118 The Basics of financial economeTrics

Thus far we have assumed that there is no interaction between the cat-
egorical and the quantitative variable, that is, the slope of the regression is the
same for the two categories. This means that the effects of variables are addi-
tive; that is, the effect of one variable is added regardless of the value taken
by the other variable. In many applications, this is an unrealistic assumption.
Using dummy variables, the treatment is the same as that applied to inter-
cepts. Consider the regression equation (6.1) and write two regression equa-
tions for the two categories as we did above where β 10 , β 11 are the slopes

(^) y

XD

i XD

ii

ii

=

++ =

++

ββ ε

ββ ε

01011

01111

if 0

if ==





^1

(6.4)

We can couple these two equations in a single equation as follows:

(^) YXii=+ββ 010 ++δε()DXii i (6.5)
where δ = β 11 – β 10. In fact, equation (6.5) is identical to the first of two
equations in equation (6.4) for Di = 0 and to the second for Di = 1. This
regression can be estimated with the usual OLS methods.
In practice, it is rarely appropriate to consider only interactions and not
the intercept, which is the main effect. We refer to the fact that the interac-
tion effect is marginal with respect to the main effect as marginalization.
However, we can easily construct a model that combines both effects. In fact
we can write the following regression adding two variables, the dummy D
and the interaction DX:
(^) YDii=+βγ 01 ++βδXDii()Xii+ε (6.6)
This regression equation, which now includes three regressors, combines
both effects.
The above process of introducing dummy variables can be generalized
to regressions with multiple variables. Consider the following regression:
YXijij i iT
j
N
=+ +=

ββ 0 ∑ ε
1

1, ..., (6.7)

where data can be partitioned in two categories with the use of a dummy
variable:

X=





















DX X

DX X

N

TT TN

1111

1

1

1