  We test for departures from normal and independent and identically
distributed (NIID) returns, when returns under the alternative hypothesis are
self-affine. Self-affine returns are either fractionally integrated and
long-range dependent, or drawn randomly from an L-stable distribution with
infinite higher-order moments. The finite sample performance of estimators of
the two forms of self-affinity is explored in a simulation study which
demonstrates that, unlike rescaled range analysis and other conventional
estimation methods, the variant of fluctuation analysis that considers finite
sample moments only is able to identify either form of self-affinity. However,
when returns are self-affine and long-range dependent under the alternative
hypothesis, rescaled range analysis has greater power than fluctuation
analysis. The finite-sample properties of the estimators when returns exhibit
either form of self-affinity can be exploited to determine the source of
self-affinity in empirical returns data. The techniques are illustrated by
means of an analysis of the fractal properties of the daily logarithmic returns
for the indices of 11 stock markets.

Statistical Finance