13-Fat Tails-Scaling-Stabl Page 379 Wednesday, February 4, 2004 1:00 PM
Fat Tails, Scaling, and Stable Laws 379through state-space models or Markov switching models. In this way,
stationarity is recovered but at the price of a more complex, serially
autocorrelated model.
EVT for multivariate models with complex patterns of serial corre-
lations loses its generality and becomes model-dependent. One has to
evaluate each model in terms of its behavior as regards extremes. In this
section we will explore a number of models that have been proposed for
modeling financial time series: ARCH and GARCH models and, more in
general, state-space models. First, however, a number of methodological
considerations are in order.
In the context of IID sequences, EVT tries to answer the question of
how to estimate a distribution with heavy tails given only a limited
amount of data. The model is the simplest (i.e., a sequence of IID vari-
ables) and the question is how to extrapolate from finite samples to the
entire tail. In the context of IID distributions, conditional and uncondi-
tional distributions coincide. However, if we release the IID assumption,
we have to specify the model and to estimate the entire model—not just
the tail of one variable. Conditional and unconditional distributions no
longer coincide. For instance, there are families of models that are con-
ditionally normal and unconditionally fat-tailed.
Here difficulties begin as model estimation might be complex. In
addition, estimation of some specific tail might not be the primary con-
cern in model estimation. In the context of variables with a dependence
structure, EVT can be thought of as a methodology to estimate the tails
of the unconditional distribution, leaving aside the question of full
model estimation.
An important methodological question is whether fat-tailedness is
generated by the transformation of a sequence of zero-mean, finite vari-
ance IID variables (i.e., white noise) or whether innovations themselves
have fat tails (i.e., so-called colored noise). For instance, as we will see,
GARCH models entail fat-tailed return distributions as the result of the
transformation of white noise. On the other hand, one might want to
estimate an Autoregressive Moving Average (ARMA) model under the
assumption of innovations with infinite variance.
Understanding how power laws and, more in general, fat tails are
generated from normal variables has been a primary concern of econo-
metrics and econophysics. Given the universality of power laws in eco-
nomics, it is clearly important to understand how they are generated.
These questions go well beyond the statistical analysis of heavy-tailed
processes and involve questions of economic theories. Essentially, one
wants to understand how the decisions of a large number of economic
agents do not average out but produce cascading and amplification phe-
nomena.