13-Fat Tails-Scaling-Stabl Page 385 Wednesday, February 4, 2004 1:00 PM
Fat Tails, Scaling, and Stable Laws 385IID variables. Note that if variables are fat-tailed, we cannot say that they
are serially autocorrelated as moments of second order generally do not
exist. Therefore we have to make some hypothesis on the DGP.
There is no general theory of estimation under arbitrary DGP. Both
theoretical and simulation work are limited to specific DGPs. ARMA
models have been extensively studied. EVT holds for ARMA models
under general non-clustering conditions.^21
Often only simulation results are available. A fairly ample set of
results are available for GARCH(1,1) models. For these models Resnick
and Starica^22 showed that the Hill estimator is a consistent estimator of
the tail index. Wagner and Marsh compared the performance of the Hill
estimator and of the moment ratio estimator for three model classes: IID
α-stable returns, IID symmetric student, and GARCH(1,1) with student-
t innovation. They found that, in an adoptive framework, the moment
ratio estimator generally yields results superior to the Hill estimator.Scaling and Self-Similarity
The concept of scaling is now quite frequently evoked in economics and
finance. Let’s begin by making a distinction between scaling and self-
similarity and some of the properties associated with inverse power laws
within or outside the Levy-stable scaling regime. These concepts have
different, and not equivalent, definitions.
The concepts of scaling and self-similarity apply to distributions,
processes or structures. Self-similarity was introduced as a property that
applies to geometrical self-similar objects (i.e., fractal structures). In this
context, self-similarity means that a structure can be put into a one-to-
one correspondence with a part of itself. Note that no finite structure
can have this property; self-similarity is the mark of infinite structures.
Self-similarity entails scaling: If a fractal structure is expanded by a
given factor, its measure expands by a power of the same factor.^23 The
notion of scaling is often expressed as absence of scale, meaning that a
scaling object looks the same at any scale, large or small: It is impossible
to ascertain the size of a portion of a scaling object by looking at its
shape. The classical illustration is a Norwegian coastline with its fjords
and fjords within fjords that look the same regardless of the scale.(^21) See Embrechts, Kluppelberg, and Mikosch, Modelling Extremal Events for Insur -
ance and Finance.
(^22) S. Resnick and C. Starica, “Tail Index Estimation for Dependent Data,” Annals of
Applied Probability 8 (1998), pp. 1156–1183.
(^23) For an introduction to fractals, see J. Falconer, Fractal Geometry (Chichester,
U.K.: John Wiley & Sons, 1990).