07350015%2E2014%2E885436

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Comment
Philip L. H. Yu & Guodong Li
To cite this article: Philip L. H. Yu & Guodong Li (2014) Comment, Journal of Business &
Economic Statistics, 32:2, 166-167, DOI: 10.1080/07350015.2014.885436
To link to this article: http://dx.doi.org/10.1080/07350015.2014.885436

Published online: 16 May 2014.

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Journal of Business & Economic Statistics, April 2014

easy to implement. The dimension-reduction is often effective
when k is large.
Inspired by the idea in the article, I propose an alternative
way to define PVC below. I also make an observation on the
link between PVC and volatility factor models.
An Alternative Definition for PVC. An alternative approach
would be to replace Ŵ ∞ by


[E{(yt y′t − )I (B)}]2 ,
M=
B∈Bt

where Bt is a π -class and the σ -algebra generated by Bt is equal
to the filtration Ft−1 = σ (yt−1 , yt−2 , . . .). Then for any constant
vector b such that b′ Mb = 0, it holds that
var(b′ yt |Ft−1 ) = var(b′ yt ),
that is, b′ yt does not have conditional heteroscedasticity. Hence,
the PVC can be defined as a′1 yt , . . . , a′r yt , where a1 , . . . , ar are
the orthonormal eigenvectors of matrix M with the corresponding eigenvalues nonzero. This approach requires some mild moment conditions, and is free from model assumptions such as
(1). Note that the PVC are not necessarily independent with
each other.
We refer to Fan, Wang, and Yao (2008, sec. 2.2.1) for how to
choose Bt in practice.
Link to Factor Models. There is an innate connection between
the PVC approach and factor models for volatility. In fact, PVC
can be viewed as latent factsors that drive the dynamics of
conditional heteroscedasticity.

Let A = (a1 , . . . , ar ). Then A′ A = Ir . Let Ŵ = (A, B) be a
k × k orthogonal matrix, and
xt = A′ yt ,

and

ε t = BB′ yt .

Then xt is r-variate factor exhibiting conditional heteroscedasticity, and ε t is a vector white-noise satisfying
var(ε t |Ft−1 ) = var(εt ) =  ε .
Furthermore, it holds that
yt = ŴŴ ′ yt = Axt + ε t .
Let  y (t) = var(yt |Ft−1 ) and  x (t) = var(xt |Ft−1 ). Then it
holds that
 y (t) = A x (t)A′ +  ε .
This is the standard form of volatility factor models; see Tao
et al. (2011, sec. 2.3) and the references within. If some initial
estimates for  y (t) can be obtained, for example, from using
high-frequency data, A can be easily identified and estimated
based on a simple eigenanalysis. See Tao et al. (2011) for further

details on this approach.
REFERENCES
Fan, J., Wang, M., and Yao, Q. (2008), “Modelling Multivariate Volatilities via
Conditionally Uncorrelated Components,” Journal of the Royal Statistical
Society, Series B, 70, 679–702. [166]
Tao, M., Wang, Y., Yao, Q., and Zou, J. (2011), “Large Volatility
Matrix Inference via Combining Low-Frequency and High-Frequency
Approaches,” Journal of the American Statistical Association, 106,
1025–1040. [166]

Comment
Philip L. H. YU and Guodong LI
Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, China (plhyu@hku.hk;
gdli@hku.hk)

This article adopts the idea of principal component analysis
(PCA) to model multivariate volatility, and the principal volatility component (PVC) analysis is then proposed to search for
common volatility components among many financial time series. We congratulate Professors Tsay and Hu for this nice work,
and some of our thoughts are given as follows.
First, from the viewpoint of applications, the idea of PVC

is more like that of cointegration. We attempt to identify the
comovements of sequences {yt } in cointegration, where {yt } are
I (1) sequences, and the linearly transformed time series {m′ yt }
are stationary. While the PVC attempts to find the common
volatility components of {yt }, where {yt } are stationary and
have a time varying conditional variance matrix, the linearly
transformed time series {m′ yt } have a constant conditional variance matrix. As argued by the authors, it will be useful for
carry trade or hedging purposes if we can successfully identify
the common volatility components. Note that these common
volatility components actually are portfolios in finance, and we

here would like to mention another two applications in this aspect. First, it should be more accurate to estimate the volatility
based on historical data since it is not time varying, and we then
can better manage the risk of this portfolio. Second, suppose
that a portfolio with constant conditional variance is a factor
(e.g., the market portfolio), and we would like to construct a
portfolio which is neutral to this factor. We then can include an
additional constraint into the portfolio optimization, where the
constraint is that the optimized portfolio is uncorrelated with this
factor.

In the literature of multivariate conditional heteroscedasticity, there are several dimension reduction methods available,
and they include orthogonal GARCH models (Alexander 2001),
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.885436

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Yu and Li: Comment

167

independent component analysis GARCH (ICA-GARCH) models (Wu, Yu, and Li 2006), conditionally uncorrelated components (CUC) models (Fan, Wang, and Yao 2008), and dynamic
orthogonal components (DOC) models (Matteson and Tsay
2011). Typically, they decompose the multivariate time series
into a linear combination of (conditional) uncorrelated components which are then modeled by various univariate GARCHtype models. This can greatly reduce both the number of parameters and estimation costs. Unlike the aforementioned models,
the authors propose a new method to identify uncorrelated components, and it is based on the generalized kurtosis matrix
γℓ =

k 
k


cov2 ( yt y′t , xij,t−ℓ ),

i=1 j =i

where xij,t−ℓ is a function of yi,t−ℓ yj,t−ℓ . A similar idea can
be found in the joint approximate diagonalization of eigenmatrices (JADE), proposed by Cardoso and Souloumiac (1993),
to identify independent components. The JADE approach takes
into account the fourth-order cumulants,

where the first term is the same as the (p, q)th entry of γℓ,ij
matrix when xij,t−ℓ = yi,t−ℓ yj,t−ℓ . Adopting the assumption
used in the article that E( yt |Ft−1 ) = 0, it is easy to see that
cum(zpt , zqt , yi,t−ℓ , yj,t−ℓ ) = cov(zpt zqt , yi,t−ℓ yj,t−ℓ ).
 ′ yt has no
Finally, it is important to check whether 
et = M

1

ARCH effect in this article, where M1 is a k × s matrix consisting of eigenvectors associated with the s smallest eigenvalues
et , and 
xt = g(yt ), where 
V is the
εt = 
e′t 
V−1
of 
Ŵ ∞ . Denote 
variance matrix, and g(·) is a function. The authors construct
two tests for ARCH effects, and they both depend on the samxt−j with j > 0. It
ple correlation coefficients between 
εt and 
may also be of interest to consider tests based on the autocorrelation function (ACF) of {
εt }, that is, the sample correlation
coefficients between 
εt and 
εt−j with j > 0. Some commonly

used portmanteau tests can then be applied; see Li (2004), Li
and Li (2005), etc. On the other hand, some information will be
missed when we transform the multivariate sequence {
et } into
a sequence of scalar variables {
εt }, and it may be of interest to
consider a multivariate portmanteau test here; see Mahdi and
McLeod (2012).

cum(yit , yj t , ypt , yqt )
= E(yit yj t ypt yqt ) − E(yit yj t )E(ypt yqt )
− E(yit ypt )E(yj t yqt ) − E(yit yqt )E(yj t ypt ),
and the fourth-order cumulant tensor,
Qij (M) =

k 
k


mpq cum(yit , yj t , ypt , yqt ).

p=1 q=1

The independent components can be found by using the eigenmatrices of the above tensor. Both the JADE and the PVC share
a few similarities. First of all, both made use of fourth-order moments in their component estimation. Second, for z t = M ′ yt ,
we have that
cum(zpt , zqt , yi,t−ℓ , yj,t−ℓ ) = cov(zpt zqt , yi,t−ℓ yj,t−ℓ )
− cov(zpt , yi,t−ℓ )cov(zqt , yj,t−ℓ )
− cov(zpt , yj,t−ℓ )cov(zqt , yi,t−ℓ ),

REFERENCES
Alexander, C. O. (2001), “Orthogonal GARCH,” in Mastering Risk (Vol. 2), ed.
C. O. Alexander, Harlow, UK: Financial Times-Prentice Hall. [166]
Cardoso, J. F., and Souloumiac, A. (1993), “Blind Beamforming for NonGaussian Signals,” IEE Proceedings-F, 140, 362–370. [167]
Fan, J., Wang, M., and Yao, Q. (2008), “Modelling Multivariate Volatilties via
Conditionally Uncorrelated Components,” Journal of the Royal Statistical
Society, Series B, 70, 679–702. [167]
Li, G., and Li, W. K. (2005), “Diagnostic Checking for Time Series Models With
Conditional Heteroscedasticity Estimated by the Least Absolute Deviation
Approach,” Biometrika, 92, 691–701. [167]

Li, W. K. (2004), Diagnostic Checks in Time Series, Boca Raton, FL: Chapman
& Hall. [167]
Mahdi, E., and McLeod, A. I. (2012), “Improved Multivariate Portmanteau
Test,” Journal of Time Series Analysis, 33, 211–222. [167]
Matteson, D. S., and Tsay, R. S. (2011), “Dynamic Orthgonal Components for
Multivariate Time Series,” Journal of the American Statistical Association,
106, 1450–1463. [167]
Wu, E. H. C., Yu, P. L. H., and Li, W. K. (2006), “Value at Risk Estimation Using
Ica-Garch Models,” International Journal of Neural Systems, 16, 371–382.
[167]