07350015%2E2014%2E887016
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Shiqing Ling
To cite this article: Shiqing Ling (2014) Comment, Journal of Business & Economic Statistics,
32:2, 165-165, DOI: 10.1080/07350015.2014.887016
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887016
Published online: 16 May 2014.
Submit your article to this journal
Article views: 171
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: 20:39
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:39 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 ŷit defined as follows:
if yit > a
a
−a ≤ yit ≤ a
ŷit = 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
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Shiqing Ling
To cite this article: Shiqing Ling (2014) Comment, Journal of Business & Economic Statistics,
32:2, 165-165, DOI: 10.1080/07350015.2014.887016
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887016
Published online: 16 May 2014.
Submit your article to this journal
Article views: 171
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: 20:39
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:39 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 ŷit defined as follows:
if yit > a
a
−a ≤ yit ≤ a
ŷit = 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