Empirically based modeling in financial economics and beyond, and spurious stylized facts
The discovery of the dynamics of a time series requires construction of the transition density. We explain why 1-point densities and scaling exponents cannot determine the class of stochastic dynamics. Time series require some sort of underlying statistical regularity to provide a basis for analysis, and we construct an exhaustive list of such tests. The condition for stationary increments, not scaling, determines the existence of long time pair autocorrelations. We conjecture that for a selfsimilar process neither the pair correlations nor the 2-point density scales in both times t and s except in a pathological case, and give examples using three well-known Gaussian processes. An incorrect assumption of stationary increments can generate spurious stylized facts, including fat tails. When a sliding window is applied to nonstationary, uncorrelated increments then a Hurst exponent Hs = 1 / 2 is generated by that procedure even if the underlying model scales with a Hurst exponent H [not equal to] 1/2. We explain how this occurs dynamically. The nonstationarity arises from systematic unevenness in the traders' behavior in real time. Spurious stylized facts arise mathematically from using a log increment with a 'sliding window' to read the series. In addition, we show that nonstationary processes are generally not globally transformable to stationary ones. We also present a more detailed explanation of our recent FX data analysis and modeling.
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- J. L. McCauley & G. H. Gunaratne & K. E. Bassler, 2006. "Martingale Option Pricing," Papers physics/0606011, arXiv.org, revised Feb 2007.
- McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2006. "Hurst exponents, Markov processes, and fractional Brownian motion," MPRA Paper 2154, University Library of Munich, Germany.
- Bassler, Kevin E. & McCauley, Joseph L. & Gunaratne, Gemunu H., 2006. "Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets," MPRA Paper 2126, University Library of Munich, Germany.
- McCauley, J.L. & Gunaratne, G.H. & Bassler, K.E., 2007. "Martingale option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 351-356.
- McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2007. "Martingale option pricing," MPRA Paper 2151, University Library of Munich, Germany.
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