Ricerc@Sapienza

A new method is proposed to estimate the self-similarity exponent. Instead of applying finite moment(s) methods, a goodness-of-fit statistic is designed to test whether two rescaled sequences are drawn from the same distribution, which is the definition of self-similarity. The test is the empirical likelihood ratio, which is robust with respect to processes with dependence. We provide a closed formula for fractional Brownian motion and prove that the distance between two rescaled sequences is zero iff the scaling exponent equals the true one.

A nonparametric method is developed to detect self-similarity among the rescaled distributions of the log-price variations over a number of time scales. The procedure allows to test the statistical significance of the scaling expo- nent that possibly characterizes each pair of time scales and to analyze the link between self-similarity and liquidity, the core assumption of the fractal mar- ket hypothesis. The method can support financial operators in the selection of the investment horizons as well as regulators in the adoption of guidelines to improve the stability of markets.