13-Fat Tails-Scaling-Stabl Page 384 Wednesday, February 4, 2004 1:00 PM
384 The Mathematics of Financial Modeling and Investment Managementrally leads to the consideration of subordinated models. Subordinated
models generate unconditional fat-tailed distributions.Markov Switching Models
The GARCH family of models is not the only family of serially corre-
lated models able to produce fat tails starting from normally distributed
innovations. State-space models and Markov-switching models present
the same feature. The basic ideas of state-space models and Markov
switching models is to split the model into two parts: (1) a regressive
model that regresses the model variable over a hidden variable and (2)
an autoregressive model that describes the hidden variables.
In its simplest linear form, a state-space model is written as follows:Xt = αZt + εtZt =βZt – 1 + δtwhere εt, δt are normally distributed independent white noises. State-
space models can also be written in a multiplicative form:Xt =αZt – 1 + εtαt = βαt – 1 + δtIf the second equation is a Markov chain, the model is called a
Markov-switching model. A well-known example of Markov-switching
models is the Hamilton model in which a two-state Markov chain drives
the switch between two different regressions.
Purely linear state-space models exhibit fat tails only if innovations
are fat-tailed. However, multiplicative state-space models and Markov-
switching models can exhibit fat tails even if innovations are normally
distributed. There is a growing literature on Markov-switching and mul-
tiplicative state-space models and a relatively large number of different
models, univariate as well as multivariate, have been proposed. Stochas-
tic volatility models are the continuous-time version of multiplicative
state-space models.Estimation
Let’s now go back to the question of model estimation in a non-IID frame-
work. Suppose that we want to estimate the tail index of the unconditional
distribution of a set of empirical observations in the general setting of non-