354 The Basics of financial economeTrics
density with a low α = 1.5.^4 The density graphs are obtained by fitting the
distributions to the same sample data of arbitrarily generated numbers. The
parameter α is related to the parameter ξ of the Pareto distribution resulting
in the tails of the density functions of α-stable random variables to vanish at
a rate proportional to the Pareto tail.
The tails of the Pareto as well as the α-stable distribution decay at a rate
with fixed power α, Cx–α (i.e., power law) where C is a positive constant,
which is in contrast to the normal distribution whose tails decay at an expo-
nential rate (i.e., roughly xe−−1/x^2
2
).
The parameter β indicates skewness where negative values represent left
skewness while positive values indicate right skewness. The scale parameter
σ has a similar interpretation to the standard deviation. Finally, the param-
eter μ indicates location of the distribution. Its interpretability depends on
the parameter α. If the latter is between 1 and 2, then μ is equal to the mean.
(^4) In the figure, the parameters for the normal distribution are μ = 0.14 and σ = 4.23.
The parameters for the stable distribution are α = 1.5, β = 0, σ = 1, and μ = 0. Note
that symbols common to both distributions have different meanings.
FigURe B.7 Comparison of the Normal (Dash-Dotted) and α-Stable (Solid)
Density Functions
−15 −10 −5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
f(
x)
x
Higher peak Heavier tails
Excess kurtosis: