شواهد تجربی از همبستگی بلند مدت در بازده سهام

A major issue in financial economics is the behaviour of stock returns over long horizons. This study provides empirical evidence of the long-range behaviour of various speculative returns. Using different techniques such as R/S and modified R/S analysis, detrended fluctuation analysis (DFA), fractional differencing test (GPH) and ARFIMA maximum likelihood estimation, we find little evidence of long memory in returns themselves, by strong evidence of persistence in volatility measured as squared returns or absolute returns. These results allow us to conclude that any stock market model should show no temporal dependence in returns and long-range correlation in conditional volatility

The long-range dependence, also known as long memory, is characterized by hyperbolically decaying autocovariance function, by a spectral density that tends to infinity as the frequencies tend to zero and by the self-similarity of aggregated summands. The intensity of these phenomena can be measured either by a parameter d, used as a differencing parameter in the ARFIMA model, or by the parameter H, that is a scaling parameter. Both parameters are related, in the case of finite variance processes by , and in the case of infinite variance processes by H=d+1/α [1]. Time series with long-range dependence are usually modelled with the ARFIMA (p,d,q). This model is given by equation(1) where L is the lag operator, d is the fractional differencing parameter and all the roots of φ(L) and θ(L) lie outside the unit circle. For any real number d, the fractional difference operator (1−L)d is defined through a binomial expansion equation(2) and for −0.5<d<0.5 the process is stationary. There is a growing literature in financial economics that analyses 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 became 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 behaviour 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.

Our investigation on the behaviour of the returns of six stock indexes using non- parametric, semiparametric and parametric statistical tests shows no evidence of long-term dependence, or to a very little degree if we attend to the results of the maximum likelihood estimation. In the case of the absolute and squared returns, long memory is found stronger for absolute returns than for squared returns. We have found important anomalies to this behaviour in the study of the FTSE index. Another re- markable result is that the agreement of the four methods employed decreases if we use the whole series of the Dow Jones and the SP500, which can be a consequence of the structural changes that would produce time-varying parameters or regime switches. The pattern of dependence in volatility may not remain constant over time, so any attempt to model the long dependence should reproduce these results. These empirical ndings open a new direction on the research of modelling stock markets.