116 The Basics of financial economeTrics
Independent Categorical Variables
Categorical input variables are used to cluster input data into different
groups.^1 That is, suppose we are given a set of input-output data and a parti-
tion of the data set in a number of subsets Ai so that each data point belongs
to one and only one set. The Ai represent a categorical input variable. In finan-
cial econometrics, categories might represent, for example, different market
regimes, economic states, credit ratings, countries, industries, or sectors.
We cannot, per se, mix quantitative input variables and categorical vari-
ables. For example, we cannot sum yield spreads and their credit ratings.
However, we can perform a transformation that allows the mixing of cat-
egorical and quantitative variables. Let’s see how. Suppose first that there is
only one categorical input variable that we denote by D, one quantitative
input variable X, and one quantitative output variable Y. Consider our set
of quantitative data, that is quantitative observations. We organize data,
residuals, and parameters in matrix form as usual:
Y
Y
Y
X
X
TTX
=
=
11111
1
EEB
T=
=
εεβ
β1
0
1Suppose data belong to two categories. An explanatory variable that
distinguishes only two categories is called a dichotomous variable. The key
is to represent a dichotomous categorical variable as a numerical variable D,
called a dummy variable, that can assume the two values 0,1. We can now
add the variable D to the input variables to represent membership in one or
the other group:
X=
D
D
X
TTX
11111
1
(^1) We can also say that categorical input variables represent qualitative inputs. This
last expression, however, can be misleading, insofar as categorical variables repre-
sent only the final coding of qualitative inputs in different categories. For example,
suppose we want to represent some aspect of market psychology, say confidence
level. We can categorize confidence in a number of categories, for example euphoria,
optimism, neutrality, fear, or panic. The crucial question is how we can operationally
determine the applicable category and whether this categorization makes sense. A
categorical variable entails the ability to categorize, that is, to determine member-
ship in different categories. If and how categorization is useful is a crucial problem
in many sciences, especially economics and the social sciences.