The effect of round-off error on long memory processes

We study how the round-off (or discretization) error changes the statistical properties of a Gaussian long memory process. We show that the autocovariance and the spectral density of the discretized process are asymptotically rescaled by a factor smaller than one, and we compute exactly this scaling factor. Consequently, we find that the discretized process is also long memory with the same Hurst exponent as the original process. We consider the properties of two estimators of the Hurst exponent, namely the local Whittle (LW) estimator and the Detrended Fluctuation Analysis (DFA). By using analytical considerations and numerical simulations we show that, in presence of round-off error, both estimators are severely negatively biased in finite samples. Under regularity conditions we prove that the LW estimator applied to discretized processes is consistent and asymptotically normal. Moreover, we compute the asymptotic properties of the DFA for a generic (i.e. non Gaussian) long memory process and we apply the result to discretized processes.Comment: 44 pages, 4 figures, 4 table

How are rescaled range analyses affected by different memory and distributional properties? A Monte Carlo study

In this paper, we present the results of Monte Carlo simulations for two popular techniques of long-range correlations detection - classical and modified rescaled range analyses. A focus is put on an effect of different distributional properties on an ability of the methods to efficiently distinguish between short and long-term memory. To do so, we analyze the behavior of the estimators for independent, short-range dependent, and long-range dependent processes with innovations from 8 different distributions. We find that apart from a combination of very high levels of kurtosis and skewness, both estimators are quite robust to distributional properties. Importantly, we show that R/S is biased upwards (yet not strongly) for short-range dependent processes, while M-R/S is strongly biased downwards for long-range dependent processes regardless of the distribution of innovations.Comment: 15 pages, 6 table

Behavior of realized volatility and correlation in exchange markets

We study time-varying realized volatility and related correlation measures as proxies for the true volatility and correlation. We investigate measures of Two-Scale realized Absolute Volatility (TSAV) and correlation (TSACORxy) which are helpful to cope effectively with the problem of market microstructure effects at very high frequency financial time series. The measures are constructed based on subsampling and averaging method so that they possess rather less bias even in presence of market microstructure noise. Absolute transformation of return values has been proved in literature to be more robust than squared transformation when considering large values. With respect to some stylized facts of markets, realized squared correlation does not display dynamic behavior. Motivated by robustness of realized absolute volatility, we study an alternative measure of correlation, built on absolute-transformed volatility. This measure of correlation exhibits experimentally some dynamics and hence some predictability capability on minute-by-minute frequency exchange market data. We show that the distribution of realized correlation series computed based on TSACORxy tends to comply a rightward asymmetric shape implying that upside co-movements are greater than downside ones. Moreover we study the association between realized volatility and correlation. According to the two-scale measure, our findings empirically suggest that when returns in Euro/USD exchange rate are highly volatile, the relation between Euro/USD and Euro/GBP exchange markets is strong, and when Euro/USD calms down, the relationship relaxes.Realized Volatility and Correlation, Long Memory, Scaling Law, Self-Similarity Dimension, Market Microstructure Effects.

Leverage effect in energy futures

We propose a comprehensive treatment of the leverage effect, i.e. the relationship between returns and volatility of a specific asset, focusing on energy commodities futures, namely Brent and WTI crude oils, natural gas and heating oil. After estimating the volatility process without assuming any specific form of its behavior, we find the volatility to be long-term dependent with the Hurst exponent on a verge of stationarity and non-stationarity. Bypassing this using by using the detrended cross-correlation and the detrending moving-average cross-correlation coefficients, we find the standard leverage effect for both crude oil. For heating oil, the effect is not statistically significant, and for natural gas, we find the inverse leverage effect. Finally, we also show that none of the effects between returns and volatility is detected as the long-term cross-correlated one. These findings can be further utilized to enhance forecasting models and mainly in the risk management and portfolio diversification.Comment: 19 pages, 2 figures, 5 table

Multi-scale correlations in different futures markets

In the present work we investigate the multiscale nature of the correlations for high frequency data (1 minute) in different futures markets over a period of two years, starting on the 1st of January 2003 and ending on the 31st of December 2004. In particular, by using the concept of "local" Hurst exponent, we point out how the behaviour of this parameter, usually considered as a benchmark for persistency/antipersistency recognition in time series, is largely time-scale dependent in the market context. These findings are a direct consequence of the intrinsic complexity of a system where trading strategies are scale-adaptive. Moreover, our analysis points out different regimes in the dynamical behaviour of the market indices under consideration.Comment: 14 pages and 25 figure

Application of Multifractal Measures to Tehran Price Index

We report an empirical study of Tehran Price Index (TEPIX). To analyze our data we use various methods like as, rescaled range analysis ($R/S$), modified rescaled range analysis (Lo's method), Detrended Fluctuation Analysis (DFA) and generalized Hurst exponents analysis. Based on numerical results, the scaling range of TEPIX returns is specified, long memory effect or long range correlation property in this market is investigated, characteristic exponent for probability distribution function of TEPIX returns is derived and finally the stage of development in Tehran Stock Exchange is determined.Comment: 19 pages, 6 figure