flexibility of the RMB exchange rate. Moreover, we find that short-term bidirectional Granger causal relationship exists, but no long-run equilibrium relationship between the time-varying Hurst exponents of the Chinese currency and stock markets was found based on the Granger causality and cointegration tests, respectively.

Applying the statistical hypothesis testing, we investigate several nonlinear properties embedded in the return series of the Chinese Fund Market (CFM), which suggests the series is non-normal, auto-correlative and heteroskedastic. We hereby analyze the Hurst exponent of the return series in different timescales on the basis of the detrended fluctuation analysis (DFA) algorithm, and discuss the fractal behavior of the CFM. Furthermore, by studying the correlation of different weights in the volatility, we find the persistent long-range power-law correlation exists in the time series. We also provide hints that the above statistical properties are insensitive to the funds kind, and may be irrelevant to the market phases. Our work may reveal the self-similarity characteristics of the financial market and show a better understanding of the CFM.

Two formal languages are f-equivalent if their symmetric difference L₁ Δ L₂ is a finite set — that is, if they differ on only finitely many words. The study of f-equivalent languages, and particularly the DFAs that accept them, was recently introduced [3]. First, we restate the fundamental results in this new area of research. Second, we introduce our main result, which is a faster algorithm for the natural minimization problem: given a starting DFA D, find the smallest (by number of states) DFA D′ such that L(D) and L(D′) are f-equivalent. Finally, we suggest a technique that combines this hyper-minimization with the well-studied notion of a deterministic finite cover automaton [2, 4, 5], or DFCA, thereby extending the applicability of DFCAs from finite to infinite regular languages.

Analyzing the statistical features of fluctuation is remarkably significant for financial risk identification and measurement. In this study, the asymmetric detrended fluctuation analysis (A-DFA) method was applied to evaluate asymmetric multifractal scaling behaviors in the Shanghai and New York gold markets. Our findings showed that the multifractal features of the Chinese and international gold spot markets were asymmetric. The gold return series persisted longer in an increasing trend than in a decreasing trend. Moreover, the asymmetric degree of multifractals in the Chinese and international gold markets decreased with the increase in fluctuation range. In addition, the empirical analysis using sliding window technology indicated that multifractal asymmetry in the Chinese and international gold markets was characterized by its time-varying feature. However, the Shanghai and international gold markets basically shared a similar asymmetric degree evolution pattern. The American subprime mortgage crisis (2008) and the European debt crisis (2010) enhanced the asymmetric degree of the multifractal features of the Chinese and international gold markets. Furthermore, we also make statistical tests for the results of multifractatity and asymmetry, and discuss the origin of them. Finally, results of the empirical analysis using the threshold autoregressive conditional heteroskedasticity (TARCH) and exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models exhibited that good news had a more significant effect on the cyclical fluctuation of the gold market than bad news. Moreover, good news exerted a more significant effect on the Chinese gold market than on the international gold market.

In this paper, the complexity-entropy causality plane approach is applied to analyze traffic data. The R/S analysis and detrended fluctuation analysis (DFA) methods are also used to compare with this approach. Moreover, based on the concept of entropy, we propose to use permutation to calculate the probability distribution of the time series when applying the representation plane. The empirical results indicate that traffic dynamics exhibit different levels of traffic congestion and demonstrate that this statistical method can give a more refined classification of traffic states than the R/S analysis and DFA.

This chapter uses four alternative ways to evaluate mutual fund performance and rank the domestic equity funds of three leading mutual fund companies: DFA, Fidelity, and Vanguard, while splitting the Fidelity and Vanguard funds into indexed and managed portfolios.

First, we use the Fama–French factors to show the extent to which various equally-weighted fund portfolios out-returned their empirically determined benchmarks, and characterize the funds according to size and book-to-market value. Our measure tweaks alpha, the extent to which a mutual fund outperforms its benchmark, and we offer a novel interpretation of alpha. Second, we use Sharpe’s (1962) style analysis to explore the robustness of our analysis. Third, we ask whether funds out-return the market. Fourth, we compare the fund returns with their published benchmarks and use Fama–French factors to adjust for bias in the benchmarks.

Return relative to the stock market reflects style choice, benchmark choice, fees, transaction efficiency, and income from lending to short sellers. Fama–French and Sharpe style adjustment abstract from style choice. Comparison with style-adjusted published benchmarks further abstracts from benchmark choice. Each calculation answers a different question.

The Fama–French and Sharpe methods, recorded in Table 6, show that after style adjustment for Fidelity 2001–2018 the indexed portfolio out-returned the managed portfolio. The indexed and managed portfolios from Vanguard and the managed portfolio from Fidelity 2001–2019 out-returned the U.S. stock market as did the Fidelity indexed portfolio 2005–2018. In comparison with the style-adjusted published benchmarks, the Vanguard managed portfolio is the best and the Fidelity managed portfolio is the worst.