The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 381 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 381

tion of the eventual tail index is part of a larger effort. However,

empirical data are a sequence of samples characterized by an uncondi-

tional distribution. One might want to understand if estimation proce-

dures used for IID sequences can be applied in this more general setting.

For instance, one might want to understand if tail-index estimators such

as the Hill estimator can be used in the case of serially correlated

sequences generated by a generic DGP.

From a practical standpoint, this question is quite important as one

wants to estimate the tails even if one does not know exactly what

model generated the sequence. Clearly, there is no general answer to this

problem. However, the behavior of a number of estimators under differ-

ent DGPs has been explored through simulation as explained in the fol-

lowing section.

Heavy-Tailed ARMA Processes

Let’s first consider the infinite moving average representation of a

univariate stationary series:

xt =

∞

∑ hiε ti– + m

i = 0

under the assumption that innovations are IID α -stable laws of tail

index α. By the properties of stable distributions it can be demonstrated

that the finite-dimensional distributions of the process x are α -stable.

However, restrictions on the coefficients need to be imposed. It can be

demonstrated that a sufficient condition to ensure that the process x

exists and is stationary is the following:

∞

∑ hi

α ∞ <

i = 0

As we have seen in the previous section, a general univariate

ARMA(p,q) model is written as follows:

Xt =

p

∑ α^ iXti– +^

q

∑ α^ jZtj–

i = 1 j = 1

where the Z are IID variables.

Using the Lag Operator—L—notation, Li represents the variable at

i lags, the ARMA(p,q) model is written as follows: