07350015%2E2014%2E903652
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Juergen Franke
To cite this article: Juergen Franke (2014) Comment, Journal of Business & Economic Statistics,
32:2, 171-172, DOI: 10.1080/07350015.2014.903652
To link to this article: http://dx.doi.org/10.1080/07350015.2014.903652
Published online: 16 May 2014.
Submit your article to this journal
Article views: 99
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http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RA JA ALI HA JI
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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.
REFERENCES
Aı̈t-Sahalia, Y., Karaman, M., and Mancini, L. (2012), “The Term Structure of
Variance Swaps, Risk Premia and the Expectation Hypothesis,” discussion
paper, Princeton University, Princeton, NJ. [168]
Amengual, D. (2009), “The Term Structure of Variance Risk Premia,” discussion
paper, Princeton University, Princeton, NJ. [168]
Anderson, H., and Vahid-Araghi, F. (2007), “Forecasting the Volatility of Australian Stock Returns: Do Common Factors Help?,” Journal of Business and
Economic Statistics, 25, 76–90. [168]
Andreou, E., and Ghysels, E. (2002), “Rolling Sample Volatility Estimators:
Some New Theoretical, Simulation and Empirical Results,” Journal of Business and Economic Statistics, 20, 363–376. [170]
——— (2013) “What Drives the VIX and the Volatility Risk Premium?”,
discussion paper, UCY and UNC. [168,170]
Bai, J., and Ng, S. (2002), “Determining the Number of Factors in Approximate
Factor Models,” Econometrica, 70, 191–221. [171]
Barigozzi, M., Brownlees, C. T., Gallo, G. M., and Veredas, D. (2013),
“Disentangling Systematic and Idiosyncratic Dynamics in Panels of
Volatility Measures,” available at http://ssrn.com/abstract=1618565 or
http://dx.doi.org/10.2139/ssrn.1618565. [168]
Bollerslev, T., and Engle, R. F. (1986), “Modelling the Persistence of Conditional
Variances,” Econometric Reviews, 1, 1–50. [xxxx]
Bühler, H. (2006), “Consistent Variance Curve Models,” Finance and Stochastics, 10, 178–203. [168]
Connor, G., Korajczyk, R. A., and Linton, O. (2006), “The Common and Specific
Components of Dynamic Volatility,” Journal of Econometrics, 132, 231–
255. [168]
Diebold, F. X., and Nerlove, M. (1989), “The Dynamics of Exchange Rate
Volatility: a Multivariate Latent Factor ARCH Model,” Journal of Applied
Econometrics, 4, 1–21. [168]
Egloff, D., Leippold, M., and Wu, L. (2010), “The Term Structure of Variance
Swap Rates and Optimal Variance Swap Investments,” Journal of Financial
and Quantitative Analysis, 45, 1279–1310. [168]
Engle, R. F. (1987), “Multivariate GARCH With Factor Structures—
Cointegration in Variance,” unpublished manuscript, Department of Economics, UCSD. [168]
Engle, R. F., Ng, V. K., and Rothschild, M. (1990), “Asset Pricing With a FactorARCH Covariance Structure: Empirical Estimates for Treasury Bills,” Journal of Econometrics, 45, 213–237. [168]
Ghysels, E. (2013), “Factor Analysis With Large Panels of Volatility Proxies,”
discussion paper, UNC. [168,170,171]
Hansen, P. R., and Lunde, A. (2006), “Realized Variance and Market Microstructure Noise,” Journal of Business and Economic Statistics, 24, 127–161. [170]
Mykland, P. A., and Zhang, L. (2008), “Inference for Volatility-Type Objects
and Implications for Hedging,” Statistics and its Interface, 1, 255–278. [170]
Ng, V. K., Engle, R. F., and Rothschild, M. (1992), “A Multi-Dynamic-Factor
Model for Stock Returns,” Journal of Econometrics, 52, 245–266. [168]
Stock, J., and Watson, M. (2002), “Forecasting Using Principal Components
From a Large Number of Predictors,” Journal of the American Statistical
Association, 97, 1167–1179. [171]
Zhang, L. (2001), “From Martingales to ANOVA: Implied and Realized Volatility,” Ph.D. thesis, Department of Statistics, University of Chicago, Chicago,
IL. [170]
Zhang, L., Mykland, P., and Aı̈t-Sahalia, Y. (2005), “A Tale of Two Time Scales:
Determining Integrated Volatility With Noisy High-Frequency Data,” Journal of the American Statistical Association, 100, 1394–1411. [170]
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
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:41 11 January 2016
172
Journal of Business & Economic Statistics, April 2014
conditional covariance matrices given the past, they develop
an analog to the principal component analysis for conditional
means which allows us to identify dependencies between the
volatilities of the single asset time series. In addition to estimation procedures for such principal volatility components, they
also develop tests for vanishing ARCH effects in linear transformations of the time series which may be used for dimension
reduction. The estimation and testing procedures can be implemented in a rather straightforward manner such that they are
very useful for analyzing financial data in practice. Hu and Tsay
themselves give a nice illustration of this approach to an FX
portfolio of dimension seven. Their PVCA approach is in particular valuable as it is flexible and can be easily modified to
cope with specific issues like the two following ones.
MOMENT CONDITIONS AND ROBUSTNESS
The tests developed in Section 4.2 are based on the original
Ling–Li statistic. The asymptotic results of Theorems 2 and 3 require the existence of the sixth order moments of the underlying
time series yt . For financial data with their well-known heavytailedness, this may seem to be a rather restrictive assumption. It
is needed for Lemma 2 where the existence of the second order
moments of ǫt xt−j is used. Here, xt is a standardized version of
hy,t having mean 0 and variance 1, where
hy,t = ρ(u)
with
u2 = y′t −1 yt /k
using Huber’s ρ-function
ρ(u) = u2 ,
if 0 ≤ u ≤ c,
2
= 2cu − c ,
if u > c,
(compare (13), where the sample version ĥy,t is defined). Hence,
xt is of the order yt , whereas ǫt is of the order et 2 , that is,
yt 2 .
As an alternative, one could use Hampel’s ρ function which
has a redescending, piecewise linear and continuous derivative
(ψ = ρ ′ , keeping to the notation of robust M-estimates):
ψ(u) = 2u,
if 0 ≤ u ≤ a,
= 2a,
if a < u ≤ b,
= 2a (c − x)/(c − b),
if b < u ≤ c,
= 0,
if u > c.
As the corresponding ρ(u) is bounded, xt would be bounded
too, and a superficial glance at the proofs of Theorems 2 and 3
indicates that the moment condition on yt could be relaxed to the
existence of fourth moments which one would expect anyhow
for an asymptotic analysis of test statistics based on empirical
covariances.
Beyond that more theoretically motivated viewpoint, it might
be advantageous also from practical considerations to use
Hampel’s ρ function or another one with redescending derivative like Andrew’s sine wave or Tukey’s biweight (compare
Sec. 4.8. of Huber 1981) as this would contribute to the robustness of PVCA against isolated outliers. Of course, for a
fully robust version, one would have to take into account the
other part of the sample correlations ρ̂ℓ,s determining the test
statistics Td,s , that is, the ǫ̂t ’s. Developing a version of PVCA
which is robust against isolated extreme events in single coordinates of yt would be interesting as that kind of outlier may
mask the heteroscedasticity which is common to all the component time series and which is a main target of the PVCA. That
would require quite some additional research similar to the robustification of classical principal component analysis (compare
Hubert, Rousseeuw, and vanden Branden 2005, and references
therein).
NUMERICAL PROBLEMS IN HIGHER DIMENSIONS
The sample version ĥy,t of hy,t requires inversion of the sam of the data yt . Additionally, the covariple covariance matrix
also has to be inverted for calculating the Ling–Li
ance matrix V
test statistic. For higher dimensions, for example, around 20, that
is a tricky problem as sample covariance matrices, then, tend
to be ill-conditioned. Fiecas et al. (2012) gave examples of the
rather dramatic effects on inference for simulated data as well as
for real financial data caused by that problem. A solution which
can be easily implemented in calculating the test statistics of
Section 4.2 is shrinkage as discussed, for example, by Ledoit
and Wolf (2004).
REFERENCES
Fiecas, M., Franke, J., von Sachs, R., and Tadjuidje Kamgaing, J. (2012),
“Shrinkage Estimation for Multivariate Hidden Markov Mixture Models,”
ISBA Discussion Paper 16, Université Catholique de Louvain. Available
at
http://www.uclouvain.be/cps/ucl/doc/stat/documents/DP2012 16.pdf
[172]
Huber, P. J. (1981), Robust Statistics, New York: Wiley. [172]
Hubert, M., Rousseeuw, P. J., and vanden Branden, K. (2005), “ROBPCA: A
New Approach to Robust Principal Component Analysis,” Technometrics,
47, 64–79. [172]
Ledoit, O., and Wolf, M. (2004), “A Well-Conditioned Estimator for LargeDimensional Covariance Matrices,” Journal of Multivariate Analysis, 88,
365–411. [172]
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Juergen Franke
To cite this article: Juergen Franke (2014) Comment, Journal of Business & Economic Statistics,
32:2, 171-172, DOI: 10.1080/07350015.2014.903652
To link to this article: http://dx.doi.org/10.1080/07350015.2014.903652
Published online: 16 May 2014.
Submit your article to this journal
Article views: 99
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
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.
REFERENCES
Aı̈t-Sahalia, Y., Karaman, M., and Mancini, L. (2012), “The Term Structure of
Variance Swaps, Risk Premia and the Expectation Hypothesis,” discussion
paper, Princeton University, Princeton, NJ. [168]
Amengual, D. (2009), “The Term Structure of Variance Risk Premia,” discussion
paper, Princeton University, Princeton, NJ. [168]
Anderson, H., and Vahid-Araghi, F. (2007), “Forecasting the Volatility of Australian Stock Returns: Do Common Factors Help?,” Journal of Business and
Economic Statistics, 25, 76–90. [168]
Andreou, E., and Ghysels, E. (2002), “Rolling Sample Volatility Estimators:
Some New Theoretical, Simulation and Empirical Results,” Journal of Business and Economic Statistics, 20, 363–376. [170]
——— (2013) “What Drives the VIX and the Volatility Risk Premium?”,
discussion paper, UCY and UNC. [168,170]
Bai, J., and Ng, S. (2002), “Determining the Number of Factors in Approximate
Factor Models,” Econometrica, 70, 191–221. [171]
Barigozzi, M., Brownlees, C. T., Gallo, G. M., and Veredas, D. (2013),
“Disentangling Systematic and Idiosyncratic Dynamics in Panels of
Volatility Measures,” available at http://ssrn.com/abstract=1618565 or
http://dx.doi.org/10.2139/ssrn.1618565. [168]
Bollerslev, T., and Engle, R. F. (1986), “Modelling the Persistence of Conditional
Variances,” Econometric Reviews, 1, 1–50. [xxxx]
Bühler, H. (2006), “Consistent Variance Curve Models,” Finance and Stochastics, 10, 178–203. [168]
Connor, G., Korajczyk, R. A., and Linton, O. (2006), “The Common and Specific
Components of Dynamic Volatility,” Journal of Econometrics, 132, 231–
255. [168]
Diebold, F. X., and Nerlove, M. (1989), “The Dynamics of Exchange Rate
Volatility: a Multivariate Latent Factor ARCH Model,” Journal of Applied
Econometrics, 4, 1–21. [168]
Egloff, D., Leippold, M., and Wu, L. (2010), “The Term Structure of Variance
Swap Rates and Optimal Variance Swap Investments,” Journal of Financial
and Quantitative Analysis, 45, 1279–1310. [168]
Engle, R. F. (1987), “Multivariate GARCH With Factor Structures—
Cointegration in Variance,” unpublished manuscript, Department of Economics, UCSD. [168]
Engle, R. F., Ng, V. K., and Rothschild, M. (1990), “Asset Pricing With a FactorARCH Covariance Structure: Empirical Estimates for Treasury Bills,” Journal of Econometrics, 45, 213–237. [168]
Ghysels, E. (2013), “Factor Analysis With Large Panels of Volatility Proxies,”
discussion paper, UNC. [168,170,171]
Hansen, P. R., and Lunde, A. (2006), “Realized Variance and Market Microstructure Noise,” Journal of Business and Economic Statistics, 24, 127–161. [170]
Mykland, P. A., and Zhang, L. (2008), “Inference for Volatility-Type Objects
and Implications for Hedging,” Statistics and its Interface, 1, 255–278. [170]
Ng, V. K., Engle, R. F., and Rothschild, M. (1992), “A Multi-Dynamic-Factor
Model for Stock Returns,” Journal of Econometrics, 52, 245–266. [168]
Stock, J., and Watson, M. (2002), “Forecasting Using Principal Components
From a Large Number of Predictors,” Journal of the American Statistical
Association, 97, 1167–1179. [171]
Zhang, L. (2001), “From Martingales to ANOVA: Implied and Realized Volatility,” Ph.D. thesis, Department of Statistics, University of Chicago, Chicago,
IL. [170]
Zhang, L., Mykland, P., and Aı̈t-Sahalia, Y. (2005), “A Tale of Two Time Scales:
Determining Integrated Volatility With Noisy High-Frequency Data,” Journal of the American Statistical Association, 100, 1394–1411. [170]
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
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:41 11 January 2016
172
Journal of Business & Economic Statistics, April 2014
conditional covariance matrices given the past, they develop
an analog to the principal component analysis for conditional
means which allows us to identify dependencies between the
volatilities of the single asset time series. In addition to estimation procedures for such principal volatility components, they
also develop tests for vanishing ARCH effects in linear transformations of the time series which may be used for dimension
reduction. The estimation and testing procedures can be implemented in a rather straightforward manner such that they are
very useful for analyzing financial data in practice. Hu and Tsay
themselves give a nice illustration of this approach to an FX
portfolio of dimension seven. Their PVCA approach is in particular valuable as it is flexible and can be easily modified to
cope with specific issues like the two following ones.
MOMENT CONDITIONS AND ROBUSTNESS
The tests developed in Section 4.2 are based on the original
Ling–Li statistic. The asymptotic results of Theorems 2 and 3 require the existence of the sixth order moments of the underlying
time series yt . For financial data with their well-known heavytailedness, this may seem to be a rather restrictive assumption. It
is needed for Lemma 2 where the existence of the second order
moments of ǫt xt−j is used. Here, xt is a standardized version of
hy,t having mean 0 and variance 1, where
hy,t = ρ(u)
with
u2 = y′t −1 yt /k
using Huber’s ρ-function
ρ(u) = u2 ,
if 0 ≤ u ≤ c,
2
= 2cu − c ,
if u > c,
(compare (13), where the sample version ĥy,t is defined). Hence,
xt is of the order yt , whereas ǫt is of the order et 2 , that is,
yt 2 .
As an alternative, one could use Hampel’s ρ function which
has a redescending, piecewise linear and continuous derivative
(ψ = ρ ′ , keeping to the notation of robust M-estimates):
ψ(u) = 2u,
if 0 ≤ u ≤ a,
= 2a,
if a < u ≤ b,
= 2a (c − x)/(c − b),
if b < u ≤ c,
= 0,
if u > c.
As the corresponding ρ(u) is bounded, xt would be bounded
too, and a superficial glance at the proofs of Theorems 2 and 3
indicates that the moment condition on yt could be relaxed to the
existence of fourth moments which one would expect anyhow
for an asymptotic analysis of test statistics based on empirical
covariances.
Beyond that more theoretically motivated viewpoint, it might
be advantageous also from practical considerations to use
Hampel’s ρ function or another one with redescending derivative like Andrew’s sine wave or Tukey’s biweight (compare
Sec. 4.8. of Huber 1981) as this would contribute to the robustness of PVCA against isolated outliers. Of course, for a
fully robust version, one would have to take into account the
other part of the sample correlations ρ̂ℓ,s determining the test
statistics Td,s , that is, the ǫ̂t ’s. Developing a version of PVCA
which is robust against isolated extreme events in single coordinates of yt would be interesting as that kind of outlier may
mask the heteroscedasticity which is common to all the component time series and which is a main target of the PVCA. That
would require quite some additional research similar to the robustification of classical principal component analysis (compare
Hubert, Rousseeuw, and vanden Branden 2005, and references
therein).
NUMERICAL PROBLEMS IN HIGHER DIMENSIONS
The sample version ĥy,t of hy,t requires inversion of the sam of the data yt . Additionally, the covariple covariance matrix
also has to be inverted for calculating the Ling–Li
ance matrix V
test statistic. For higher dimensions, for example, around 20, that
is a tricky problem as sample covariance matrices, then, tend
to be ill-conditioned. Fiecas et al. (2012) gave examples of the
rather dramatic effects on inference for simulated data as well as
for real financial data caused by that problem. A solution which
can be easily implemented in calculating the test statistics of
Section 4.2 is shrinkage as discussed, for example, by Ledoit
and Wolf (2004).
REFERENCES
Fiecas, M., Franke, J., von Sachs, R., and Tadjuidje Kamgaing, J. (2012),
“Shrinkage Estimation for Multivariate Hidden Markov Mixture Models,”
ISBA Discussion Paper 16, Université Catholique de Louvain. Available
at
http://www.uclouvain.be/cps/ucl/doc/stat/documents/DP2012 16.pdf
[172]
Huber, P. J. (1981), Robust Statistics, New York: Wiley. [172]
Hubert, M., Rousseeuw, P. J., and vanden Branden, K. (2005), “ROBPCA: A
New Approach to Robust Principal Component Analysis,” Technometrics,
47, 64–79. [172]
Ledoit, O., and Wolf, M. (2004), “A Well-Conditioned Estimator for LargeDimensional Covariance Matrices,” Journal of Multivariate Analysis, 88,
365–411. [172]