07350015%2E2014%2E898585
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Wolfgang Karl Härdle & Weining Wang
To cite this article: Wolfgang Karl Härdle & Weining Wang (2014) Comment, Journal of Business
& Economic Statistics, 32:2, 173-174, DOI: 10.1080/07350015.2014.898585
To link to this article: http://dx.doi.org/10.1080/07350015.2014.898585
Published online: 16 May 2014.
Submit your article to this journal
Article views: 119
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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]
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Hardle
and Wang: Comment
¨
173
Comment
¨
Wolfgang Karl HARDLE
Center for Applied Statistics and Economics, Humboldt-Universitat
¨ zu Berlin, Unter den Linden 6,
10099 Berlin, Germany; Singapore Management University, 50 Stamford Road, Singapore 178899
(haerdle@wiwi.hu-berlin.de)
Weining WANG
Center for Applied Statistics & Economics School of Business and Economics, Humboldt-Universitat
¨
zu Berlin, Unter den Linden 6, 10099 Berlin, Germany (wangwein@cms.hu-berlin.de)
The authors are to be congratulated for a timely and important contribution. The article proposes a novel principal volatility component (PVC) technique based on a generalized kurtosis matrix in a time series context. The proposed test statistics
allow deep insight into higher moments and tail behavior of
multivariate time series. The article considers a weak stationary
multivariate time series yt (k × 1) with finite fourth moments,
the lag l generalized kurtosis matrix is defined as
def
def
⊤
γl,ij γl,ij
, (1)
γl =
cov2 yt y⊤
t , xij,t−l =
i
j
i
j
where
def
γl,ij = cov yt y⊤
t , xij,t−l .
(2)
m⊤
v yt ,
where
The PVCs are then defined as linear combinations
the mv ’s are the vth eigenvectors of the cumulative generalized kurtosis matrix Γ∞ for general multivariate GARCH-type
def
models
(Γm for ARCH(m) effects in yt ) with Γ∞ = ∞
l=1 γl
def
(Γm = m
γ
).
Note
that
x
is
a
function
of
y
y
.
l
ij,t−l
i,t−l
j,t−l
l=1
The kurtosis matrix indicates the correlations and crosscorrelations between the current variance–covariance matrix
and its lagged one, and thus would be a four-dimensional object
(k × k × k × k). Nevertheless, the authors consider a k × k generalized kurtosis matrix, which sums up all the effects of a lagged
variance–covariance matrix. In some cases, one might like to
look at the componentwise effects, which requires alternatives
of defining a generalized kurtosis matrix. For example, one can
analyze the variance–covariance matrix between vec(yt y⊤
t ) and
2
2
vec(yt−l y⊤
),
whose
dimension
is
k
×
k
matrix.
Moreover,
t−l
to generalize the idea
of impulse response functions, one can
look at the matrix j cov2 (yt y⊤
t , xi0 j,t−l ) to isolate the lagged
variable i0 ’s(i0 = 1, . . . , k) contribution.
The article employs Huber’s function which is symmetric. One might by an asymmetric clip function address the
well-known leverage effect, which means that negative returns
increase future volatility by a larger amount than positive returns of the same magnitude. In particular, to model asymmetry
in the ARCH process, for example, as in GJRGARCH models
introduced by Glosten, Jagannathan, and Runkle (1993). For
−
−
may serve this propose,
yj,t−l
instance, setting xij,t−l = yi,t−l
−
where yj,t−l = yj,t−l only when yj,t−l < 0 (negative part of
yj,t−l ).
The idea of PVC is a decomposition of a (mixed) moment
matrix. In PCA, one considers the variance–covariance matrix,
which falls short on modeling a nonlinear and asymmetric
multivariate distribution. This fact reminds us of a strand of
literature on independent component analysis (ICA); see, for
example, Chen, H¨ardle, and Spokoiny (2007); Chen et al.
(2014). ICA looks for a projection that maximizes a nonGaussianity measure. Similarly, a generalized kurtosis matrix
in PVC is connected to measuring non-Gaussianity. However,
kurtosis does not provide the whole picture of a distribution
function, and therefore other perspectives of the conditional
distribution (e.g., conditional skewness and conditional quantile) may also be of interest, see, for example, Lanne and Pentti
(2007).
Another issue is possible nonstationarity in yt . In this situation, the nonstationarity can be modeled via switching parameters of a stationary model, see H¨ardle, Herwartz, and Spokoiny
(2003). The eigenvectors mv ’s would then be time varying
mvt . Accordingly, at time t one can adopt local adaptive techniques (see Spokoiny, Wang, and H¨ardle 2013) to identify a
local homogeneous interval [t − t0 , t], in which one may apply
PVC.
Once more we would like to congratulate the authors for
this great advance. We are sure that this work will create a
new strand of literature with implications on asset allocation:
portfolio choice and factor models. If one is interested in the
factors that have no ARCH effects, one can certainly employ
the presented technique. The factors isolated can be used as
factors in asset pricing model, taking into account of rare events
as in Martin (2013).
ACKNOWLEDGMENTS
The financial support From the Deutsche Forschungsge¨
meinschaft via SFB 649 “Okonomisches
Risiko,” HumboldtUniversit¨at zu Berlin and IRTG 1972 “High Dimensional Non
Stationary Time Series” is gratefully acknowledged.
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.898585
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:41 11 January 2016
174
Journal of Business & Economic Statistics, April 2014
REFERENCES
Chen, Y., Chen, R.-B., and H¨ardle, W. K. (2014), “TVICA—Time Varying
Independent Component Analysis and Its Application to Financial Data,”
Journal of Computational Statistics and Data Analysis, forthcoming, DOI:
http://dx.doi.org/10.1016/j.csda.2014.01.002. [173]
Chen, Y., H¨ardle, W., and Spokoiny, V. (2007), “Portfolio Value at Risk Based on
Independent Component Analysis,” Journal of Computational and Applied
Mathematics, 205, 594–607. [173]
Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993), “On the Relation
Between the Expected Value and the Volatility of the Nominal Excess Return
on Stocks,” The Journal of Finance, 48, 1779–1801. [173]
H¨ardle, W., Herwartz, H., and Spokoiny, V. (2003), “Time Inhomogeneous
Multiple Volatility Modeling,” Journal of Financial Econometrics, 1, 55–
95. [173]
Lanne, M., and Pentti, S. (2007), “Modeling Conditional Skewness in Stock
Returns,” The European Journal of Finance, 13, 691–704. [173]
Martin, I. W. R. (2013), “Consumption-Based Asset Pricing With Higher Cumulants,” Review of Economic Studies, 80, 745–773. [173]
Spokoiny, V., Wang, W., and H¨ardle, W. K. (2013), “Local Quantile Regression”
(with discussion), Journal of Statistical Planning and Inference, 143, 1109–
1129. [173]
Comment
Michael MCALEER
Department of Quantitative Finance, College of Technology Management, National Tsing Hua University, Taiwan;
Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, The Netherlands;
Tinbergen Institute, The Netherlands;
Department of Quantitative Economics, Complutense University of Madrid, Spain
DISCUSSION
It is a pleasure to provide some comments on the excellent
and topical article by Hu and Tsay (2014).
The article extends principal component analysis (PCA) to
principal volatility component analysis (PVCA), and should
prove to be an invaluable addition to the existing multivariate
models for dynamic covariances and correlations that are essential for sensible risk and portfolio management, including
dynamic hedging.
One of the key obstacles to developing multivariate covariance and correlation models is the “curse of dimensionality,”
namely the number of underlying parameters to be estimated,
the article is concerned with dimension reduction through the
use of PCA, which is possible if there are some common volatility components in the time series.
In particular, the method searches for linear combinations
of a vector time series for which there are no time-varying
conditional variances or covariances, and hence no time-varying
conditional correlations.
The authors extend PCA to PCVA in a clear, appealing, and
practical manner. Specifically, they use a spectral analysis of a
cumulative generalized kurtosis matrix to summarize the volatility dependence of multivariate time series and define the principal volatility components for dimension reduction.
The technical part of the article starts in Section 2 with a
vectorization of the volatility matrix, and a connection to the
BEKK model of Engle and Kroner (1995).
However, because a primary purpose of PCVA is to search
for the absence of multivariate time-varying conditional heteroskedasticity in vector time series, it would have been helpful
to see how PCVA might be connected to the conditional covariances arising from various specializations of BEKK (for further
details, see below).
Theorem 1 assumes the existence of fourth moments of
a weakly stationary vector time series, but Theorems 2 and
3 assume the existence of sixth moments. The latter two
theorems beg the question as to whether the assumption is
testable.
Interestingly, in the simulation study, the four simple ARCH
models considered are understood to “not satisfy the moment
condition of Theorems 2 and 3,” with a reference to Box and
Tiao (1977) that traditional PCA works well in finite samples
for nonstationary time series.
However, the purported connection between PCA for nonstationary time series, on the one hand, and time-varying conditional covariances and correlations for a weakly stationary
vector time series, on the other, is not entirely clear.
The empirical analysis considers a dataset of weekly log returns of seven exchange rates against the U.S. dollar from March
2000 to October 2011, giving 605 observations, which would
be considered a relatively small number of observations for purposes of estimating dynamic vector covariance and correlation
matrices.
The simple GARCH(1,1) model is used to estimate the conditional volatility models for the first six principal volatility
components. It would have been useful to compare the GARCH
estimates with the univariate asymmetric GJR and (possibly)
leverage-based EGARCH alternatives.
A simple comparison is made of the PVCA results with the
varying conditional correlation (VCC) model of Tse and Tsui
(2002), though VCC is referred to as a “dynamic conditional
correlation (DCC) model” (see Engle 2002).
As the effect of “news” in the VCC model has an estimated
coefficient of 0.013 and a standard error of 0.004, it is stated
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.898584
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Wolfgang Karl Härdle & Weining Wang
To cite this article: Wolfgang Karl Härdle & Weining Wang (2014) Comment, Journal of Business
& Economic Statistics, 32:2, 173-174, DOI: 10.1080/07350015.2014.898585
To link to this article: http://dx.doi.org/10.1080/07350015.2014.898585
Published online: 16 May 2014.
Submit your article to this journal
Article views: 119
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: 20:41
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:41 11 January 2016
Hardle
and Wang: Comment
¨
173
Comment
¨
Wolfgang Karl HARDLE
Center for Applied Statistics and Economics, Humboldt-Universitat
¨ zu Berlin, Unter den Linden 6,
10099 Berlin, Germany; Singapore Management University, 50 Stamford Road, Singapore 178899
(haerdle@wiwi.hu-berlin.de)
Weining WANG
Center for Applied Statistics & Economics School of Business and Economics, Humboldt-Universitat
¨
zu Berlin, Unter den Linden 6, 10099 Berlin, Germany (wangwein@cms.hu-berlin.de)
The authors are to be congratulated for a timely and important contribution. The article proposes a novel principal volatility component (PVC) technique based on a generalized kurtosis matrix in a time series context. The proposed test statistics
allow deep insight into higher moments and tail behavior of
multivariate time series. The article considers a weak stationary
multivariate time series yt (k × 1) with finite fourth moments,
the lag l generalized kurtosis matrix is defined as
def
def
⊤
γl,ij γl,ij
, (1)
γl =
cov2 yt y⊤
t , xij,t−l =
i
j
i
j
where
def
γl,ij = cov yt y⊤
t , xij,t−l .
(2)
m⊤
v yt ,
where
The PVCs are then defined as linear combinations
the mv ’s are the vth eigenvectors of the cumulative generalized kurtosis matrix Γ∞ for general multivariate GARCH-type
def
models
(Γm for ARCH(m) effects in yt ) with Γ∞ = ∞
l=1 γl
def
(Γm = m
γ
).
Note
that
x
is
a
function
of
y
y
.
l
ij,t−l
i,t−l
j,t−l
l=1
The kurtosis matrix indicates the correlations and crosscorrelations between the current variance–covariance matrix
and its lagged one, and thus would be a four-dimensional object
(k × k × k × k). Nevertheless, the authors consider a k × k generalized kurtosis matrix, which sums up all the effects of a lagged
variance–covariance matrix. In some cases, one might like to
look at the componentwise effects, which requires alternatives
of defining a generalized kurtosis matrix. For example, one can
analyze the variance–covariance matrix between vec(yt y⊤
t ) and
2
2
vec(yt−l y⊤
),
whose
dimension
is
k
×
k
matrix.
Moreover,
t−l
to generalize the idea
of impulse response functions, one can
look at the matrix j cov2 (yt y⊤
t , xi0 j,t−l ) to isolate the lagged
variable i0 ’s(i0 = 1, . . . , k) contribution.
The article employs Huber’s function which is symmetric. One might by an asymmetric clip function address the
well-known leverage effect, which means that negative returns
increase future volatility by a larger amount than positive returns of the same magnitude. In particular, to model asymmetry
in the ARCH process, for example, as in GJRGARCH models
introduced by Glosten, Jagannathan, and Runkle (1993). For
−
−
may serve this propose,
yj,t−l
instance, setting xij,t−l = yi,t−l
−
where yj,t−l = yj,t−l only when yj,t−l < 0 (negative part of
yj,t−l ).
The idea of PVC is a decomposition of a (mixed) moment
matrix. In PCA, one considers the variance–covariance matrix,
which falls short on modeling a nonlinear and asymmetric
multivariate distribution. This fact reminds us of a strand of
literature on independent component analysis (ICA); see, for
example, Chen, H¨ardle, and Spokoiny (2007); Chen et al.
(2014). ICA looks for a projection that maximizes a nonGaussianity measure. Similarly, a generalized kurtosis matrix
in PVC is connected to measuring non-Gaussianity. However,
kurtosis does not provide the whole picture of a distribution
function, and therefore other perspectives of the conditional
distribution (e.g., conditional skewness and conditional quantile) may also be of interest, see, for example, Lanne and Pentti
(2007).
Another issue is possible nonstationarity in yt . In this situation, the nonstationarity can be modeled via switching parameters of a stationary model, see H¨ardle, Herwartz, and Spokoiny
(2003). The eigenvectors mv ’s would then be time varying
mvt . Accordingly, at time t one can adopt local adaptive techniques (see Spokoiny, Wang, and H¨ardle 2013) to identify a
local homogeneous interval [t − t0 , t], in which one may apply
PVC.
Once more we would like to congratulate the authors for
this great advance. We are sure that this work will create a
new strand of literature with implications on asset allocation:
portfolio choice and factor models. If one is interested in the
factors that have no ARCH effects, one can certainly employ
the presented technique. The factors isolated can be used as
factors in asset pricing model, taking into account of rare events
as in Martin (2013).
ACKNOWLEDGMENTS
The financial support From the Deutsche Forschungsge¨
meinschaft via SFB 649 “Okonomisches
Risiko,” HumboldtUniversit¨at zu Berlin and IRTG 1972 “High Dimensional Non
Stationary Time Series” is gratefully acknowledged.
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.898585
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:41 11 January 2016
174
Journal of Business & Economic Statistics, April 2014
REFERENCES
Chen, Y., Chen, R.-B., and H¨ardle, W. K. (2014), “TVICA—Time Varying
Independent Component Analysis and Its Application to Financial Data,”
Journal of Computational Statistics and Data Analysis, forthcoming, DOI:
http://dx.doi.org/10.1016/j.csda.2014.01.002. [173]
Chen, Y., H¨ardle, W., and Spokoiny, V. (2007), “Portfolio Value at Risk Based on
Independent Component Analysis,” Journal of Computational and Applied
Mathematics, 205, 594–607. [173]
Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993), “On the Relation
Between the Expected Value and the Volatility of the Nominal Excess Return
on Stocks,” The Journal of Finance, 48, 1779–1801. [173]
H¨ardle, W., Herwartz, H., and Spokoiny, V. (2003), “Time Inhomogeneous
Multiple Volatility Modeling,” Journal of Financial Econometrics, 1, 55–
95. [173]
Lanne, M., and Pentti, S. (2007), “Modeling Conditional Skewness in Stock
Returns,” The European Journal of Finance, 13, 691–704. [173]
Martin, I. W. R. (2013), “Consumption-Based Asset Pricing With Higher Cumulants,” Review of Economic Studies, 80, 745–773. [173]
Spokoiny, V., Wang, W., and H¨ardle, W. K. (2013), “Local Quantile Regression”
(with discussion), Journal of Statistical Planning and Inference, 143, 1109–
1129. [173]
Comment
Michael MCALEER
Department of Quantitative Finance, College of Technology Management, National Tsing Hua University, Taiwan;
Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, The Netherlands;
Tinbergen Institute, The Netherlands;
Department of Quantitative Economics, Complutense University of Madrid, Spain
DISCUSSION
It is a pleasure to provide some comments on the excellent
and topical article by Hu and Tsay (2014).
The article extends principal component analysis (PCA) to
principal volatility component analysis (PVCA), and should
prove to be an invaluable addition to the existing multivariate
models for dynamic covariances and correlations that are essential for sensible risk and portfolio management, including
dynamic hedging.
One of the key obstacles to developing multivariate covariance and correlation models is the “curse of dimensionality,”
namely the number of underlying parameters to be estimated,
the article is concerned with dimension reduction through the
use of PCA, which is possible if there are some common volatility components in the time series.
In particular, the method searches for linear combinations
of a vector time series for which there are no time-varying
conditional variances or covariances, and hence no time-varying
conditional correlations.
The authors extend PCA to PCVA in a clear, appealing, and
practical manner. Specifically, they use a spectral analysis of a
cumulative generalized kurtosis matrix to summarize the volatility dependence of multivariate time series and define the principal volatility components for dimension reduction.
The technical part of the article starts in Section 2 with a
vectorization of the volatility matrix, and a connection to the
BEKK model of Engle and Kroner (1995).
However, because a primary purpose of PCVA is to search
for the absence of multivariate time-varying conditional heteroskedasticity in vector time series, it would have been helpful
to see how PCVA might be connected to the conditional covariances arising from various specializations of BEKK (for further
details, see below).
Theorem 1 assumes the existence of fourth moments of
a weakly stationary vector time series, but Theorems 2 and
3 assume the existence of sixth moments. The latter two
theorems beg the question as to whether the assumption is
testable.
Interestingly, in the simulation study, the four simple ARCH
models considered are understood to “not satisfy the moment
condition of Theorems 2 and 3,” with a reference to Box and
Tiao (1977) that traditional PCA works well in finite samples
for nonstationary time series.
However, the purported connection between PCA for nonstationary time series, on the one hand, and time-varying conditional covariances and correlations for a weakly stationary
vector time series, on the other, is not entirely clear.
The empirical analysis considers a dataset of weekly log returns of seven exchange rates against the U.S. dollar from March
2000 to October 2011, giving 605 observations, which would
be considered a relatively small number of observations for purposes of estimating dynamic vector covariance and correlation
matrices.
The simple GARCH(1,1) model is used to estimate the conditional volatility models for the first six principal volatility
components. It would have been useful to compare the GARCH
estimates with the univariate asymmetric GJR and (possibly)
leverage-based EGARCH alternatives.
A simple comparison is made of the PVCA results with the
varying conditional correlation (VCC) model of Tse and Tsui
(2002), though VCC is referred to as a “dynamic conditional
correlation (DCC) model” (see Engle 2002).
As the effect of “news” in the VCC model has an estimated
coefficient of 0.013 and a standard error of 0.004, it is stated
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.898584