07350015%2E2014%2E887014
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Qiwei Yao
To cite this article: Qiwei Yao (2014) Comment, Journal of Business & Economic Statistics, 32:2,
165-166, DOI: 10.1080/07350015.2014.887014
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887014
Published online: 16 May 2014.
Submit your article to this journal
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Ling and Yao: Comments
165
Comment
Shiqing LING
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong, China (maling@ust.hk)
Congratulations to the authors for their very interesting work.
The authors define a cumulative generalized kurtosis matrix to
summarize the volatility dependence of multivariate time series
in (2) and (3). It is the first generalization of the autocovariancematrix in the field of time series. Its importance lies in that it not
only can measure the dependence of the ARCH effect but also
can reduce the moment condition of yt . A lot of evidence shows
that the fourth moment of financial time series does not exist,
see ,for example, Zhu and Ling (2011). The classical statistical
inference does not work for the ARCH-type time series if its
fourth moment does not exist; see Zhu and Ling (2013). This
concept opens a new direction to study the financial time series
in the future.
Theorem 1 gives an important fact that the linear combination
of several time series may not have the ARCH effect even if each
individual time series has an ARCH effect. This means that one
can select a stable portfolios in funding management such that
its volatilities do not depend on the time horizon. Using this,
one can also reduce the dimension of the parameter space in
modeling vector ARCH-type time series. Its importance may be
comparable to the concept of co-integration in the field of time
series.
The authors proposed a principal volatility component approach to identify the linear combination of yt such that it does
not have the ARCH effect. Huber’s function is used in (8) to
reduce the moment condition. It turns out that the test statistic
in Ling and Li (1997) and its generalization can be used to test
if a linear combination of yt has the ARCH effect.
Based on the authors’ idea, I believe that there exist some
other approaches to identify the linear combinations or make
a dimension reduction. More research can be done after this
excellent work. For example, if we replace the component yit
of yt in (9) with yˆit defined as follows:
⎧
if yit > a
⎪
⎨ a
−a ≤ yit ≤ a
yˆit = yit
⎪
⎩
−a
if yit < −a,
then no moment condition is required, where a is a prespecified
constant. Is this possible, and how does it affect the kurtosis
matrix of yt ?
ACKNOWLEDGMENT
The author thanks the funding support in part from Hong
Kong RGC Grants (numbered HKUST641912 and 603413).
REFERENCES
Ling, S., and Li, W. K. (1997), “Diagnostic Checking of Nonlinear Multivariate Time Series With Multivariate ARCH Errors,” Journal of Time Series
Analysis, 18, 447–464. [165]
Zhu, K., and Ling, S. (2011), “Global Self-Weighted and Local Quasi-Maximum
Exponential Likelihood Estimators for ARMA-GARCH/IGARCH Models,” The Annals of Statistics, 39, 2131–2163. [165]
——— (2013), “Inference for ARMA Models With Unknown-Form and HeavyTailed G/ARCH-Type Noises,” Working paper, Department of Mathematics,
HKUST. [165]
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.887016
Comment
Qiwei YAO
Department of Statistics, London School of Economics, Houghton Street WC2A 2AE, London (q.yao@lse.ac.uk)
The authors are to be congratulated for tackling a challenging
statistical problem with important financial applications, that
is, modeling multivariate volatility processes via dimensionreduction. By introducing the so-called principal volatility components (PVC), they are able to identify a lower-dimensional
space within which the dynamics of conditional heteroscedasticity confines.
Technically the authors look at the correlations between
yt y′t − and its lagged values in terms of the so-called gen-
eralized kurtosis matrices. To link those correlations to the conditional heteroscedasticity, they assume a vectorized ARCH(∞)
model (1). The Huber truncation (8) is employed to refrain the
moment condition required. The whole approach is simple and
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.887014
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:40 11 January 2016
166
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
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Qiwei Yao
To cite this article: Qiwei Yao (2014) Comment, Journal of Business & Economic Statistics, 32:2,
165-166, DOI: 10.1080/07350015.2014.887014
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887014
Published online: 16 May 2014.
Submit your article to this journal
Article views: 121
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RA JA ALI HA JI
TANJUNGPINANG, KEPULAUAN RIAU]
Date: 11 January 2016, At: 20:40
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:40 11 January 2016
Ling and Yao: Comments
165
Comment
Shiqing LING
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong, China (maling@ust.hk)
Congratulations to the authors for their very interesting work.
The authors define a cumulative generalized kurtosis matrix to
summarize the volatility dependence of multivariate time series
in (2) and (3). It is the first generalization of the autocovariancematrix in the field of time series. Its importance lies in that it not
only can measure the dependence of the ARCH effect but also
can reduce the moment condition of yt . A lot of evidence shows
that the fourth moment of financial time series does not exist,
see ,for example, Zhu and Ling (2011). The classical statistical
inference does not work for the ARCH-type time series if its
fourth moment does not exist; see Zhu and Ling (2013). This
concept opens a new direction to study the financial time series
in the future.
Theorem 1 gives an important fact that the linear combination
of several time series may not have the ARCH effect even if each
individual time series has an ARCH effect. This means that one
can select a stable portfolios in funding management such that
its volatilities do not depend on the time horizon. Using this,
one can also reduce the dimension of the parameter space in
modeling vector ARCH-type time series. Its importance may be
comparable to the concept of co-integration in the field of time
series.
The authors proposed a principal volatility component approach to identify the linear combination of yt such that it does
not have the ARCH effect. Huber’s function is used in (8) to
reduce the moment condition. It turns out that the test statistic
in Ling and Li (1997) and its generalization can be used to test
if a linear combination of yt has the ARCH effect.
Based on the authors’ idea, I believe that there exist some
other approaches to identify the linear combinations or make
a dimension reduction. More research can be done after this
excellent work. For example, if we replace the component yit
of yt in (9) with yˆit defined as follows:
⎧
if yit > a
⎪
⎨ a
−a ≤ yit ≤ a
yˆit = yit
⎪
⎩
−a
if yit < −a,
then no moment condition is required, where a is a prespecified
constant. Is this possible, and how does it affect the kurtosis
matrix of yt ?
ACKNOWLEDGMENT
The author thanks the funding support in part from Hong
Kong RGC Grants (numbered HKUST641912 and 603413).
REFERENCES
Ling, S., and Li, W. K. (1997), “Diagnostic Checking of Nonlinear Multivariate Time Series With Multivariate ARCH Errors,” Journal of Time Series
Analysis, 18, 447–464. [165]
Zhu, K., and Ling, S. (2011), “Global Self-Weighted and Local Quasi-Maximum
Exponential Likelihood Estimators for ARMA-GARCH/IGARCH Models,” The Annals of Statistics, 39, 2131–2163. [165]
——— (2013), “Inference for ARMA Models With Unknown-Form and HeavyTailed G/ARCH-Type Noises,” Working paper, Department of Mathematics,
HKUST. [165]
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.887016
Comment
Qiwei YAO
Department of Statistics, London School of Economics, Houghton Street WC2A 2AE, London (q.yao@lse.ac.uk)
The authors are to be congratulated for tackling a challenging
statistical problem with important financial applications, that
is, modeling multivariate volatility processes via dimensionreduction. By introducing the so-called principal volatility components (PVC), they are able to identify a lower-dimensional
space within which the dynamics of conditional heteroscedasticity confines.
Technically the authors look at the correlations between
yt y′t − and its lagged values in terms of the so-called gen-
eralized kurtosis matrices. To link those correlations to the conditional heteroscedasticity, they assume a vectorized ARCH(∞)
model (1). The Huber truncation (8) is employed to refrain the
moment condition required. The whole approach is simple and
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.887014
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:40 11 January 2016
166
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