07350015%2E2014%2E902238
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Elena Andreou & Eric Ghysels
To cite this article: Elena Andreou & Eric Ghysels (2014) Comment, Journal of Business &
Economic Statistics, 32:2, 168-171, DOI: 10.1080/07350015.2014.902238
To link to this article: http://dx.doi.org/10.1080/07350015.2014.902238
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Journal of Business & Economic Statistics, April 2014
Comment
Elena ANDREOU
Department of Economics, University of Cyprus, P.O. Box 537, CY 1678 Nicosia, Cyprus
(elena.andreou@ucy.ac.cy)
Eric GHYSELS
Department of Economics, University of North Carolina, Gardner Hall CB 3305, Chapel Hill, NC 27599;
Department of Finance, Kenan-Flagler Business School, Chapel Hill, NC 27599 (eghysels@unc.edu)
1.
INTRODUCTION
The article by Hu and Tsay (2013) relates to an important area
of research on analyzing multivariate time series with dynamic
heteroscedasticity and addresses the dimensionality problem of
traditional multivariate models for financial asset returns volatilities via common factor analysis. This article provides an alternative approach to estimate the common volatility factor for
multivariate dynamic heteroscedastic processes. The principal
volatility component (PVC) proposed is based on the cumulative generalized kurtosis matrix which summarizes the higher
order dynamics of multivariate volatility processes. The spectral
analysis decomposition of the cumulative generalized kurtosis
matrix is used to estimate the PVC. Given that one of the factors
may not have dynamic heteroscedasticity, a generalized test is
proposed for the presence of no ARCH type dependence. Simulation and empirical results provide additional evidence of the
PVC procedure and test properties. Hu and Tsay (2013) worked
on a fundamental question of extracting common volatility factors which are especially relevant for financial time series. We
are therefore pleased to contribute some comments related to the
volatility factor estimation some of which are also addressed
in our own related recent work (Andreou and Ghysels 2013;
Ghysels 2013).
The idea to estimate the common volatility factor from a
cross-section of volatilities for financial time series is not new
but has been in an area of recent research. Early contributions
in the common volatility factor analysis involving small crosssections of assets includes various ARCH factor models, see, for
example, Engle (1987); Engle, Ng, and Rothschild (1990); and
Ng, Engle, and Rothschild (1992), among others. Diebold and
Nerlove (1989) suggested a closely related latent factor model.
For larger cross-sections of financial returns with dynamic heteroscedasticity, Connor, Korajczyk, and Linton (2006) developed a dynamic approximate factor model and relied on large
panel data asymptotics to estimate a common volatility component. Their analysis involves a semiparametric ARCH-type
volatility filter. Instead, Anderson and Vahid-Araghi (2007) and
Barigozzi et al. (2013) used large panels of realized volatilities
to extract the common volatility factor. In the context of risk neutral pricing there are also examples of factor analysis. Bühler
(2006), Amengual (2009), Egloff, Leippold, and Wu (2010),
and Aı̈t-Sahalia, Karaman, and Mancini (2012) applied principal component analysis (PCA) to panels of variance swap rates
and found that two factors—which can be interpreted as level
and slope—explain close to 100% of the variation in variance
swap rates for the S&P 500.
In view of some of the above literature, we first briefly describe the PVC estimation approach in Hu and Tsay (2013)
in the next section. In the following section, we revisit their
simulation design to evaluate the role of the finite higher order
moment condition in their PVC procedure. In the last section,
we discuss some further results on common volatility factor estimation which are related to our recent work and complement
the results of Hu and Tsay (2013).
2.
THE PRINCIPAL VOLATILITY COMPONENT (PVC)
PROCEDURE
Let the multivariate stochastic process yt be α-mixing such
as for instance an ARCH-type process. Hu and Tsay (2013) assumed that yt is a k-dimensional weakly stationary time series
with finite fourth moments. Hence, the generalized kurtosis matrix γl of yt captures the dynamics of the fourth power of yt y′t
and yt−i y′t−i , i > 0 given by Equations (1) and (2):
γl =
k
k
i=1 j =1
cov2 (yt y′t , xij,t−l ) ≡
k
k
′
γl,ij γl,ij
,
i=1 j =1
for l ≥ 0, (1)
γl,i j = cov(yt y′t , xij,t−l ) = E[(yt y′t − )(xij,t−l − E(xij,t ))],
(2)
where xij,t−l can be, for example, the Huber’s function
⎧
if |yi,t−l yj,t−l | ≤ c2
⎪
⎨ yi,t−l yj,t−l
√
if yi,t−l yj,t−l > c2
(3)
xij,t−l = 2c yi,t−l yj,t−l − c2
⎪
⎩ 2
2
c − 2c |yi,t−l yj,t−l | if yi,t−l yj,t−l < −c
and c is a prespecified constant. The matrix γl,i j is a k × k generalized covariance matrix and the cumulative matrix is given
by:
Ŵm =
m
γl .
(4)
l=1
In Equation (4), Ŵm is estimated by
m
k
k
l 2
1−
Ŵ̂m =
c
ov2 (yt y′t − Ȳ, xij,t−l − x̄ij )
n
l=1 i=1 j =1
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.902238
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Andreou and Ghysels: Comment
169
√
such that Ŵ̂m = Ŵm + O(1/ n) assuming yt has finite sixth
moments.
The idea of the common volatility factor or more precisely
the PVC in Hu and Tsay (2013) is based on the spectral analysis
of the generalized kurtosis matrix of the multivariate observed
financial processes, Ŵ̂m in (4). The novel aspect of the Hu and
Tsay (2013) procedure compared to the papers mentioned in
the Introduction is related to the generalized kurtosis matrix
estimation and the principal component estimation based on
this matrix. One advantage of this procedure is that it is based
on the observed process, yt . Yet the squared cross-products of
yt involved in γl can be noisy estimates of volatility. This point
is discussed further in the last section given it relates to an
alternative approach to estimate common volatility factors.
3.
PVC AND FINITE MOMENT CONDITIONS
The objective of this section is to examine the finiteness of
the higher moments condition required in the PVC theory of Hu
and Tsay (2013). Examples of such financial processes where
the fourth moments are not finite are empirically relevant for financial time series such as weekly processes (as presented in the
empirical analysis of the article) as well as higher frequencies.
ARCH-type models with highly leptokurtic distributions or the
Integrated GARCH (IGARCH) model are found in empirical
research. Although Hu and Tsay (2013) provided simulation
results for highly persistent normal ARCH models that do not
satisfy the finite fourth moment condition we present additional
simulation evidence in this section.
First, we follow the simulation design in Hu and Tsay (2013)
and extend it to study the performance of the PVC in various other cases where the individual ARCH-type processes, ft
defined below, do not necessarily satisfy the conditions of finiteness of higher order moments in ARCH-type processes. In their
Monte Carlo design, Hu and Tsay (2013) assumed that the multivariate factor structure is driven by multivariate normal errors.
Therefore, in the second step we extend the simulation results
to the case where the multivariate ARCH-type processes are
driven by highly leptokurtic error distributions and reexamine
the PVC procedure.
We simulate the following data-generating processes (DGPs)
to further examine the performance of the PVC when the moments of the ARCH-type processes are not finite following models ft = σt et where we first assume that volatility follows an
integrated GARCH model:
2
2
,
+ βσt−1
σt2 = ω + αft−1
(5)
with α + β = 1. Second, we assume the ARCH models are
driven by Student’s-t errors, et , with 3 and 5 degrees of freedom1
2
σt2 = ω + αft−1
.
(6)
Third, we consider the GJR-ARCH model
2
2
2
+ βσt−1
+ τft−1
It−1 ,
σt2 = ω + αft−1
1It
(7)
is worth noting that we used the standardized Student’s-t distribution for the
t-ARCH DGP (Bollerslev and Engle 1986) which is not the same as that in the
build-in ARCH routine in Matlab that uses unstandardized Student’s-t errors.
with high persistence and normal or t(3) errors where τ is the
threshold parameter and It−1 = 0 if et−1 ≥ 0, and It−1 = 1 if
et−1 < 0. Finally, we consider an AR-ARCH model
2
ft = ρft−1 + σt et , σt2 = ω + αft−1
(8)
where ρ is set to be equal to 0.1 or 0.9 with et ∼ normal. In the
AR-ARCH model, we consider both ft and
ft in our analysis.
Note that for completeness we also simulate the high persistence
(HP) Normal ARCH models as given in Hu and Tsay (2013) as
well as the corresponding low-persistence (LP) normal ARCH
for which the finite fourth moment conditions are satisfied.
The multivariate ARCH factor model is given by
yt = Hft + ǫ t ,
(9)
where H is the k × r loading matrix, ft = (f1t , . . . , frt ) are
independent conditional heteroscedastic processes and ǫ t is the
k × T noise term, a sequence of independent and identically distributed random vectors with zero mean and constant positivedefinite covariance matrix t . In the first step, we assume ǫ t
follows a normal distribution as in Hu and Tsay (2013) and
ft is either of the ARCH-type models in (5)–(8). These results
are reported in Table 1. In the second step, we assume ǫ t follows a t(3) distribution and report the corresponding results in
Table 2.
Table 1 reports the results for the correlations coefficients R1
and R2 given by
R1 =
|m̂′1 H(H ′ H)−1 H ′ m̂1 |
, R2 = cor2 (m̂′1 yt , M1′ yt ). (10)
|m̂′1 m̂1 |
Here, M1 = (0, −1, 1, −1, 1)′ , T = 250, 500, 1000 and k = 5
as in Hu and Tsay (2013). The choice of c = 2.5 and m =
5 for the Huber’s function follows from the relatively good
performance of the simulation results reported in Hu and Tsay
(2013). The case of m = 10 was also applied in the simulations
and gave similar results.
The general conclusion from the results reported in Table 1 is
that even if the ARCH-type processes in ft do not satisfy the finite fourth moment conditions, the PVC behaves well according
to criteria R1 and R2 which are expected to be close to zero and
one, respectively. Hence, our results in Table 1 provide further
support in favor of the simulation evidence presented in Hu and
Tsay (2013) for different DGPs. Table 2 reports the results for
some of the ARCH processes in ft as above, but now ǫ t in Equation (9) follows the standardized Student’s t-(3) distribution. In
contrast to the results in Table 1, those in Table 2 show that
the PVC performs relatively less well when the noise term is
t(3) distributed. In particular, the R2 criterion falls and the R1
criterion increases especially for both the normal and Student’st(3) high-persistent ARCH-type models, ft . This result is valid
even for large samples T for the highly persistent ARCH models. Therefore extending the simulation analysis of Hu and Tsay
(2013), we find that there are cases especially when the error
distribution of the multivariate factor ARCH model in (9) is
highly leptokurtic that the relative performance of the PVC, via
the measures R1 and R2 , is weaker compared to the case where
ǫ t is normal. It is also worth noting that the results on PVC
become worse when ǫ t in Equation (9) follows an asymmetric
distribution such as the log–normal.
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170
Journal of Business & Economic Statistics, April 2014
Table 1. Principal volatility component analysis summary statistics, yt = Hft + ǫ t , ǫ t ∼ N (0, 1)
N-LP-ARCH
T
250
500
1000
t(3)-HP-ARCH
t(5)-HP-ARCH
N-IGARCH
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.033
0.019
0.011
0.890
0.932
0.954
0.037
0.019
0.050
0.839
0.889
0.839
0.080
0.075
0.052
0.795
0.824
0.839
0.046
0.041
0.037
0.831
0.857
0.889
0.0004
0.0003
0.0003
0.924
0.938
0.942
N-GJR-ARCH
250
500
1000
N-HP-ARCH
AR-ARCH
for ft (ρ = 0.1)
AR-ARCH
for ft (ρ = 0.9)
AR-ARCH
for
ft (ρ = 0.1)
AR-ARCH
for
ft (ρ = 0.9)
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.030
0.032
0.011
0.842
0.845
0.954
0.026
0.024
0.023
0.856
0.860
0.870
0.023
0.002
0.002
0.944
0.949
0.955
0.040
0.026
0.028
0.631
0.668
0.628
0.052
0.073
0.062
0.663
0.650
0.654
NOTES: The table shows the summary statistics of the Hu and Tsay (2013) principal volatility component analysis for various k = five-dimensional series for ft . The multivariate factor
ARCH model is given by Equation (5) where ǫ t ∼ N(0, 1). The performance measures R1 and R2 in (10) are calculated using 1000 iterations in Matlab. For the Ŵ̂m matrix m = 5 is
used and for the Huber’s function c = 2.5. We used Hu and Tsay (2013) high-persistent HP-ARCH components for all the ARCH, GJR-ARCH and AR-ARCH models, that is (ω, α)
= (1,0.9), (2,0.8), (3,0.7), and (1,0.95). For the low-persistent LP-ARCH models the ARCH parameter of the 4 ARCH(1) components are α = 0.1, 0.2, 0.3, 0.4, respectively. In the
IGARCH model, (ω, α, β) = (1,0.09,0.9), (2,0.06,0.93), (3,0.05,0.94), (1,0.02,0.97). N, t(3), and t(5) denote the standard normal and the standardized Student’s t-(3) and Student’s t-(5)
innovation’s distribution. ρ is the AR coefficient of the AR(1)–ARCH(1) models.
4.
AN ALTERNATIVE APPROACH FOR COMMON
VOLATILITY FACTOR ESTIMATION
mate common volatility factors directly from panels of filtered
volatilities. Specifically we consider the following model repi
i
, i = 1, . . . , N)′ , where V̂[h:t]
are
resentation for X[h:t] = (V̂[h:t]
0
the filtered volatilities. Let Ft be the r × 1 vector of the true
f
population volatility factors with either: Ft0 = σ (Xt )2 , when
t
f
panels of spot volatilities are consider, and Ft0 = t−d σ (Xτ )2 dτ
f
with integrated volatility proxies, where Xt are the latent factors of a continuous time affine diffusion model. Moreover, for
factor loadings 0 = (λ01 , . . . , λ0N )′ , we have the following factor
representation:
The analysis of Hu and Tsay (2013) does not address certain
aspects of common volatility factor estimation namely the large
cross-sectional dimension, the alternative volatility filters and
the use of high-frequency data, as well as the selection of the
number of volatility factors.
These aspects are addressed in our recent work (Andreou and
Ghysels 2013; Ghysels 2013). We provide the asymptotic analysis of common volatility risk factor estimation using large panels
of filtered volatilities such as (1) parametric spot volatility filters
(e.g., ARCH-type models) or (2) data-driven realized volatilities (e.g., Zhang 2001; Andreou and Ghysels 2002; Mykland
and Zhang 2008) and data-driven spot volatilities (e.g., Zhang,
Mykland and Aı̈t-Sahalia 2005; Hansen and Lunde 2006). Unlike Hu and Tsay (2013), who consider the cumulative generalized kurtosis matrix based on cross-products of yt , we esti-
X[h:t] = 0 Ft0 + e[h:t] ,
(11)
i
(e[h:t]
,i
= 1, . . . , N)′ . We use principal compowhere e[h:t] =
nent analysis with panels of filtered volatilities.
The common volatility factor estimation analysis is based
on three types of asymptotic expansions: the cross-section of
volatility estimates at each point in time, that is i = 1, . . . , N
Table 2. Principal volatility component analysis summary statistics, yt = Hft + ǫ t , ǫ t ∼ t(3)
N-LP-ARCH
T
250
500
1000
t(3)-HP-ARCH
t(5)-HP-ARCH
N-IGARCH
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.061
0.058
0.033
0.788
0.815
0.839
0.124
0.102
0.111
0.712
0.759
0.761
0.127
0.090
0.069
0.716
0.776
0.789
0.052
0.026
0.042
0.782
0.843
0.868
0.001
0.001
0.001
0.926
0.937
0.947
N-GJR-ARCH
250
500
1000
N-HP-ARCH
AR-ARCH
for ft (ρ = 0.1)
AR-ARCH
for ft (ρ = 0.9)
AR-ARCH
for
ft (ρ = 0.1)
AR-ARCH
for
ft (ρ = 0.9)
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.083
0.054
0.076
0.757
0.822
0.808
0.043
0.031
0.025
0.828
0.849
0.856
0.002
0.003
0.002
0.939
0.930
0.935
0.025
0.031
0.028
0.710
0.717
0.720
0.078
0.023
0.015
0.677
0.653
0.660
NOTES: The table shows the summary statistics of the Hu and Tsay (2013) principal volatility component analysis for various k =five-dimensional series for ft . The multivariate factor
ARCH model is given by Equation (5) where ǫ t ∼ t(3). The performance measures R1 and R2 in (10) are calculated using 1000 iterations in Matlab. For the Ŵ̂m matrix m = 5 is used and
for the Huber’s function c = 2.5. We used Hu and Tsay (2013) high-persistent HP-ARCH components for all the models, that is (ω, α) = (1,0.9), (2,0.8), (3,0.7), and (1,0.95). For the low
persistent LP-ARCH models the ARCH parameter of the 4 ARCH(1) components are α = 0.1, 0.2, 0.3, 0.4, respectively. In the IGARCH model, (ω, α, β) = (1,0.09,0.9), (2,0.06,0.93),
(3,0.05,0.94), (1,0.02,0.97). N, t(3), and t(5) denote the standard normal and the standardized Student’s-t(3) and Student’s-t(5) innovation’s distribution. ρ is the AR coefficient of the
AR(1)–ARCH(1) models.
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Franke: Comment
171
observed over a time span i = 1, . . . , T and the sampling frequency h of the data used to compute the volatility estimates
which rely on data collected at increasing frequency, h ↓ 0. The
continuous record or in-fill asymptotics, h ↓ 0, have a key role
in that they can control the cross-sectional and serial correlation among the idiosyncratic errors of the panel of volatilities.
Therefore under suitable regularity conditions, the traditional
principal component analysis yields super-consistent estimates
of the common volatility factors at each point in time. The intuition behind the super-consistency result is because the highfrequency sampling scheme of the filtered volatilities is tied to
the size of the cross-section, boosting the rate of convergence.
Consequently, the super-consistency of the common volatility
factor extracted from the panel asymptotic arguments can also
improve upon the individual volatility estimates.
In Hu and Tsay (2013), the analysis is based on the first PVC.
An extension of the current analysis could address the choice
of the number of factors. This question has been addressed
in the context of linear factor models which do not involve
common volatility factors (e.g., Stock and Watson 2002; Bai
and Ng 2002). Namely, in the traditional factor models Bai and
Ng (2002) among others proposed various information criteria
that depend on the asymptotic rates of N and T of the panel.
In contrast, Ghysels (2013) showed that the standard crosssectional criteria suffice for consistent estimation of the number
of factors in large cross-sections of filtered volatilities, which
is different from the traditional panel data results. This result
on common volatility factor selection criteria draws from the
super-consistency.
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Comment
Juergen FRANKE
Universität Kaiserslautern, Kaiserslautern, Germany (franke@mathematik.uni-kl.de)
Financial data, in particular in the context of managing
portfolios and quantifying their risk, come in the form of
high-dimensional time series. The evolution of the coordinate
processes, representing asset prices or returns, all depend
on common market factors in addition to specific events
concerning the value of the underlying asset only. An important
task of financial time series analysis with major practical implications is the identification and description of such common
factors.
In the present article, Hu and Tsay focus on the volatility
aspect of financial time series and proposes an innovative way
to structure the interdependencies between the fluctuations of
the assets in a portfolio. For the volatility matrices, that is, the
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.903652
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Elena Andreou & Eric Ghysels
To cite this article: Elena Andreou & Eric Ghysels (2014) Comment, Journal of Business &
Economic Statistics, 32:2, 168-171, DOI: 10.1080/07350015.2014.902238
To link to this article: http://dx.doi.org/10.1080/07350015.2014.902238
Published online: 16 May 2014.
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Article views: 125
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168
Journal of Business & Economic Statistics, April 2014
Comment
Elena ANDREOU
Department of Economics, University of Cyprus, P.O. Box 537, CY 1678 Nicosia, Cyprus
(elena.andreou@ucy.ac.cy)
Eric GHYSELS
Department of Economics, University of North Carolina, Gardner Hall CB 3305, Chapel Hill, NC 27599;
Department of Finance, Kenan-Flagler Business School, Chapel Hill, NC 27599 (eghysels@unc.edu)
1.
INTRODUCTION
The article by Hu and Tsay (2013) relates to an important area
of research on analyzing multivariate time series with dynamic
heteroscedasticity and addresses the dimensionality problem of
traditional multivariate models for financial asset returns volatilities via common factor analysis. This article provides an alternative approach to estimate the common volatility factor for
multivariate dynamic heteroscedastic processes. The principal
volatility component (PVC) proposed is based on the cumulative generalized kurtosis matrix which summarizes the higher
order dynamics of multivariate volatility processes. The spectral
analysis decomposition of the cumulative generalized kurtosis
matrix is used to estimate the PVC. Given that one of the factors
may not have dynamic heteroscedasticity, a generalized test is
proposed for the presence of no ARCH type dependence. Simulation and empirical results provide additional evidence of the
PVC procedure and test properties. Hu and Tsay (2013) worked
on a fundamental question of extracting common volatility factors which are especially relevant for financial time series. We
are therefore pleased to contribute some comments related to the
volatility factor estimation some of which are also addressed
in our own related recent work (Andreou and Ghysels 2013;
Ghysels 2013).
The idea to estimate the common volatility factor from a
cross-section of volatilities for financial time series is not new
but has been in an area of recent research. Early contributions
in the common volatility factor analysis involving small crosssections of assets includes various ARCH factor models, see, for
example, Engle (1987); Engle, Ng, and Rothschild (1990); and
Ng, Engle, and Rothschild (1992), among others. Diebold and
Nerlove (1989) suggested a closely related latent factor model.
For larger cross-sections of financial returns with dynamic heteroscedasticity, Connor, Korajczyk, and Linton (2006) developed a dynamic approximate factor model and relied on large
panel data asymptotics to estimate a common volatility component. Their analysis involves a semiparametric ARCH-type
volatility filter. Instead, Anderson and Vahid-Araghi (2007) and
Barigozzi et al. (2013) used large panels of realized volatilities
to extract the common volatility factor. In the context of risk neutral pricing there are also examples of factor analysis. Bühler
(2006), Amengual (2009), Egloff, Leippold, and Wu (2010),
and Aı̈t-Sahalia, Karaman, and Mancini (2012) applied principal component analysis (PCA) to panels of variance swap rates
and found that two factors—which can be interpreted as level
and slope—explain close to 100% of the variation in variance
swap rates for the S&P 500.
In view of some of the above literature, we first briefly describe the PVC estimation approach in Hu and Tsay (2013)
in the next section. In the following section, we revisit their
simulation design to evaluate the role of the finite higher order
moment condition in their PVC procedure. In the last section,
we discuss some further results on common volatility factor estimation which are related to our recent work and complement
the results of Hu and Tsay (2013).
2.
THE PRINCIPAL VOLATILITY COMPONENT (PVC)
PROCEDURE
Let the multivariate stochastic process yt be α-mixing such
as for instance an ARCH-type process. Hu and Tsay (2013) assumed that yt is a k-dimensional weakly stationary time series
with finite fourth moments. Hence, the generalized kurtosis matrix γl of yt captures the dynamics of the fourth power of yt y′t
and yt−i y′t−i , i > 0 given by Equations (1) and (2):
γl =
k
k
i=1 j =1
cov2 (yt y′t , xij,t−l ) ≡
k
k
′
γl,ij γl,ij
,
i=1 j =1
for l ≥ 0, (1)
γl,i j = cov(yt y′t , xij,t−l ) = E[(yt y′t − )(xij,t−l − E(xij,t ))],
(2)
where xij,t−l can be, for example, the Huber’s function
⎧
if |yi,t−l yj,t−l | ≤ c2
⎪
⎨ yi,t−l yj,t−l
√
if yi,t−l yj,t−l > c2
(3)
xij,t−l = 2c yi,t−l yj,t−l − c2
⎪
⎩ 2
2
c − 2c |yi,t−l yj,t−l | if yi,t−l yj,t−l < −c
and c is a prespecified constant. The matrix γl,i j is a k × k generalized covariance matrix and the cumulative matrix is given
by:
Ŵm =
m
γl .
(4)
l=1
In Equation (4), Ŵm is estimated by
m
k
k
l 2
1−
Ŵ̂m =
c
ov2 (yt y′t − Ȳ, xij,t−l − x̄ij )
n
l=1 i=1 j =1
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.902238
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Andreou and Ghysels: Comment
169
√
such that Ŵ̂m = Ŵm + O(1/ n) assuming yt has finite sixth
moments.
The idea of the common volatility factor or more precisely
the PVC in Hu and Tsay (2013) is based on the spectral analysis
of the generalized kurtosis matrix of the multivariate observed
financial processes, Ŵ̂m in (4). The novel aspect of the Hu and
Tsay (2013) procedure compared to the papers mentioned in
the Introduction is related to the generalized kurtosis matrix
estimation and the principal component estimation based on
this matrix. One advantage of this procedure is that it is based
on the observed process, yt . Yet the squared cross-products of
yt involved in γl can be noisy estimates of volatility. This point
is discussed further in the last section given it relates to an
alternative approach to estimate common volatility factors.
3.
PVC AND FINITE MOMENT CONDITIONS
The objective of this section is to examine the finiteness of
the higher moments condition required in the PVC theory of Hu
and Tsay (2013). Examples of such financial processes where
the fourth moments are not finite are empirically relevant for financial time series such as weekly processes (as presented in the
empirical analysis of the article) as well as higher frequencies.
ARCH-type models with highly leptokurtic distributions or the
Integrated GARCH (IGARCH) model are found in empirical
research. Although Hu and Tsay (2013) provided simulation
results for highly persistent normal ARCH models that do not
satisfy the finite fourth moment condition we present additional
simulation evidence in this section.
First, we follow the simulation design in Hu and Tsay (2013)
and extend it to study the performance of the PVC in various other cases where the individual ARCH-type processes, ft
defined below, do not necessarily satisfy the conditions of finiteness of higher order moments in ARCH-type processes. In their
Monte Carlo design, Hu and Tsay (2013) assumed that the multivariate factor structure is driven by multivariate normal errors.
Therefore, in the second step we extend the simulation results
to the case where the multivariate ARCH-type processes are
driven by highly leptokurtic error distributions and reexamine
the PVC procedure.
We simulate the following data-generating processes (DGPs)
to further examine the performance of the PVC when the moments of the ARCH-type processes are not finite following models ft = σt et where we first assume that volatility follows an
integrated GARCH model:
2
2
,
+ βσt−1
σt2 = ω + αft−1
(5)
with α + β = 1. Second, we assume the ARCH models are
driven by Student’s-t errors, et , with 3 and 5 degrees of freedom1
2
σt2 = ω + αft−1
.
(6)
Third, we consider the GJR-ARCH model
2
2
2
+ βσt−1
+ τft−1
It−1 ,
σt2 = ω + αft−1
1It
(7)
is worth noting that we used the standardized Student’s-t distribution for the
t-ARCH DGP (Bollerslev and Engle 1986) which is not the same as that in the
build-in ARCH routine in Matlab that uses unstandardized Student’s-t errors.
with high persistence and normal or t(3) errors where τ is the
threshold parameter and It−1 = 0 if et−1 ≥ 0, and It−1 = 1 if
et−1 < 0. Finally, we consider an AR-ARCH model
2
ft = ρft−1 + σt et , σt2 = ω + αft−1
(8)
where ρ is set to be equal to 0.1 or 0.9 with et ∼ normal. In the
AR-ARCH model, we consider both ft and
ft in our analysis.
Note that for completeness we also simulate the high persistence
(HP) Normal ARCH models as given in Hu and Tsay (2013) as
well as the corresponding low-persistence (LP) normal ARCH
for which the finite fourth moment conditions are satisfied.
The multivariate ARCH factor model is given by
yt = Hft + ǫ t ,
(9)
where H is the k × r loading matrix, ft = (f1t , . . . , frt ) are
independent conditional heteroscedastic processes and ǫ t is the
k × T noise term, a sequence of independent and identically distributed random vectors with zero mean and constant positivedefinite covariance matrix t . In the first step, we assume ǫ t
follows a normal distribution as in Hu and Tsay (2013) and
ft is either of the ARCH-type models in (5)–(8). These results
are reported in Table 1. In the second step, we assume ǫ t follows a t(3) distribution and report the corresponding results in
Table 2.
Table 1 reports the results for the correlations coefficients R1
and R2 given by
R1 =
|m̂′1 H(H ′ H)−1 H ′ m̂1 |
, R2 = cor2 (m̂′1 yt , M1′ yt ). (10)
|m̂′1 m̂1 |
Here, M1 = (0, −1, 1, −1, 1)′ , T = 250, 500, 1000 and k = 5
as in Hu and Tsay (2013). The choice of c = 2.5 and m =
5 for the Huber’s function follows from the relatively good
performance of the simulation results reported in Hu and Tsay
(2013). The case of m = 10 was also applied in the simulations
and gave similar results.
The general conclusion from the results reported in Table 1 is
that even if the ARCH-type processes in ft do not satisfy the finite fourth moment conditions, the PVC behaves well according
to criteria R1 and R2 which are expected to be close to zero and
one, respectively. Hence, our results in Table 1 provide further
support in favor of the simulation evidence presented in Hu and
Tsay (2013) for different DGPs. Table 2 reports the results for
some of the ARCH processes in ft as above, but now ǫ t in Equation (9) follows the standardized Student’s t-(3) distribution. In
contrast to the results in Table 1, those in Table 2 show that
the PVC performs relatively less well when the noise term is
t(3) distributed. In particular, the R2 criterion falls and the R1
criterion increases especially for both the normal and Student’st(3) high-persistent ARCH-type models, ft . This result is valid
even for large samples T for the highly persistent ARCH models. Therefore extending the simulation analysis of Hu and Tsay
(2013), we find that there are cases especially when the error
distribution of the multivariate factor ARCH model in (9) is
highly leptokurtic that the relative performance of the PVC, via
the measures R1 and R2 , is weaker compared to the case where
ǫ t is normal. It is also worth noting that the results on PVC
become worse when ǫ t in Equation (9) follows an asymmetric
distribution such as the log–normal.
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170
Journal of Business & Economic Statistics, April 2014
Table 1. Principal volatility component analysis summary statistics, yt = Hft + ǫ t , ǫ t ∼ N (0, 1)
N-LP-ARCH
T
250
500
1000
t(3)-HP-ARCH
t(5)-HP-ARCH
N-IGARCH
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.033
0.019
0.011
0.890
0.932
0.954
0.037
0.019
0.050
0.839
0.889
0.839
0.080
0.075
0.052
0.795
0.824
0.839
0.046
0.041
0.037
0.831
0.857
0.889
0.0004
0.0003
0.0003
0.924
0.938
0.942
N-GJR-ARCH
250
500
1000
N-HP-ARCH
AR-ARCH
for ft (ρ = 0.1)
AR-ARCH
for ft (ρ = 0.9)
AR-ARCH
for
ft (ρ = 0.1)
AR-ARCH
for
ft (ρ = 0.9)
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.030
0.032
0.011
0.842
0.845
0.954
0.026
0.024
0.023
0.856
0.860
0.870
0.023
0.002
0.002
0.944
0.949
0.955
0.040
0.026
0.028
0.631
0.668
0.628
0.052
0.073
0.062
0.663
0.650
0.654
NOTES: The table shows the summary statistics of the Hu and Tsay (2013) principal volatility component analysis for various k = five-dimensional series for ft . The multivariate factor
ARCH model is given by Equation (5) where ǫ t ∼ N(0, 1). The performance measures R1 and R2 in (10) are calculated using 1000 iterations in Matlab. For the Ŵ̂m matrix m = 5 is
used and for the Huber’s function c = 2.5. We used Hu and Tsay (2013) high-persistent HP-ARCH components for all the ARCH, GJR-ARCH and AR-ARCH models, that is (ω, α)
= (1,0.9), (2,0.8), (3,0.7), and (1,0.95). For the low-persistent LP-ARCH models the ARCH parameter of the 4 ARCH(1) components are α = 0.1, 0.2, 0.3, 0.4, respectively. In the
IGARCH model, (ω, α, β) = (1,0.09,0.9), (2,0.06,0.93), (3,0.05,0.94), (1,0.02,0.97). N, t(3), and t(5) denote the standard normal and the standardized Student’s t-(3) and Student’s t-(5)
innovation’s distribution. ρ is the AR coefficient of the AR(1)–ARCH(1) models.
4.
AN ALTERNATIVE APPROACH FOR COMMON
VOLATILITY FACTOR ESTIMATION
mate common volatility factors directly from panels of filtered
volatilities. Specifically we consider the following model repi
i
, i = 1, . . . , N)′ , where V̂[h:t]
are
resentation for X[h:t] = (V̂[h:t]
0
the filtered volatilities. Let Ft be the r × 1 vector of the true
f
population volatility factors with either: Ft0 = σ (Xt )2 , when
t
f
panels of spot volatilities are consider, and Ft0 = t−d σ (Xτ )2 dτ
f
with integrated volatility proxies, where Xt are the latent factors of a continuous time affine diffusion model. Moreover, for
factor loadings 0 = (λ01 , . . . , λ0N )′ , we have the following factor
representation:
The analysis of Hu and Tsay (2013) does not address certain
aspects of common volatility factor estimation namely the large
cross-sectional dimension, the alternative volatility filters and
the use of high-frequency data, as well as the selection of the
number of volatility factors.
These aspects are addressed in our recent work (Andreou and
Ghysels 2013; Ghysels 2013). We provide the asymptotic analysis of common volatility risk factor estimation using large panels
of filtered volatilities such as (1) parametric spot volatility filters
(e.g., ARCH-type models) or (2) data-driven realized volatilities (e.g., Zhang 2001; Andreou and Ghysels 2002; Mykland
and Zhang 2008) and data-driven spot volatilities (e.g., Zhang,
Mykland and Aı̈t-Sahalia 2005; Hansen and Lunde 2006). Unlike Hu and Tsay (2013), who consider the cumulative generalized kurtosis matrix based on cross-products of yt , we esti-
X[h:t] = 0 Ft0 + e[h:t] ,
(11)
i
(e[h:t]
,i
= 1, . . . , N)′ . We use principal compowhere e[h:t] =
nent analysis with panels of filtered volatilities.
The common volatility factor estimation analysis is based
on three types of asymptotic expansions: the cross-section of
volatility estimates at each point in time, that is i = 1, . . . , N
Table 2. Principal volatility component analysis summary statistics, yt = Hft + ǫ t , ǫ t ∼ t(3)
N-LP-ARCH
T
250
500
1000
t(3)-HP-ARCH
t(5)-HP-ARCH
N-IGARCH
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.061
0.058
0.033
0.788
0.815
0.839
0.124
0.102
0.111
0.712
0.759
0.761
0.127
0.090
0.069
0.716
0.776
0.789
0.052
0.026
0.042
0.782
0.843
0.868
0.001
0.001
0.001
0.926
0.937
0.947
N-GJR-ARCH
250
500
1000
N-HP-ARCH
AR-ARCH
for ft (ρ = 0.1)
AR-ARCH
for ft (ρ = 0.9)
AR-ARCH
for
ft (ρ = 0.1)
AR-ARCH
for
ft (ρ = 0.9)
R1
R2
R1
R2
R1
R2
R1
R2
R1
R2
0.083
0.054
0.076
0.757
0.822
0.808
0.043
0.031
0.025
0.828
0.849
0.856
0.002
0.003
0.002
0.939
0.930
0.935
0.025
0.031
0.028
0.710
0.717
0.720
0.078
0.023
0.015
0.677
0.653
0.660
NOTES: The table shows the summary statistics of the Hu and Tsay (2013) principal volatility component analysis for various k =five-dimensional series for ft . The multivariate factor
ARCH model is given by Equation (5) where ǫ t ∼ t(3). The performance measures R1 and R2 in (10) are calculated using 1000 iterations in Matlab. For the Ŵ̂m matrix m = 5 is used and
for the Huber’s function c = 2.5. We used Hu and Tsay (2013) high-persistent HP-ARCH components for all the models, that is (ω, α) = (1,0.9), (2,0.8), (3,0.7), and (1,0.95). For the low
persistent LP-ARCH models the ARCH parameter of the 4 ARCH(1) components are α = 0.1, 0.2, 0.3, 0.4, respectively. In the IGARCH model, (ω, α, β) = (1,0.09,0.9), (2,0.06,0.93),
(3,0.05,0.94), (1,0.02,0.97). N, t(3), and t(5) denote the standard normal and the standardized Student’s-t(3) and Student’s-t(5) innovation’s distribution. ρ is the AR coefficient of the
AR(1)–ARCH(1) models.
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Franke: Comment
171
observed over a time span i = 1, . . . , T and the sampling frequency h of the data used to compute the volatility estimates
which rely on data collected at increasing frequency, h ↓ 0. The
continuous record or in-fill asymptotics, h ↓ 0, have a key role
in that they can control the cross-sectional and serial correlation among the idiosyncratic errors of the panel of volatilities.
Therefore under suitable regularity conditions, the traditional
principal component analysis yields super-consistent estimates
of the common volatility factors at each point in time. The intuition behind the super-consistency result is because the highfrequency sampling scheme of the filtered volatilities is tied to
the size of the cross-section, boosting the rate of convergence.
Consequently, the super-consistency of the common volatility
factor extracted from the panel asymptotic arguments can also
improve upon the individual volatility estimates.
In Hu and Tsay (2013), the analysis is based on the first PVC.
An extension of the current analysis could address the choice
of the number of factors. This question has been addressed
in the context of linear factor models which do not involve
common volatility factors (e.g., Stock and Watson 2002; Bai
and Ng 2002). Namely, in the traditional factor models Bai and
Ng (2002) among others proposed various information criteria
that depend on the asymptotic rates of N and T of the panel.
In contrast, Ghysels (2013) showed that the standard crosssectional criteria suffice for consistent estimation of the number
of factors in large cross-sections of filtered volatilities, which
is different from the traditional panel data results. This result
on common volatility factor selection criteria draws from the
super-consistency.
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Comment
Juergen FRANKE
Universität Kaiserslautern, Kaiserslautern, Germany (franke@mathematik.uni-kl.de)
Financial data, in particular in the context of managing
portfolios and quantifying their risk, come in the form of
high-dimensional time series. The evolution of the coordinate
processes, representing asset prices or returns, all depend
on common market factors in addition to specific events
concerning the value of the underlying asset only. An important
task of financial time series analysis with major practical implications is the identification and description of such common
factors.
In the present article, Hu and Tsay focus on the volatility
aspect of financial time series and proposes an innovative way
to structure the interdependencies between the fluctuations of
the assets in a portfolio. For the volatility matrices, that is, the
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.903652