07350015%2E2014%2E902236
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Rejoinder
Yu-Pin Hu & Ruey S. Tsay
To cite this article: Yu-Pin Hu & Ruey S. Tsay (2014) Rejoinder, Journal of Business & Economic
Statistics, 32:2, 176-177, DOI: 10.1080/07350015.2014.902236
To link to this article: http://dx.doi.org/10.1080/07350015.2014.902236
Published online: 16 May 2014.
Submit your article to this journal
Article views: 417
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: 20:42
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:42 11 January 2016
176
Journal of Business & Economic Statistics, April 2014
Rejoinder: Principal Volatility Component
Analysis
Yu-Pin HU
Department of International Business Studies, National Chi Nan University, Taiwan (huyp@ncnu.edu.tw)
Ruey S. TSAY
Booth School of Business, University of Chicago, 5807 South Woodlawn Avenue, Chicago, IL 60637
(ruey.tsay@chicagobooth.edu)
We express our sincere thanks to all the discussants for their
constructive discussions and encouragement. In particular, we
appreciate their proposals of alternative approaches to finding
common volatility factors and their suggestions for improving
the estimation of the generalized kurtosis matrix. For obvious
reasons, we cannot answer satisfactorily every point or question
raised in the discussions. We shall address some common and
important issues of the discussions. We use the same notation
as that in the article.
1.
ROBUSTNESS AND MOMENT CONDITIONS
We share the discussants’ concerns about the higher-order
moment condition imposed in the article. Different conditions
are used in estimating the generalized kurtosis matrix and in
testing the ARCH effects. The test statistics require existence of
the sixth moments, whereas estimation uses the fourth moment.
Our use of the Huber transform is just a first attempt to relax the
moment condition. Thus, we are encouraged by the suggestion
of Professors Franke and Ling on using a bounded transformation and intend to investigate further this important issue. We
are also glad to see the expanded simulation study by Professors
Andreou and Ghysels. Their use of integrated volatility series,
heavy-tailed innovations, and GJR-ARCH models is informative. As expected, the performance of the proposed PVC analysis deteriorates when the moment condition is violated. On the
other hand, their simulation result also provides encouragement
for further investigation in using a more robust transformation
in estimating the generalized kurtosis matrix.
Following the suggestion of Professor Franke, we consider
the Hampel filter
h(xij,t−ℓ ) = σij sign(xij,t−ℓ )ρ(|xij,t−ℓ |/σij ),
(1)
where sign(x) denotes the sign of the real number x, xij,t−ℓ =
yi,t−ℓ yj,t−ℓ , σij is the sample standard deviation of xij,t−ℓ , and
the Hampel’s ρ(.) is defined as
⎧1
⎪
z2
for z ≤ a
⎪
⎪
⎪
2
⎪
⎪
⎪
1 2
⎪
⎪
for a < z ≤ b
⎪
⎨ az − 2 a
2
ρ(z) = a(cz − 0.5z )
⎪
⎪
⎪
⎪
c−b
⎪
⎪
⎪
⎪
+
0.5a(ab − ac − b2 )/(c − b) for b < z ≤ c
⎪
⎪
⎩
0.5a(b + c − a)
for z > c,
where a = 1.5, b = 3.5, and c = 8. Using the Hampel filter, we
apply the PVC analysis to the seven exchange series considered in the article and obtain similar results. Figure 1 shows the
time plots of the seventh principal volatility component of the
exchange rate series with and without the Hampel filter. From
the plots, the impact of using the filter on the estimation of noARCH linear combination is negligible As a matter of fact, the
correlation between the two sample principal volatility components of Figure 1 is 0.992. The filter, however, affects some of
the PVC associated with large eigenvalues. Since the Hampel
filter is bounded, we only need the finite fourth moments for
the existence of the generalized kurtosis matrix Ŵ ∞ . Note that
the Hampel filter or the Huber transform is applied to xij,t−ℓ
only in estimating the generalized kurtosis matrix; otherwise,
the sufficient part of Theorem 1 does not hold. In general, we
agree with the discussants that additional research similar to the
robustification of the classical principal component analysis is
needed.
2.
ALTERNATIVE APPROACHES
Let yt be a k-dimensional financial time series. We agree
with Professors Yao, H¨ardle, and Wang that the conditional
heteroscedasticity of yt implies that yt y′t is correlated with its
lagged values yt−ℓ y′t−ℓ for some ℓ > 0. Our use of Equation
(1) is simply a parametric form to show the stated linear dependence. Alternative formulations are possible. The matrix M of
Professor Yao is more general and interesting. One can indeed
define
γ x = cov( yt y′t , x) = E[( yt y′t − )x],
(2)
where x is a univariate random variable generated by F t−1 ,
and consider the null space of γ x . In particular, let E be the
intersection of the null space of γ x for all x ⊆ F t−1 . Then, E
is spanned by a full-rank k × (k − r) matrix M 1 if and only if
the transformed series M ′1 yt has no ARCH effects. Using this
result and the suggestion of Professor Yao, one can define an
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.902236
177
−0.02
0.01
4. INDEPENDENT COMPONENTS ANALYSIS (ICA)
AND JOINT APPROXIMATE DIAGONALIZATION OF
EIGEN-MATRICES (JADE)
0
100
200
300
weeks
400
500
600
0
100
200
300
weeks
400
500
600
−0.02
0.01
pvc7−Hampel
pvc7
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:42 11 January 2016
Hu and Tsay: Rejoinder
Figure 1. Time plots of the 7th sample principal volatility components for the weekly log returns of seven foreign exchange rates with
and without using the Hampel filter. The upper plot is with the Hampel
filter.
alternative generalized kurtosis matrix as
γ x γ ′x .
Ŵ ∗∞ =
x∈F t−1
In practice, one may choose x to represent a proper partition
of the data to obtain a sample estimate of Ŵ ∗∞ for eigenvalue–
eigenvector analysis. The efficacy of this alternative approach
deserves a careful study. Like Professors H¨ardle and Wang,
we have considered the covariances between vec( yt y′t ) and its
lagged values. However, the resulting linear combinations often involve the cross-product terms of yt y′t . As such, these
linear combinations may not form proper portfolios in financial applications. On the other hand, the suggestion of using an
asymmetric clip is valuable and deserves a careful investigation
as leverage effect is important in financial applications; see the
discussion of Professors Andreou and Ghysels.
3.
COMMON FEATURES AND FACTOR MODELS
As noted by Professors Yu, Li, and Ling, finding linear combinations of asset returns that have no conditional heteroscedasticity belongs to the general concept of common features or
co-movements in multivariate time series analysis. The basic
idea is similar to that of co-integration, even though the tools
used and the statistical distributions involved are different. The
issues of interest are the definition of common features and ways
to detect them. In the article, we provide a definition of conditional heteroscedasticity and use generalized kurtosis matrix to
extract the common feature. The concept of principal volatility
component (PVC) analysis is also highly related to common
factors. The discussion of Professor Yao concerning the link
between PVC and common factors is insightful. Our concern
lies in the assumption that the matrix A is of lower rank, that
is, r < k. It seems that some test statistics need to be derived to
check the rank of the sample estimate of A.
Professors H¨ardle, Wang, Yu, and Li all mentioned the use
of independent components in multivariate volatility modeling.
This is indeed an interesting idea and several related approaches
have been proposed in the literature. We have pursued related
research in recent years; see Matteson and Tsay (2011). Our
limited experience shows that it is important in practice to check
the existence of independent components before applying most
of the available ICA-based methods. We thank Professors Yu
and Li for pointing out the link between PVC and JADE. The
connection would be useful in understanding properties of the
proposed PVC analysis.
5.
MULTIVARIATE VOLATILITY MODELS
We thank Professor McAleer for his pursuit of perfection in
multivariate volatility models. His points should be taken seriously. Indeed, many properties of some widely used multivariate volatility models are yet to be developed. From a practical
viewpoint, one often makes a compromise between simplicity and theoretical completeness. Our use of the dynamic (or
time-varying) conditional correlation models in the empirical
demonstration is driven by its simplicity. We believe that the
theory for useful multivariate volatility models will be fully developed in due course. On the other hand, his point about the
connection between PCA for unit-root nonstationary time series and PVC for weakly stationary time series is well taken.
Note that one can study properties of the subspace of no ARCH
effects in yt , if exists, without investigating properties of the
subspace of yt that has ARCH effects. In spirit, this is similar
to studying the co-integrating series, which is stationary, of a
co-integrated system.
6.
USE OF ADDITIONAL DATA
Professors Andreou, Gyhsels, and Yao all mentioned using
high-frequency data to estimate common volatility factors. This
is yet another sensible approach. It would be interesting to compare various approaches to extracting common volatility factors
and to quantify the contributions of realized volatility and covariances. When the realized volatility series are nonstationary,
the traditional PCA tends to provide a few dominating linear
combinations. It is then not surprising to see that a small number of principal components explain a high percentage of the
variations in volatility. However, this phenomenon could be
misleading in real applications similar to a high R 2 in linear
regressions involving unit-root time series.
REFERENCE
Matteson, D. S., and Tsay, R. S. (2011), “Dynamic Orthogonal Components for
Multivariate Time Series,” Journal of the American Statistical Association,
106, 1450–1463. [177]
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Rejoinder
Yu-Pin Hu & Ruey S. Tsay
To cite this article: Yu-Pin Hu & Ruey S. Tsay (2014) Rejoinder, Journal of Business & Economic
Statistics, 32:2, 176-177, DOI: 10.1080/07350015.2014.902236
To link to this article: http://dx.doi.org/10.1080/07350015.2014.902236
Published online: 16 May 2014.
Submit your article to this journal
Article views: 417
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: 20:42
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:42 11 January 2016
176
Journal of Business & Economic Statistics, April 2014
Rejoinder: Principal Volatility Component
Analysis
Yu-Pin HU
Department of International Business Studies, National Chi Nan University, Taiwan (huyp@ncnu.edu.tw)
Ruey S. TSAY
Booth School of Business, University of Chicago, 5807 South Woodlawn Avenue, Chicago, IL 60637
(ruey.tsay@chicagobooth.edu)
We express our sincere thanks to all the discussants for their
constructive discussions and encouragement. In particular, we
appreciate their proposals of alternative approaches to finding
common volatility factors and their suggestions for improving
the estimation of the generalized kurtosis matrix. For obvious
reasons, we cannot answer satisfactorily every point or question
raised in the discussions. We shall address some common and
important issues of the discussions. We use the same notation
as that in the article.
1.
ROBUSTNESS AND MOMENT CONDITIONS
We share the discussants’ concerns about the higher-order
moment condition imposed in the article. Different conditions
are used in estimating the generalized kurtosis matrix and in
testing the ARCH effects. The test statistics require existence of
the sixth moments, whereas estimation uses the fourth moment.
Our use of the Huber transform is just a first attempt to relax the
moment condition. Thus, we are encouraged by the suggestion
of Professors Franke and Ling on using a bounded transformation and intend to investigate further this important issue. We
are also glad to see the expanded simulation study by Professors
Andreou and Ghysels. Their use of integrated volatility series,
heavy-tailed innovations, and GJR-ARCH models is informative. As expected, the performance of the proposed PVC analysis deteriorates when the moment condition is violated. On the
other hand, their simulation result also provides encouragement
for further investigation in using a more robust transformation
in estimating the generalized kurtosis matrix.
Following the suggestion of Professor Franke, we consider
the Hampel filter
h(xij,t−ℓ ) = σij sign(xij,t−ℓ )ρ(|xij,t−ℓ |/σij ),
(1)
where sign(x) denotes the sign of the real number x, xij,t−ℓ =
yi,t−ℓ yj,t−ℓ , σij is the sample standard deviation of xij,t−ℓ , and
the Hampel’s ρ(.) is defined as
⎧1
⎪
z2
for z ≤ a
⎪
⎪
⎪
2
⎪
⎪
⎪
1 2
⎪
⎪
for a < z ≤ b
⎪
⎨ az − 2 a
2
ρ(z) = a(cz − 0.5z )
⎪
⎪
⎪
⎪
c−b
⎪
⎪
⎪
⎪
+
0.5a(ab − ac − b2 )/(c − b) for b < z ≤ c
⎪
⎪
⎩
0.5a(b + c − a)
for z > c,
where a = 1.5, b = 3.5, and c = 8. Using the Hampel filter, we
apply the PVC analysis to the seven exchange series considered in the article and obtain similar results. Figure 1 shows the
time plots of the seventh principal volatility component of the
exchange rate series with and without the Hampel filter. From
the plots, the impact of using the filter on the estimation of noARCH linear combination is negligible As a matter of fact, the
correlation between the two sample principal volatility components of Figure 1 is 0.992. The filter, however, affects some of
the PVC associated with large eigenvalues. Since the Hampel
filter is bounded, we only need the finite fourth moments for
the existence of the generalized kurtosis matrix Ŵ ∞ . Note that
the Hampel filter or the Huber transform is applied to xij,t−ℓ
only in estimating the generalized kurtosis matrix; otherwise,
the sufficient part of Theorem 1 does not hold. In general, we
agree with the discussants that additional research similar to the
robustification of the classical principal component analysis is
needed.
2.
ALTERNATIVE APPROACHES
Let yt be a k-dimensional financial time series. We agree
with Professors Yao, H¨ardle, and Wang that the conditional
heteroscedasticity of yt implies that yt y′t is correlated with its
lagged values yt−ℓ y′t−ℓ for some ℓ > 0. Our use of Equation
(1) is simply a parametric form to show the stated linear dependence. Alternative formulations are possible. The matrix M of
Professor Yao is more general and interesting. One can indeed
define
γ x = cov( yt y′t , x) = E[( yt y′t − )x],
(2)
where x is a univariate random variable generated by F t−1 ,
and consider the null space of γ x . In particular, let E be the
intersection of the null space of γ x for all x ⊆ F t−1 . Then, E
is spanned by a full-rank k × (k − r) matrix M 1 if and only if
the transformed series M ′1 yt has no ARCH effects. Using this
result and the suggestion of Professor Yao, one can define an
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.902236
177
−0.02
0.01
4. INDEPENDENT COMPONENTS ANALYSIS (ICA)
AND JOINT APPROXIMATE DIAGONALIZATION OF
EIGEN-MATRICES (JADE)
0
100
200
300
weeks
400
500
600
0
100
200
300
weeks
400
500
600
−0.02
0.01
pvc7−Hampel
pvc7
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:42 11 January 2016
Hu and Tsay: Rejoinder
Figure 1. Time plots of the 7th sample principal volatility components for the weekly log returns of seven foreign exchange rates with
and without using the Hampel filter. The upper plot is with the Hampel
filter.
alternative generalized kurtosis matrix as
γ x γ ′x .
Ŵ ∗∞ =
x∈F t−1
In practice, one may choose x to represent a proper partition
of the data to obtain a sample estimate of Ŵ ∗∞ for eigenvalue–
eigenvector analysis. The efficacy of this alternative approach
deserves a careful study. Like Professors H¨ardle and Wang,
we have considered the covariances between vec( yt y′t ) and its
lagged values. However, the resulting linear combinations often involve the cross-product terms of yt y′t . As such, these
linear combinations may not form proper portfolios in financial applications. On the other hand, the suggestion of using an
asymmetric clip is valuable and deserves a careful investigation
as leverage effect is important in financial applications; see the
discussion of Professors Andreou and Ghysels.
3.
COMMON FEATURES AND FACTOR MODELS
As noted by Professors Yu, Li, and Ling, finding linear combinations of asset returns that have no conditional heteroscedasticity belongs to the general concept of common features or
co-movements in multivariate time series analysis. The basic
idea is similar to that of co-integration, even though the tools
used and the statistical distributions involved are different. The
issues of interest are the definition of common features and ways
to detect them. In the article, we provide a definition of conditional heteroscedasticity and use generalized kurtosis matrix to
extract the common feature. The concept of principal volatility
component (PVC) analysis is also highly related to common
factors. The discussion of Professor Yao concerning the link
between PVC and common factors is insightful. Our concern
lies in the assumption that the matrix A is of lower rank, that
is, r < k. It seems that some test statistics need to be derived to
check the rank of the sample estimate of A.
Professors H¨ardle, Wang, Yu, and Li all mentioned the use
of independent components in multivariate volatility modeling.
This is indeed an interesting idea and several related approaches
have been proposed in the literature. We have pursued related
research in recent years; see Matteson and Tsay (2011). Our
limited experience shows that it is important in practice to check
the existence of independent components before applying most
of the available ICA-based methods. We thank Professors Yu
and Li for pointing out the link between PVC and JADE. The
connection would be useful in understanding properties of the
proposed PVC analysis.
5.
MULTIVARIATE VOLATILITY MODELS
We thank Professor McAleer for his pursuit of perfection in
multivariate volatility models. His points should be taken seriously. Indeed, many properties of some widely used multivariate volatility models are yet to be developed. From a practical
viewpoint, one often makes a compromise between simplicity and theoretical completeness. Our use of the dynamic (or
time-varying) conditional correlation models in the empirical
demonstration is driven by its simplicity. We believe that the
theory for useful multivariate volatility models will be fully developed in due course. On the other hand, his point about the
connection between PCA for unit-root nonstationary time series and PVC for weakly stationary time series is well taken.
Note that one can study properties of the subspace of no ARCH
effects in yt , if exists, without investigating properties of the
subspace of yt that has ARCH effects. In spirit, this is similar
to studying the co-integrating series, which is stationary, of a
co-integrated system.
6.
USE OF ADDITIONAL DATA
Professors Andreou, Gyhsels, and Yao all mentioned using
high-frequency data to estimate common volatility factors. This
is yet another sensible approach. It would be interesting to compare various approaches to extracting common volatility factors
and to quantify the contributions of realized volatility and covariances. When the realized volatility series are nonstationary,
the traditional PCA tends to provide a few dominating linear
combinations. It is then not surprising to see that a small number of principal components explain a high percentage of the
variations in volatility. However, this phenomenon could be
misleading in real applications similar to a high R 2 in linear
regressions involving unit-root time series.
REFERENCE
Matteson, D. S., and Tsay, R. S. (2011), “Dynamic Orthogonal Components for
Multivariate Time Series,” Journal of the American Statistical Association,
106, 1450–1463. [177]