همبستگی قدرت-قانون بلند مدت در بازده سهام

This study investigates long-range power-law correlations in US, UK, Japanese, German, French and Spanish stock markets using daily data and applying a recently developed residual analysis termed detrended fluctuation analysis (DFA). We quantify correlations for the returns, absolute value of returns and square of returns. The results show that there is little evidence of long-range correlations in returns but there is strong evidence of long-range correlation in absolute and squared returns. For the absolute returns, a cross-over of approximately 41 days is found.

There is a growing literature in financial economics that analyzes the temporal dependence of stock returns. The random walk hypothesis states that returns are serially random; in other words, that today returns are independent of previous periods stock returns. So the research on either short or long-term dependence has become somehow relevant. For example, the existence of long memory in financial data would affect the investment horizon of portfolio decisions. Furthermore, many empirical studies that are based on short memory statistical techniques would have to be revised. On the other hand, the literature of mean reversion in financial prices assumes the existence of some mechanism which works over long time horizons, because the mean-reverting behavior of stock prices corresponds to the idea that a given change in prices will be followed, in long time horizons, by changes with the opposite sign. Finally, the bases of the development of ARCH type family of stochastic models are the findings of significant autocorrelations in volatility measures, such as squared returns or absolute returns. Previous approaches to long-memory analysis are the application of non-parametric statistic tests sensible to the persistence over long periods. The Hurst method [1] and the rescaled range analysis (R/S) proposed by Mandelbrot and Wallis [2], and Mandelbrot [3] have been applied to many financial series (e.g. [4], [5], [6] and [7]). But the findings of long memory in stock returns using R/S analysis have been disputed because this type of analysis might be biased due to the presence of short-term dependence. More recently, Lo [8] developed a more refined technique of R/S analysis to identify long-term dependence. The modified R/S analysis is robust to serial correlation and some forms of non-stationarity. Some applications of this method to financial data sets are [9], [10], [11] and [12]; in most of the studies little evidence is found of long memory in returns. Another test of long-memory hypothesis is the GPH [13], which is highly related to one of the dominant parametric discrete-time models that present hyperbolic decay of the autocorrelation function, the ARFIMA (fractional integrated autoregressive moving average) model introduced in [14] and [15]. This methodology captures the long-range correlations with the fractional difference parameter or d-parameter, which describes the higher order correlation structure of the series. This approach and the R/S analysis have been used together in some empirical analyses, for example in [9], [12] and [16]. In this paper we will use a new methodology, a non-parametric approach that can be applied without a detailed assumption of the structure of the underlying model.

Our investigation on the behavior of the returns of six stock indexes using DFA analysis shows no evidence of long-term dependence in returns. The squared returns show persistent power-law correlations. For absolute returns, the DFA method shows two regions characterized by di=erent power-law behaviors with a cross-over at approximately 41 days. The behavior of all the indexes are quite similar, with small di=erences in the intensity on the long-range correlations.