Multifractality of US Dollar/Deutsche Mark Exchange Rates - Studies2

their distribution is usually specified Student t rather than Gaussian. In simulation, we assume
independence for lack of an explicit alternative, and use the Gaussian for ut. These assumptions
should not qualitatively affect our conclusions.
rewrite
where
For FIGARCH(1,d,0), (12) can be written more intuitively. Starting with
(1 − L) d ε 2 t = ω +[1− β1L] vt,
σ 2 t =
ω − β1vt−1 + 1 − (1 − L) d
ε 2 t
=
∞
ω − β1vt−1 − ψkε 2 t−k .
ψk =
k=1
Γ(k − d)
Γ(k +1)Γ(−d)
The ARMA representation in volatility clarifies how simulation proceeds, and how long memory
arises. (Plotting ψk will help readers gain further intuition for the process.) Because the Gamma
function is smooth, the path of volatility is also smooth. This is the property that leads to clumping
of like-sized returns.
Like ARFIMA, FIGARCH is a hybrid from the scaling point of view. For ARFIMA, intuition
is clear since it is asymptotically equivalent in increments to Fractional Gaussian Noise. Short
memory components are non-scaling, but become negligible over long sampling increments. This
intuition disappears for FIGARCH since squared increments are asymptotically scaling, not incre-
ments themselves.
An alternative intuition is the literature on diffusion limits of ARCH sequences. The continuous
time limit of discrete GARCH sequences is an Itô diffusion or jump-diffusion in certain cases.
(Nelson, 1990; Drost and Nijman, 1996). Intuitively, GARCH is not scaling since Itô diffusions are
not scaling. 37 Continuous time limits of FIGARCH, on the other hand, are not known. 38
Twenty simulated partition functions, appearing in Figure 19, are used to investigate the scaling
properties of FIGARCH. Each simulations consists of 100,000 observations, allowing like comparison
37
A reference is not available, but intuitively, it results from smoothness of volatility.
38
Nelson (1990) notes that his method of demonstrating convergence to a continuous time diffusion does not apply
to long memory processes.
27