for nonlinear Granger causality is illustrated in two examples. In the first example, we examine the main explanations of the
asymmetric volatility phenomenon using high-frequency data on SP 500 Index futures contracts and find evidence of a
nonlinear leverage and volatility feedback effects. In the sec- ond application, we investigate the Granger causality between
SP 500 Index returns and trading volume. We find convincing evidence of linear and nonlinear feedback effects from stock
returns to volume, but a weak evidence of nonlinear feedback effect from volume to stock returns.
The rest of the article is organized as follows. The condi- tional independence test using the Hellinger distance and the
Bernstein copula is introduced in Section 2. Section 3 provides the test statistic and its asymptotic properties. In Section 4, we
investigate the finite sample size and power properties of our test and we compare with Su and White’s
2008 test. Section
5 contains the two applications described above. Section 6 con- cludes. The proofs of the asymptotic results are presented in the
Technical Appendix, which is available online.
2. NULL HYPOTHESIS, HELLINGER DISTANCE, AND
THE BERNSTEIN COPULA
Let {X
′ t
, Y
′ t
, Z
′ t
′
∈ R
d
1
× R
d
2
× R
d
3
, t = 1, . . . , T } be a sample of stochastic processes in R
d
, where d = d
1
+ d
2
+ d
3
, with joint distribution function F
XYZ
and density function f
XYZ
.
We wish to test the conditional independence between Y and Z conditionally on X. Formally, the null hypothesis can be written
in terms of densities as
H :
Pr{f
Y|X,Z
y | X, Z = f
Y|X
y | X} = 1, ∀y ∈ R
d
2
, or
H :
Pr{f y, X, Zf X = f y, Xf X, Z} = 1, ∀y ∈ R
d
2
, 1
and the alternative hypothesis as H
1
: Pr{f
Y|X,Z
y | X, Z = f
Y|X
y | X} 1, for some y ∈ R
d
2
, where f
·|·
·|· denotes the conditional density. As we mentioned in the introduction, Granger noncausality is a form of condi-
tional independence and to see that let us consider the following example. For Y, Z
′
a Markov process of order 1, the null hy-
pothesis that corresponds to Granger noncausality from Z to Y is given by
H :
Pr{f
Y|X,Z
y
t
| y
t −1
, z
t −1
= f
Y|X
y
t
| y
t −1
} = 1,
where in this case y = y
t
, x = y
t −1
, z = z
t −1
and d
1
= d
2
= d
3
= 1. Next, we reformulate the null hypothesis
1 in terms of cop-
ulas. This will allow us to keep only the terms that involve the dependence among the random vectors. It is well known from
Sklar 1959
that the distribution function of the joint process
X
′
, Y
′
, Z
′ ′
can be expressed via a copula F
XYZ
x, y, z = C
XYZ
¯ F
X
x, ¯ F
Y
y, ¯ F
Z
z, 2
where C
XYZ
. is a copula function defined on [0, 1]
d
that
captures the dependence of X
′
, Y
′
, Z
′ ′
, and for simplicity of notation and to keep more space we denote
¯ F
X
x =
F
X
1
x
1
, . . . , F
X
d1
x
d
1
, ¯
F
Y
y = F
Y
1
y
1
, . . . , F
Y
d2
y
d
2
, ¯
F
Z
z = F
Z
1
z
1
, . . . , F
Z
d3
z
d
3
, where
F
Q
i
., for
Q = X, Y, Z, is the marginal distribution function of the i
th element of the vector Q. If we derive Equation 2
with
respect to x
′
, y
′
, z
′ ′
, we obtain the density function of the joint
process X
′
, Y
′
, Z
′ ′
that can be expressed as f
XYZ
x, y, z =
d
1
j =1
f
X
j
x
j
×
d
2
j =1
f
Y
j
y
j
×
d
3
j =1
f
Z
j
z
j
× c
XYZ
¯ F
X
x, ¯ F
Y
y, ¯ F
Z
z, 3
where f
Q
j
., for Q = X, Y, Z, is the marginal density of the jth element of the vector Q and c
XYZ
. is a copula density defined on [0, 1]
d
of X
′
, Y
′
, Z
′ ′
. Using Equation 3
, we can show that the null hypothesis in
1 can be rewritten in terms of copula
densities as H
: Pr{c
XYZ
¯ F
X
X, ¯ F
Y
y, ¯ F
Z
Zc
X
¯ F
X
X
= c
XY
¯ F
X
X, ¯ F
Y
yc
XZ
¯ F
X
X, ¯ F
Z
Z} = 1, ∀y ∈ R
d
2
, against the alternative hypothesis
H
1
: Pr{c
XYZ
¯ F
X
X, ¯ F
Y
y, ¯ F
Z
Zc
X
¯ F
X
X
= c
XY
¯ F
X
X, ¯ F
Y
yc
XZ
¯ F
X
X, ¯ F
Z
Z} 1,
for some y ∈ R
d
2
where c
X
., c
XY
. and c
XZ
. are the copula densities of the
processes X, X
′
, Y
′ ′
, and X
′
, Z
′ ′
, respectively. Observe that
under H ,
the dependence of the vector X
′
, Y
′
, Z
′ ′
is controlled
by the dependence of X, X
′
, Y
′ ′
, and X
′
, Z
′ ′
and not that of
Y
′
, Z
′ ′
. Note also that in the typical case where d
1
= 1, we have c
X
u = 1 and therefore does not need to be estimated below.
Note that
in Equation
3, the
term c
XYZ
¯ F
X
x,
¯ F
Y
y, ¯ F
Z
z corresponds to the copula density that is de- fined on all the univariate components of X, Y, and Z. Al-
ternatively, we can equivalently rewrite Equation 3 in terms of the product of densities of the multivariate random vectors
X, Y, Z, say f
X
x × f
Y
y × f
Z
z, and the density copula
c
XYZ
F
X
x, F
Y
y, F
Z
z that is now defined in terms of the cumulative distributions of the multivariate random vectors X,
Y, Z, rather than the marginal distributions of their respec- tive univariate components. Redefining the null hypothesis H
in this way allows us to avoid the estimation of the copula of the components of X. However, this approach requiresus
to estimate nonparametrically the joint cumulative distribution functions F
X
X,F
Y
y,F
Z
Z. The null could also be written in
terms of conditional copulas, but similarly this would requireus to estimate nonparametrically conditional distributions. The dif-
ferences between these approaches will be investigated in future work.
Given the null hypothesis, our test statistic is based on the Hellinger distance between the copulas c
XYZ
u, v, wc
