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Appendix

B

Continuous probability distributions

Commonly Used

in Financial econometrics

i

n this appendix, we discuss the more commonly used continuous
probability distributions are used in financial econometrics. The four
distributions discussed are the normal distribution, the chi-square distri-
bution, the Student’s t-distribution, the Fisher’s F-distribution. It should
be emphasized that although many of these distributions enjoy wide-
spread attention in financial econometrics as well as financial theory
(e.g., the normal distribution), due to their well-known characteristics
or mathematical simplicity, the use of some of them might be ill-suited
to replicate the real-world behavior of financial returns. In particular,
the four distributions just mentioned are appealing in nature because
of their mathematical simplicity, due to the observed behavior of many
quantities in finance, there is a need for more flexible distributions com-
pared to keeping models mathematically simple. For example, although
the Student’s t-distribution that will be discussed in this appendix is able
to mimic some behavior inherent in financial data such as so-called fat
tails or heavy tails (which means that a lot of the probability mass is
attributed to extreme values),^1 it fails to capture other observed behavior
such as skewness. For this reason, there has been increased interest in a
continuous probability distribution in finance and financial econometrics
known as the α-stable distribution. We will describe this distribution at
the end of this appendix.

(^1) There are various characterizations of fat tails in the literature. In finance, typically
the tails that are heavier than those of the exponential distribution are considered
“heavy.”