Anon

328 The Basics of financial economeTrics

For a more thorough analysis of this, let’s use a more formal presenta-
tion. Let I denote the set of all class indexes i with i being some integer
value between 1 and nII= (i.e., the number of classes). Moreover, let aj
and fj denote the (absolute) frequency and relative frequency of some class
j, respectively. The cumulative frequency distribution at some upper bound,
xui, of a given class i is computed as

==∑∑+
≤≤

Fx()ui aaj a

jxx

ji

::ju ui jxuj xli

(A.1)

In words, this means that we sum up the frequencies of all classes in which
the upper bound is less than xui plus the frequency of class i itself. The cor-
responding cumulative relative frequency distribution at the same value is
then,

==∑∑ +
≤ ≤

Fxf()ui ffjjfi

jx:juxiu jx:uj xli

(A.2)

This describes the same procedure as in equation (A.1) using relative
frequencies instead of frequencies. For any value x in between the boundar-
ies of, say, class i, xli and xui, the cumulative relative frequency distribution
is defined by

=+

−

−

Fx Fx

xx

xx

ff() ()li l f

i

ui li

i^ (A.3)

In words, this means that we compute the cumulative relative frequency dis-
tribution at value x as the sum of two things. First, we take the cumulative
relative frequency distribution at the lower bound of class i. Second, we add
that share of the relative frequency of class i that is determined by the part
of the whole interval of class i that is covered by x.

Measures of Location and Spread

Once we have the data at our disposal, we now want to retrieve key num-
bers conveying specific information about the data. As key numbers we will
introduce measures for the center and location of the data as well as mea-
sures for the spread of the data.

parameters versus Statistics

Before we go further, we have to introduce a distinction that is valid for any
type of data. We have to be aware of whether we are analyzing the entire