338 The Basics of financial economeTrics

That is, for any pair (v, w), the joint frequency is the mathematical product
of their respective marginals. By the definition of the conditional frequen-
cies, we can state an equivalent definition as in the following:

fv==fvw

fvw

fw

() (|)

(,)

()

x

xy

y

, (A.8)

which, in the case of independence of x and y, has to hold for all values v
and w. Conversely, an equation equivalent to (A.8) has to be true for the
marginal frequency of y, fy(w), at any value w. In general, if one can find
one pair (v, w) where either equations (A.7) or (A.8) and, hence, both do not
hold, then x and y are dependent. So, it is fairly easy to show that x and y
are dependent by simply finding a pair violating equations (A.7) and (A.8).
Now we show that the concept of influence of x on values of y is analo-
gous. Thus, the feature of statistical dependence of two variables is mutual.
This will be shown in a brief formal way by the following. Suppose that the

frequency of the values of x depends on the values of y, in particular,^7

fv≠=

fvw

fw

() fvw

(,)

()

x xy (|)

y

, (A.9)

Multiplying each side of equation (A.9) by fy(w) yields

(^) fvxy,(,wf)(≠⋅xyvf)(w) (A.10)
which is just the definition of dependence. Dividing each side of equation
(A.10) by fvx()>^0 gives
(^) =≠
fvw
fv
fwvfw

(,)

()

xy (|)()

x

y

,

showing that the values of y depend on x. Conversely, one can demonstrate
the mutuality of the dependence of the components.

covariance

In this bivariate context, there is a measure of joint variation for quantita-
tive data. It is the (sample) covariance defined by

==∑ −−

sxy

n

cov(,) xxyy

1

xy ()ii()

i

n

,

1

(A.11)

(^7) This holds provided that fy(w) > 0.