processes. The proposed analysis, on the other hand, focuses on the dynamic dependence of the volatility. It is concerned with
the fourth moment, both serial and cross-sectional, of a vector time series. Details are given in Section
3 .
The article is organized as follows. In Section 2
, we state clearly the conditional heteroscedasticity considered in this ar-
ticle and define the proposed cumulative generalized kurtosis matrix of a k-dimensional time series. Some properties of the
proposed matrix are given. In Section 3
, we introduce PVCs and discuss their properties. Section
4 considers sample estimate of
the cumulative generalized kurtosis matrix and sample PVC. It also proposes a test statistic for verifying that an estimated linear
combination of the observed time series has constant conditional variance. Simulations are used to demonstrate the performance
of the proposed PVC analysis and the test statistic in finite sam- ples. Section
5 considers an application of the proposed PVC
analysis using weekly log returns of seven foreign exchange rates, and Section
6 concludes. All proofs are in the Appendix.
2. CONDITIONAL HETEROSCEDASTICITY
AND GENERALIZED KURTOSIS MATRIX
Let y
t
= y
1t
, . . . , y
kt ′
be a k-dimensional weakly sta- tionary time series with finite fourth moment. Let F
t −1
=
σ { y
t −1
, y
t −2
, . . .}, denoting the information available at time
t − 1. Since we focus on volatility, we assume E y
t
|F
t −1
=
0. The volatility matrix
t
of y
t
is then defined as
t
≡
cov y
t
|F
t −1
= E y
t
y
′ t
|F
t −1
. We assume that the volatility ma- trix can be written as
vec
t
= c +
∞ i=1
C
i
vec y
t −i
y
′ t −i
, 1
where vecM denotes the column-stacking vector of the matrix M, c
is a k
2
-dimensional positive constant vector, and C
i
are k
2
× k
2
constant matrices for i 0. The matrices C
i
and vector
c satisfy certain conditions to ensure that
t
is positive defi- nite for all t. The Baba–Engle–Kraft–Kroner BEKK model of
Engle and Kroner 1995
can be written in the form of Equation 1
via repeated substitutions. From Equation 1
, the process y
t
has conditional heteroscedasticity if and only if C
i
= 0 for some
i 0. For multivariate autoregressive conditional heteroscedas-
tic ARCH models, the summation in Equation 1
is truncated
at a finite lag. In this article, we say that a vector time series y
t
has ARCH effects or conditional heteroscedasticity if C
i
= 0
for some i 0. From Equation
1 , the existence of ARCH effects in y
t
im-
plies that y
t
y
′ t
is correlated with y
t −i
y
′ t −i
for some i 0. This motivates us to define the lag-ℓ generalized kurtosis matrix γ
ℓ
of y
t
as γ
ℓ
=
k i=1
k j =i
cov
2
y
t
y
′ t
, x
ij,t −ℓ
≡
k i=1
k j =i
γ
ℓ,ij
γ
′ ℓ,ij
, for ℓ ≥ 0, 2
where x
ij,t −ℓ
is a function of y
i,t −ℓ
y
j,t −ℓ
for 1 ≤ i, j ≤ k, γ
ℓ,ij
= cov y
t
y
′ t
, x
ij,t −ℓ
= E[ y
t
y
′ t
− x
ij,t −ℓ
− Ex
ij,t
], 3
and = E y
t
y
′ t
is the unconditional covariance matrix of y
t
. The matrix γ
ℓ,ij
of Equation 3
is called a generalized covari- ance matrix in the statistical literature, for example, Li
1992 .
It is a k × k symmetric matrix, but might be negative definite. However, its square matrix, which is equivalent to γ
ℓ,ij
γ
ℓ,ij ′
, is semipositive definite. This justifies the use of square in Equa-
tion 2
. From the definition, the lag-ℓ generalized kurtosis ma- trix γ
ℓ
is symmetric and semipositive definite, because it is the sum of kk + 12 symmetric and semipositive definite matri-
ces. An important property of γ
ℓ
is that γ
ℓ
= 0 if and only if y
t
y
′ t
is not correlated with y
i,t −ℓ
y
j,t −ℓ
for all i and j. For a given positive integer m, we define the cumulative gen-
eralized kurtosis matrix as Ŵ
m
=
m ℓ=1
γ
ℓ
. 4
This cumulative matrix is symmetric and semipositive definite, and we use it to measure the ARCHm effects in y
t
. For the general multivariate GARCH-type models, we consider
Ŵ
∞
=
∞ ℓ=1
γ
ℓ
, 5
which is assumed to exist. We refer to Ŵ
∞
as the cumulative
generalized kurtosis matrix of y
t
. Ŵ
∞
is also symmetric and semipositive definite.
2.1 Properties of Generalized Kurtosis Matrix
The idea of generalized covariance matrix cov y
t
y
′ t
, x
t
, where x
t
is a scalar random variable, has been used in the statistical literature. See, for instance, Li
1992 . To the best
of our knowledge, the concept has not been used in the time series analysis. In this section, we discuss some properties of
the generalized kurtosis matrix in Equation 5
that are relevant to our study.
Let M be a k × k nontrivial linear transformation matrix so that z
t
= M
′
y
t
is a transformed series. For asset returns,
columns of M may represent, after normalization, investment portfolios. Let x
t −1
be a scalar function of elements of F
t −1
, for example, x
t −1
= y
i,t −h
y
j,t −h
for some h 0. It is easy to see that the following lemma holds.
Lemma 1. For a constant k × k matrix M, let z
t
= M
′
y
t
. Then,
covz
t
z
′ t
, x
t −1
= covM
′
y
t
y
′ t
M, x
t −1
= M
′
cov y
t
y
′ t
, x
t −1
M. Let m
v
= m
1v
, . . . , m
kv ′
be the vth column of M. If z
vt
=
m
′ v
y
t
is a linear combination of y
t
that has no ARCH effects, then we have Ez
2 vt
|F
t −1
= c
2 v
, which is a constant. This implies that z
2 vt
is not correlated with y
i,t −ℓ
y
j,t −ℓ
for ℓ 0 and 1 ≤ i, j ≤ k. In other words, covz
2 vt
, y
i,t −ℓ
y
j,t −ℓ
= 0 for ℓ 0 and 1 ≤ i ≤ j ≤ k. Using Lemma 1, we see that γ
ℓ,ij
is singular for all ℓ and 1 ≤ i ≤ j ≤ k and, hence, γ
ℓ
is singular for all ℓ. Consequently, Ŵ
∞
is also singular. On the other hand, assume that Ŵ
∞
is singular and m
v
is in its null space. That is, Ŵ
∞
m
v
= 0. Since γ
ℓ
is semipositive
definite, we have m
′ v
γ
ℓ
m
v
= 0 for all ℓ. This in turn shows that
m
′ v
γ
2 ℓ,ij
m
v
= 0 for all ℓ 0 and 1 ≤ i ≤ j ≤ k. Consequently, by the symmetry of γ
ℓ,ij
, we have γ
ℓ,ij
m
v ′
γ
ℓ,ij
m
v
= 0. This implies that γ
ℓ,ij
m
v
= 0 for all ℓ 0 and 1 ≤ i ≤ j ≤ k.
Again, using Lemma 1, we see that z
2 vt
is not correlated with
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y
i,t −ℓ
y
j,t −ℓ
for ℓ 0 and 1 ≤ i ≤ j ≤ k, where z
vt
= m
′ v
y
t
. This implies that Ez
2 vt
|F
t −1
is not time varying. In other words, z
vt
does not have conditional heteroscedasticity. The above dis- cussion shows that an eigenvector of Ŵ
∞
associated with a zero
eigenvalue gives rise to a linear combination of y
t
that has no ARCH effect. We summarize the above result into a theorem.
Theorem 1. Consider a weakly stationary process y
t
with finite fourth moment and satisfying Equation
1 . Let Ŵ
∞
be the cumulative generalized kurtosis matrix defined in Equation
5 ,
where x
ij,t −ℓ
= y
i,t −ℓ
y
j,t −ℓ
in Equation 3
. Then, there exist
k − m linearly independent linear combinations of y
t
that have no ARCH effects if and only if rankŴ
∞
= m. 3.
PRINCIPAL VOLATILITY COMPONENTS Consider the spectral decomposition of the cumulative gen-
eralized kurtosis matrix Ŵ
∞
, say Ŵ
∞
M = M, where =
diag{λ
2 1
≥ λ
2 2
≥ · · · ≥ λ
2 k
} is the diagonal matrix of ordered
eigenvalues and M = [m
1
, . . . , m
k
] is the matrix of eigenvec- tors. Here, we use the notation λ
2 v
to denote eigenvalues because Ŵ
∞
is semipositive definite. We assume that the columns m
v
are
normalized with m
v
= 1.
We define the vth PVC of y
t
as z
vt
= m
′ v
y
t
. From the defini- tion and spectral decomposition of Ŵ
∞
, we have
∞ ℓ=1
k i=1
k j =i
m
′ v
γ
2 ℓ,ij
m
v
= λ
2 v
, v = 1, . . . , k.
6 Let γ
ℓ,ij
m
v
= w
ℓ,ij,v
. Then, we have
∞ ℓ=1
k i=1
k j =i
w
′ ℓ,ij,v
w
ℓ,ij,v
= λ
2 v
. Using Lemma 1, we have
m
′ v
w
ℓ,ij,v
= m
′ v
γ
ℓ,ij
m
v
= cov z
2 vt
, x
ij,t −ℓ
. 7
This result indicates that m
′ v
γ
ℓ,ij
m
v
can be regarded as a mea-
sure of the dependence of volatility of the portfolio z
vt
on the lagged cross-product term x
ij,t −ℓ
. In practice, this quantity can be negative so that we use squared matrices in Equation
6 to
construct a nonnegative dependence measure. From Equation
6 , λ
2 v
summarizes the dependence measure in Equation
7 over all combinations of i and j and over all
lags. As such, it can be considered as an approximate measure of volatility dependence of the portfolio z
vt
. A larger λ
v
is indicative of a stronger volatility dependence. Therefore, we
call z
vt
the vth PVC. On the other hand, the summation in Equation
6 distinguishes the PVC from the traditional principal
components of y
t
, which depends on the covariance matrix alone. The PVC analysis is designed to consider simultaneously
volatility dependence at all past lags. Since Ŵ
∞
is semipositive definite, its eigenvectors are or- thogonal provided that the associated eigenvalues are distinct.
Consequently, any two PVCs, z
vt
= m
′ v
y
t
and z
ut
= m
′ u
y
t
, are uncorrelated if λ
2 v
= λ
2 u
. On the other hand, for PVC z
vt
associ- ated with a nonzero λ
2 v
, z
2 vt
may still be correlated with lagged values of z
2 ut
. Our limited experience indicates that such corre- lations, if exist, are of smaller magnitudes compared with those
of the observed y
2 it
series. Like the traditional PCA, an important application of the
proposed PVC analysis is to reduce the dimension in volatility modeling. To this end, the number of zero eigenvalues of Ŵ
∞
is of special interest. By Theorem 1, we have common volatility
components if the cumulative generalized kurtosis matrix Ŵ
∞
is not of full rank. We explore possible applications of this property
in Section 5
. 4.
SAMPLE PRINCIPAL VOLATILITY COMPONENTS In this section, we consider estimation of the generalized
kurtosis matrices and obtain the sample PVCs. We establish some consistency properties of the sample estimate of Ŵ
∞
. Fur- thermore, to verify that a given portfolio does not have ARCH
effects, we propose a test statistic and establish its asymptotic distribution.
To simplify the moment restrictions on y
t
for making statisti- cal inference of Ŵ
∞
or Ŵ
m
, we follow the approach of Matteson and Tsay
2011 by employing the Huber’s function of the cross-
product variable y
i,t −ℓ
y
j,t −ℓ
: x
ij,t −ℓ
= ⎧
⎪ ⎨
⎪ ⎩
y
i,t −ℓ
y
j,t −ℓ
if |y
i,t −ℓ
y
j,t −ℓ
| ≤ c
2
2c √
y
i,t −ℓ
y
j,t −ℓ
− c
2
if y
i,t −ℓ
y
j,t −ℓ
c
2
c
2
− 2c |y
i,t −ℓ
y
j,t −ℓ
| if y
i,t −ℓ
y
j,t −ℓ
−c
2
, 8
where c is a prespecified constant. Note that the necessary part of Theorem 1 continues to hold
if we substitute y
i,t −ℓ
y
j,t −ℓ
in Ŵ
ℓ
by the Huber’s function x
ij,t −ℓ
of Equation 8
. The sufficient part of the theorem, however, may not hold. This weakness, however, can be out-weighted
by the gain obtained in relaxing the moment conditions needed for the estimation and testing discussed in Theorems 2 and 3.
In what follows, we adopt the x
ij,t −ℓ
of Equation 8
to de- fine the sample generalized kurtosis matrix using Equation
3 and the corresponding cumulative generalized kurtosis matrix.
Obviously, x
ij,t −ℓ
approaches y
i,t −ℓ
y
j,t −ℓ
as c increases. 4.1
Estimation
Given the data {r
1
, . . . , r
n
} of a stationary process r
t
, we
employ the normalized data { y
1
, . . . , y
n
}, where y
t
=
−12
r
t
with being the sample covariance matrix of r
t
. The normal- ization is taken to increase the numerical stability of estimation
because asset returns tend to be small. Let
cov y
t
y
′ t
, x
ij,t −ℓ
= 1
n
n t =ℓ+1
y
t
y
′ t
− ¯Y x
ij,t −ℓ
− ¯x
ij
, where ¯
Y and ¯ x
ij
are the sample mean of y
t
y
′ t
and x
ij,t
, respec- tively. This is the sample counterpart of the lag-ℓ generalized
kurtosis matrix with the Huber’s function of the lagged cross- product variable. We estimate Ŵ
m
by Ŵ
m
=
m ℓ=1
k i=1
k j =i
1 − ℓ
n
2
cov
2
y
t
y
′ t
, x
ij,t −ℓ
. 9
Suppose y
t
is stationary with finite sixth moment and the as- sumption
A.1 in the Appendix holds. Then, for a fixed positive
integer m ∞, Ŵ
m
= Ŵ
m
+ O
p
1 √
n .
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We estimate Ŵ
∞
by Ŵ
n
=
n−1 ℓ=1
k i=1
k j =i
w
⋆
ℓ m
n
cov
2
y
t
y
′ t
, x
ij,t −ℓ
, 10
where w
⋆
ℓm
n
is a prespecified smoothing function and m
n
is a positive real number depending on n.
Under the regularity conditions used in spectral density esti- mation Hannan
1970 , and m
n
n → 0, m
n
→ ∞ as n → ∞, we have
Ŵ
n
= Ŵ
∞
+ O
p
a
n
, 11
where a
n
is a function of m
n
and n. Some guidelines for choosing m
n
are as follows: the convergence rate of Ŵ
n
of Equation 11
depends on the minimal mean squared error E Ŵ
n
− Ŵ
∞ 2
. For a specific smoothing function w
∗
., we can select m
n
care- fully to achieve the minimal mean squared error. For exam-
ple, if w
⋆
is the Bartlett’s smoothing function, the best choice of m
n
is m
n
= n
13
such that a
n
= n
−13
, and if w
⋆
is the Daniell’s smoothing function, one can choose m
n
= n
15
such that a
n
= n
−25
. See Hannan 1970
for details. For simplicity, we use the estimate
Ŵ
m
of Equation 9
in our applications. We check the stability of the results with several
choices of m. For a given estimate Ŵ
m
, we perform eigenvalue– eigenvector analysis to obtain the sample PVCs. Specifically, the
v th sample volatility component is ˆz
vt
= m
′ v
y
t
= m
′ v
−12
r
t
, where
m
v
is the normalized eigenvector of the vth eigenvalue of Ŵ
m
. Finally, we perform hypothesis testing to check the ARCH dependence of the sample PVCs at all lags, not just the first m
lags. 4.2
Testing An important application of the proposed PVC analysis is
dimension reduction. Here dimension reduction means finding linear combinations of y
t
that have no ARCH effects. Let M
1
be a k × s matrix consisting of eigenvectors associated with the s smallest eigenvalues of the
Ŵ
m
or Ŵ
∞
matrix. In other words,
M
1
gives rise to the k − s + 1th to the kth PVCs of y
t
. Our goal here is to consider test statistics for verifying that the
transformed series ˆe
t
= M
′ 1
y
t
indeed has no ARCH effects.
Many data-generating processes DGPs of y
t
can lead to
linear combinations of y
t
that have no ARCH effects. Consider, for instance, the case of common volatility components, which
is the focus of our application in Section 5
. Assume that
y
t
= H f
t
+ ǫ
t
, 12
where H is a k × r real-valued matrix of rank r, f
t
= f
1t
, . . . , f
rt ′
consists of r independent conditional het- eroscedastic processes,
{ǫ
t
} is a sequence of independent and identically distributed random vectors with mean zero and con-
stant positive-definite covariance matrix
ǫ
, and ǫ
t
is inde-
pendent of f
t
. Based on the definition of Equation 1
, each volatility component varf
it
|F
t −1
is a nontrivial function of el-
ements of { y
t −j
y
′ t −j
|j 0}. For this particular DGP, if r k,
then the volatility of y
t
is driven by the r-dimensional common
volatility components in f
t
. Let M
1
be a k × k − r real-valued
matrix such that M
′ 1
H = 0. Let e
t
= M
′ 1
y
t
. It is easy to see that
e
t
= M
′ 1
ǫ
t
and, hence, it has no ARCH effects. There are several tests available in the literature to check
for multivariate ARCH effects. In this section, we generalize the results of Ling and Li
1997 , Duchesne and Lalancette
2003 , Duchesne and Roy
2004 , and Hong
1996 to derive
two test statistics that are applicable to the PVC analysis. The generalization is mainly to deal with the fact that
M
1
is an
estimate and the dimension s of ˆe
t
could be smaller than k. We establish the limiting distributions of the proposed test statistics.
All proofs are given in the Appendix. 4.2.1
Ling–Li Test Statistic . Following Ling and Li
1997 , we define
ˆǫ
t
=
ˆe
′ t
V
−1
ˆe
t
− s
n t =1
ˆe
′ t
V
−1
ˆe
t
− s
2
n ,
ˆ x
t −j
= ˆ
h y
,t −j
− ¯h y
n t =1
ˆ h y
,t
− ¯h y
2
n ,
and ˆ
ρ
j,s
= 1
n
n t =j +1
ˆǫ
t
ˆ x
t −j
= 1n
n t =j +1
ˆe
′ t
V
−1
ˆe
t
− s ˆ
h y
,t −j
− ¯h y
n t =1
ˆe
′ t
V
−1
ˆe
t
− s
2
n
n t =1
ˆ h y
,t
− ¯h y
2
n ,
where ˆ
h y
,t
= ⎧
⎨ ⎩
ˆy
′ t
−1
ˆy
t
k, if ˆy
′ t
−1
ˆy
t
k ≤ c
2
2c ˆy
′ t
−1
ˆy
t
k − c
2
if ˆy
′ t
−1
ˆy
t
k c
2
, 13
and ¯ h y = 1n
n t =1
¯
h y
,t
, and V and
are, respectively, the
sample covariance matrix of ˆe
t
and y
t
. Furthermore, let ǫ
t
=
e
′ t
V
−1
e
t
− s σ
ǫ
and x
t −j
=
h y
,t −j
− Eh y
,t
σ
x
, where σ
2 ǫ
= Ee
′ t
V
−1
e
t
− s
2
and σ
2 x
= Eh y
,t
− Eh y
,t 2
.
Here e
t
and h y
,t
are the theoretical counterparts of ˆe
t
and ˆ h y
,t
, respectively.
To test that ǫ
t
is uncorrelated with x
t −1
, . . . , x
t −d
for some positive integer d, consider the hypothesis testing H
: ρ
1,s
= ρ
2,s
= · · · = ρ
d,s
= 0 versus H
a
: ρ
i,s
= 0 for some 1 ≤ i ≤ d, where ρ
j,s
is the correlation between ǫ
t
and x
t −j
. The Ling–Li test statistic for the hypotheses is
T
d,s
= n R
′ d,s
−1 d,s
R
d,s
, 14
where R
d,s
= ˆ ρ
1,s
, . . . , ˆ
ρ
d,s ′
and
d,s
is an estimated covari- ance matrix of
√ n
R
d,s
. More specifically,
d,s
= [φ
ij
]
d×d
with φ
jj
= 1 − jn for 1 ≤ j ≤ d, φ
ij
= 1 −
j n
cov ˆ x
t
, ˆ
x
t −j −i
for i j,
where cov ˆ
x
t
, ˆ
x
t −h
= 1n
n t =h+1
ˆ x
t
− ¯x ˆx
t −h
− ¯x with ¯x being the sample mean of ˆ
x
t
. Note that if s = k, then covx
t
, x
t −h
= 0 for h = 0 and
d,k
is asymptotically a
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diagonal matrix. In this case, T
d,s
reduces to the original Ling and Li test statistic.
Theorem 2. Suppose that y
t
is a k-dimensional weakly sta- tionary time series with ARCH dynamic given in Equation
1 .
Assume that the sixth moment of y
t
exists and M
1
is the
full-rank transformation matrix such that e
t
= M
′ 1
y
t
has no ARCH effects. Under the Assumption
A.1 in the Appendix, if
√ na
2 n
→ 0, where a
n
is the convergence rate of M
1
to M
1
as given in Equation
11 , then
√ n
−12 d,s
R
d,s
→
d
N 0, I
d×d
, and
T
d,s
= n R
′ d,s
−1 d,s
R
d,s
→
d
χ
2 d
, where →
d
denotes convergence in distribution. 4.2.2
Generalized Test Statistic . The T
d,s
statistic is de- signed to detect the serial volatility dependence in the first d
lags. In empirical applications, one would prefer a test statistic that can account for volatility dependence in all past lags. To this
end, we adopt the idea of Hong 1996
to derive a generalized Ling–Li test statistic. The generalized test statistic is defined as
G
p
n
,s
= n
n−1 j =1
w
2
jp
n
ˆ ρ
2 j,s
− M
n
w [2
V
n
w]
12
, 15
where M
n
w =
n−1 j =1
1 − jnw
2
jp
n
, V
n
w =
n−2 j =1
1 − jn
[1 − j + 1n]w
4
jp
n
,
= 1+2
∞ h=1
cov
2
x
t
, x
t −h
, p
n
is a function of n such that p
n
→ ∞ and p
n
n → 0 as n → ∞, and w. is a symmetric kernel function.
If y
t
is a k-dimensional process of independent and iden- tically distributed random variables, then s = k. In this case,
G
p
n
,s
reduces to the Hong’s statistic in which
= 1 because cov
2
x
t
, x
t −h
= 0 for all h 0. In this sense,
is used to ad-
just for the ARCH effects in the y
t
series. This quantity can be estimated by a smoothing method as
= 1 + 2
n−1 h=1
k hs
n
cov
2
ˆ x
t
, ˆ
x
t −h
, 16
where k. is a kernel function satisfying some regularity con- ditions such that
∗
= 1 + 2
n−1 h=1
k hs
n
cov
2
x
t
, x
t −h
is a consistent estimate of
and s
n
is a function of n such that s
n
→ ∞ and s
n
n → 0 as n → ∞. In this article, we use the Daniell function gx = sinπxπx for both wx and kx.
Theorem 3. Suppose that y
t
is a k-dimensional weakly sta- tionary process with ARCH effects governed by Equation
1
and has finite sixth moment. Let M
1
be a constant full-rank
k × s transformation matrix such that e
t
= M
′ 1
y
t
has no ARCH effects. Assume that
M
1
is a consistent estimate of M
1
and
ˆe
t
= M
′ 1
y
t
. Under the Assumptions A.1
and A.2
of the Ap- pendix, if p
n
→ ∞, p
n
n → 0,
na
4 n
√ p
n
→ 0, and p
n
a
2 n
→ 0 as n → ∞, then G
p
n
,s
→
d
N 0, 1, where G
p
n
,s
is the test statis- tic in Equation
15 and a
n
is given in Equation 11
. 4.2.3
Testing for ARCH Effects . For a given dataset, let
Ŵ
m
be the sample estimate of the cumulative generalized kur- tosis matrix. Further, let
M = [ m
1
, . . . , m
k
] be the matrix of standardized eigenvectors such that
Ŵ
m
m
v
= λ
2 v
m
v
, where λ
2 1
≥ λ
2 2
≥ · · · ≥ λ
2 k
are the eigenvalues. Since zero eigenvalues of
Ŵ
m
give rise to components without ARCH effects, we con- sider the following test procedure to detect linear combinations
of y
t
that have no ARCH effects. Let s be the number of lin-
ear combinations of y
t
that have no ARCH effects. Let M
1
= [
m
k
, . . . , m
k−s+1
] be the k × s matrix consisting of the last s columns of
M, that is, consisting of the standardized eigen-
