332 The Basics of financial economeTrics
This is because, not only is some absolute deviation of interest, but the direc-
tion is as well.
Multivariate Variables and Distributions
Thus far in this appendix, we examined one variable only. However, for
applications of financial econometrics, there is typically less of a need to
analyze one variable in isolation. Instead, a typical problem is to investigate
the common behavior of several variables and joint occurrences of events. In
other words, there is the need to establish joint frequency distributions and
introduce measures determining the extent of dependence between variables.
Frequencies
As in the single variable case, we first gather all joint observations of our
variables of interest. For a better overview of occurrences of the variables,
it might be helpful to set up a table with rows indicating observations and
columns representing the different variables. This table is called the table of
observations. Thus, the cell of, say, row i and column j contains the value
that observation i has with respect to variable j. Let us express this rela-
tionship between observations and variables a little more formally by some
functional representation.
In the following, we will restrict ourselves to observations of pairs, that
is, k = 2. In this case, the observations are bivariate variables of the form x =
(x 1 ,x 2 ). The first component x 1 assumes values in the set V of possible values
while the second component x 2 takes values in W, that is, the set of possible
values for the second component.
Consider the Dow Jones Industrial Average over some period, say one
month (roughly 22 trading days). The index includes the stock of 30 com-
panies. The corresponding table of observations could then, for example,
list the roughly 22 observation dates in the columns and the individual com-
pany names row-wise. So, in each column, we have the stock prices of all
constituent stocks at a specific date. If we single out a particular row, we
have narrowed the observation down to one component of the joint obser-
vation at that specific day.
Since we are not so much interested in each particular observation’s
value with respect to the different variables, we condense the information to
the degree where we can just tell how often certain variables have occurred.^5
In other words, we are interested in the frequencies of all possible pairs with
(^5) This is reasonable whenever the components assume certain values more than once.