Long-Range Dependence in Financial Markets: a Moving Average Cluster Entropy Approach

A perspective is taken on the intangible complexity of economic and social systems by investigating the underlying dynamical processes that produce, store and transmit information in financial time series in terms of the $\mathit{\text{moving}}\mathit{\text{average}}\mathit{\text{cluster}}\mathit{\text{entropy}}$. An extensive analysis has evidenced market and horizon dependence of the $\mathit{\text{moving}}\mathit{\text{average}}\mathit{\text{cluster}}\mathit{\text{entropy}}$ in real world financial assets. The origin of the behavior is scrutinized by applying the $\mathit{\text{moving}}\mathit{\text{average}}\mathit{\text{cluster}}\mathit{\text{entropy}}$ approach to long-range correlated stochastic processes as the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Fractional Brownian motion (FBM). To that end, an extensive set of series is generated with a broad range of values of the Hurst exponent $H$ and of the autoregressive, differencing and moving average parameters $p,d,q$. A systematic relation between \textit{moving average cluster entropy}, $\mathit{\text{Market}}\mathit{\text{Dynamic}}\mathit{\text{Index}}$ and long-range correlation parameters $H$, $d$ is observed. This study shows that the characteristic behaviour exhibited by the horizon dependence of the cluster entropy is related to long-range positive correlation in financial markets. Specifically, long range positively correlated ARFIMA processes with differencing parameter $d\simeq 0.05$, $d\simeq 0.15$ and $d\simeq 0.25$ are consistent with $\mathit{\text{moving}}\mathit{\text{average}}\mathit{\text{cluster}}\mathit{\text{entropy}}$ results obtained in time series of DJIA, S&P500 and NASDAQ.