Information measure for long-range correlated time series: Quantifying horizon dependence in financial markets

Market dynamics is quantified via the cluster entropy S(τ,n)=∑jPj(τ,n)logPj(τ,n), an information measure with Pjτ,n the probability for the clusters, defined by the intersection between the price series and its moving average with window n, to occur with duration τ. The cluster entropy S(τ,n) is estimated over a broad range of temporal horizons M, for raw and sampled highest-frequency data of US markets. A systematic dependence of S(τ,n) on M emerges in agreement with price dynamics and correlation involving short and long range horizon dependence over multiple temporal scales. A comparison with the price dynamics based on Kullback–Leibler entropy simulations with different representative agent models is also reported.