2017/04/07 by Lijian Chen, Kevin E. Bassler, Joseph L. McCauley +1 · 1 citation
Economics, Econometrics and Finance · Physics and Astronomy · Mathematics · #q-fin.ST #cond-mat.stat-mech #math.DS #physics.data-an
paper · pdf · doi:10.1103/physreve.95.042141
13 pages, 4 figures, Supplementary Info provided by request to the authors
arxiv created 2017/04/07 · arxiv updated 2017/05/24
The state of a stochastic process evolving over a time t is typically assumed to lie on a normal distribution whose width scales like t1/2. However, processes where the probability distribution is not normal and the scaling exponent differs from (1)/(2) are known. The search for possible origins of such "anomalous" scaling and approaches to quantify them are the motivations for the work reported here. In processes with stationary increments, where the stochastic process is time-independent, auto-correlations between increments and infinite variance of increments can cause anomalous scaling. These sources have been referred to as the \itJoseph \iteffect the \itNoah \iteffect, respectively. If the increments are non-stationary, then scaling of increments with t can also lead to anomalous scaling, a mechanism we refer to as the \itMoses \iteffect. Scaling exponents quantifying the three effects are defined and related to the Hurst exponent that characterizes the overall scaling of the stochastic process. Methods of time series analysis that enable accurate independent measurement of each exponent are presented. Simple stochastic processes are used to illustrate each effect. Intraday Financial time series data is analyzed, revealing that its anomalous scaling is due only to the Moses effect. In the context of financial market data, we reiterate that the Joseph exponent, not the Hurst exponent, is the appropriate measure to test the efficient market hypothesis.