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The main goal of this paper is to investigate whether there is long-memory behavior in the CBOE Brazil ETF volatility index (named here VIXBR). As structural breaks may create a spurious long-range dependence, the presence of structural breaks is also gauged.

The study considers the period from October 2011 to March 2021, using daily data. To test the long-memory behavior, three empirical approaches are adopted: GPH, ELW and robust GPH (RGPH) estimator. To estimate the structural break points adopted to date the subsamples, the ICSS algorithm is used.

Results considering the total period (TP) and subsamples show that the breaks did not create a spurious long-memory behavior and together with the rolling estimation, reveal strong evidence of the long-range dependence in the CBOE Brazil ETF volatility index. The higher degree of persistent of the VIXBR series suggests an extended period of increased uncertainty that agents need consider when making their investment decision.

As possible extension of this study is to investigate the behavior of long memory and structural breaks for different frequencies (weekly, monthly, among others).

The presence of long-range dependence in the CBOE Brazil ETF volatility index reveals that the past information is important for the predictability of risks, and therefore, can help to protect against market risks, which has important implications regarding the future decisions of economic agents (for example, policy makers and investors).

Brazil is an emerging capital market (ECM) that has attracted a great deal of attention from investors and investment funds seeking to diversify its assets. This paper contributes to the empirical financial literature, by studying the long-memory behavior of the CBOE Brazil ETF volatility index, considering possible structural breaks. To the best of knowledge, this has not been done so far.

This paper seeks to characterize the behavior of the naira/dollar foreign exchange rate series over the period 1999 through 2006 to determine if the process generating the series has long memory which is a special case of fractional Brownian motion. The existence of long memory contradicts the notion of market efficiency.

The paper employs the modified rescaled range R/S test which is proposed by Lo to test the null hypothesis that daily and weekly NGN/USD exchange rates from 1999 through 2006 exhibit short‐memory process. The second test that was also employed is the Geweke‐Porter‐Hubak (GPH) test which was refined by Hurvich et al.

The results show that long memory is present in daily and weekly foreign exchange level series of the Nigerian naira for the period sampled. This evidence implies that the Nigerian foreign exchange market may not be efficient. Thus, it is possible for investors to realize abnormal profit by taking an investment position based on predicted exchange rates. The results reported in this paper are also indicative of a deviation from long‐run PPP.

This paper is the first to empirically apply the modified R/S and GPH tests to explore the existence of long‐memory process in a country study of foreign exchange series using data from Nigeria.

The purpose of this paper is to consider that the model of volatility characteristics is more reasonable and the description of volatility is more explanatory.

This paper analyzes the basic characteristics of market yield volatility based on the five-minute trading data of the Chinese CSI300 stock index futures from 2012 to 2017 by Hurst index and GPH test, A-J and J-O Jumping test and Realized-EGARCH model, respectively. The results show that the yield fluctuation rate of CSI300 stock index futures market has obvious non-linear characteristics including long memory, jumpy and asymmetry.

This paper finds that the LHAR-RV-CJ model has a better prediction effect on the volatility of CSI300 stock index futures. The research shows that CSI300 stock index futures market is heterogeneous, means that long-term investors are focused on long-term market fluctuations rather than short-term fluctuations; the influence of the short-term jumping component on the market volatility is limited, and the long jump has a greater negative influence on market fluctuation; the negative impact of long-period yield is limited to short-term market fluctuation, while, with the period extending, the negative influence of long-period impact is gradually increased.

This paper has research limitations in variable measurement and data selection.

This study is based on the high-frequency data or the application number of financial modeling analysis, especially in the study of asset price volatility. It makes full use of all kinds of information contained in high-frequency data, compared to low-frequency data such as day, weekly or monthly data. High-frequency data can be more accurate, better guide financial asset pricing and risk management, and result in effective configuration.

The existing research on the futures market volatility of high frequency data, mainly focus on single feature analysis, and the comprehensive comparative analysis on the volatility characteristics of study is less, at the same time in setting up the model for the forecast of volatility, based on the model research on the basic characteristics is less, so the construction of a model is relatively subjective, in this paper, considering the fluctuation characteristics of the model is more reasonable, characterization of volatility will also be more explanatory power. The difference between this paper and the existing literature lies in that this paper establishes a prediction model based on the basic characteristics of market return volatility, and conducts a description and prediction study on volatility.

The purpose of this paper is to focus on whether the deviations from the cointegrating relationship possess long memory and the fractional cointegration analyses may capture a wider range of mean-reversion behaviour than standard cointegration analyses.

This paper uses a fractional cointegration technique to test the purchasing power parity (PPP).

The authors found that PPP held, but very weakly, in the long run between the Argentine, Brazil, Chile, Colombia, Indonesia, Korea, Mexico, Thailand and Venezuela and US exchange rate during our floating exchange rate period but that the deviations from it did not follow a stationary process. Nevertheless, it is also found that the deviations from PPP exists and can be characterized by a fractionally integrated process in nine out of 13 countries studied.

The findings are consistent with the consensus of the empirical literature, reviewed earlier in this paper, on PPP between Argentine, Brazil, Chile, Colombia, Indonesia, Korea, Mexico, Thailand and Venezuela and the USA.

This paper, using Turkish stock index data, set outs to present long‐term memory effect using chaotic and conventional unit root tests and investigate if chaotic technique as wavelets captures long‐memory better than conventional techniques.

Haar and Daubechies as wavelet‐based OLS estimator and GPH and other classical models are applied in order to investigate the performance of long memory in the time series.

The results indicate that Daubechies wavelet analysis provide the accurate determination for long memory where conventional techniques does not.

The research results have both methodological and practical originality. On the theoretical side, the wavelet‐based OLS estimator is superior in modeling the behaviours of the stock returns in emerging markets where non‐linearities and high volatility exist due to their chaotic natures. For practical aims, on the other hand, the results show that the Istanbul Stock Exchange is not in the weak‐form efficient because the prices have memories that are not reflected in the prices, yet.

Reviews previous research on the nature of beta and investigates the stochastic structure of time‐varying beta in Hong Kong, Malaysia and Singapore using the bi‐variate GARCH‐in‐mean model and fractional tests. Develops mathematical models and applies them to 1989‐1998 daily data from all three stock markets. Presents the results, which suggest, in contrast to other findings, that all three time‐varying betas are slowly mean‐reverting (long memory).

The purpose of this paper is to examine the long memory property of equity returns and volatility of emerging equity market by focusing on the Malaysian equity market, namely the Kuala Lumpur Stock Exchange (KLSE).

The study adopts the Fractionally Integrated GARCH (FIGARCH) model and Fractionally Integrated Asymmetric Power ARCH (FIAPARCH), focusing on the Malaysian data covering the period from April 15, 2004 to April 30, 2007.

The study finds evidence of long memory property as well as asymmetric effects in the volatility of the KLSE. The traditional ARCH/GARCH is shown to be insufficient in modeling the volatility persistence. The FIAPARCH specification outperforms the FIGARCH model by capturing both asymmetry effects and long memory in the conditional variance.

The results of this study have practical implications for the investors intending to invest in the emerging markets such as Malaysia. Understanding volatility and developing the appropriate models are important since volatility can be a measure of risk which is highly relevant in forecasting the conditional volatility of returns for portfolio selection, asset pricing, and value at risk, option pricing and hedging strategies.

This study contributes in providing the empirical evidence on the long memory property of equity returns and volatility of an emerging equity market with reliable estimation models, which is currently lacking, particularly for emerging markets.

The purpose of this paper is to explore whether stock index volatility series exhibit real long memory.

The authors employ sequential procedure to test structural break in volatility series, and use DFA and 2ELW to estimate long memory parameter for the whole samples and subsamples, and further apply adaptive FIGARCH (AFIGARCH) to describe long memory and structural break.

The empirical results show that stock index volatility series are characterized by long memory and structural break, and therefore it is appropriate to use AFIGARCH to model stock index volatility process.

This study empirically investigates the properties of long memory and structural break in stock index volatility series. The conclusion has a certain reference value for understanding the properties of long memory and structural break in volatility series for academic researchers, market participants and policy makers, and for modeling and forecasting future volatility, testing market efficiency, pricing financial assets, constructing quantitative investment strategy and measuring market risk.

The purpose of this research is to analyze the Bitcoin (BTC) and Ether (ETH) long memory and conditional volatility.

The empirical approach includes ARFIMA-HYGARCH and ARFIMA-FIGARCH, both models under Student‘s t-distribution, during the period (ETH: November 9, 2017 to November 25, 2021 and BTC: September 17, 2014 to November 25, 2021).

Findings suggest that ARFIMA-HYGARCH is the best model to analyze BTC volatility, and ARFIMA-FIGARCH is the best approach to model ETH volatility. Empirical evidence also confirms the existence of long memory on returns and on BTC volatility parameters. Results evidence that the models proposed are not as suitable for modeling ETH volatility as they are for the BTC.

Findings allow to confirm the fractal market hypothesis in BTC market. The data confirm that, despite the impact of the Covid-19 crisis, the dynamics of BTC returns, and volatility maintained their patterns, i.e. the way in which they evolve, in relation to the prepandemic era, did not change, but it is rather reaffirmed. Yet, ETH conditional volatility was more affected, as it is apparently higher during Covid-19. The originality of the research lies in the focus of the analysis, the proposed methodology and the variables and periods of study.