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Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Philip L. H. Yu & Guodong Li
To cite this article: Philip L. H. Yu & Guodong Li (2014) Comment, Journal of Business & Economic Statistics, 32:2, 166-167, DOI: 10.1080/07350015.2014.885436
To link to this article: http://dx.doi.org/10.1080/07350015.2014.885436
Published online: 16 May 2014.
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166 Journal of Business & Economic Statistics, April 2014
easy to implement. The dimension-reduction is often effective whenkis large.
Inspired by the idea in the article, I propose an alternative way to define PVC below. I also make an observation on the link between PVC and volatility factor models.
An Alternative Definition for PVC.An alternative approach would be to replaceŴ∞by
M=
B∈Bt
[E{(yty′t−)I(B)}]
2,
whereBtis aπ-class and theσ-algebra generated byBtis equal to the filtrationFt−1=σ(yt−1,yt−2, . . .). Then for any constant
vectorbsuch thatb′Mb=0, it holds that
var(b′yt|Ft−1)=var(b′yt),
that is,b′ytdoes not have conditional heteroscedasticity. Hence, the PVC can be defined asa′1yt, . . . ,ar′yt, wherea1, . . . ,ar are the orthonormal eigenvectors of matrixMwith the correspond-ing eigenvalues nonzero. This approach requires some mild mo-ment conditions, and is free from model assumptions such as (1). Note that the PVC are not necessarily independent with each other.
We refer to Fan, Wang, and Yao(2008, sec. 2.2.1)for how to chooseBt in practice.
Link to Factor Models.There is an innate connection between the PVC approach and factor models for volatility. In fact, PVC can be viewed as latent factsors that drive the dynamics of conditional heteroscedasticity.
LetA=(a1, . . . ,ar). ThenA′A=Ir. LetŴ=(A,B) be a k×korthogonal matrix, and
xt =A′yt, and εt =BB′yt.
Thenxt isr-variate factor exhibiting conditional heteroscedas-ticity, andεt is a vector white-noise satisfying
var(εt|Ft−1)=var(εt)=ε.
This is the standard form of volatility factor models; see Tao et al.(2011, sec. 2.3) and the references within. If some initial estimates fory(t) can be obtained, for example, from using high-frequency data,Acan be easily identified and estimated based on a simple eigenanalysis. See Tao et al.(2011)for further details on this approach.
REFERENCES
Fan, J., Wang, M., and Yao, Q. (2008), “Modelling Multivariate Volatilities via Conditionally Uncorrelated Components,”Journal of the Royal Statistical Society, Series B, 70, 679–702. [166]
Tao, M., Wang, Y., Yao, Q., and Zou, J. (2011), “Large Volatility Matrix Inference via Combining Low-Frequency and High-Frequency Approaches,” Journal of the American Statistical Association, 106, 1025–1040. [166]
Comment
Philip L. H. Y
Uand Guodong L
IDepartment of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, China (plhyu@hku.hk; gdli@hku.hk)
This article adopts the idea of principal component analysis (PCA) to model multivariate volatility, and the principal volatil-ity component (PVC) analysis is then proposed to search for common volatility components among many financial time se-ries. We congratulate Professors Tsay and Hu for this nice work, and some of our thoughts are given as follows.
First, from the viewpoint of applications, the idea of PVC is more like that of cointegration. We attempt to identify the comovements of sequences{yt}in cointegration, where{yt}are I(1) sequences, and the linearly transformed time series{m′y
t} are stationary. While the PVC attempts to find the common volatility components of {yt}, where {yt} are stationary and have a time varying conditional variance matrix, the linearly transformed time series{m′y
t}have a constant conditional vari-ance matrix. As argued by the authors, it will be useful for carry trade or hedging purposes if we can successfully identify the common volatility components. Note that these common volatility components actually are portfolios in finance, and we
here would like to mention another two applications in this as-pect. First, it should be more accurate to estimate the volatility based on historical data since it is not time varying, and we then can better manage the risk of this portfolio. Second, suppose that a portfolio with constant conditional variance is a factor (e.g., the market portfolio), and we would like to construct a portfolio which is neutral to this factor. We then can include an additional constraint into the portfolio optimization, where the constraint is that the optimized portfolio is uncorrelated with this factor.
In the literature of multivariate conditional heteroscedastic-ity, there are several dimension reduction methods available, and they include orthogonal GARCH models (Alexander2001),
© 2014American Statistical Association Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI:10.1080/07350015.2014.885436
Yu and Li: Comment 167
independent component analysis GARCH (ICA-GARCH) mod-els (Wu, Yu, and Li2006), conditionally uncorrelated compo-nents (CUC) models (Fan, Wang, and Yao2008), and dynamic orthogonal components (DOC) models (Matteson and Tsay
2011). Typically, they decompose the multivariate time series into a linear combination of (conditional) uncorrelated compo-nents which are then modeled by various univariate GARCH-type models. This can greatly reduce both the number of param-eters and estimation costs. Unlike the aforementioned models, the authors propose a new method to identify uncorrelated com-ponents, and it is based on the generalized kurtosis matrix
γℓ= be found in the joint approximate diagonalization of eigen-matrices (JADE), proposed by Cardoso and Souloumiac (1993), to identify independent components. The JADE approach takes into account the fourth-order cumulants,
cum(yit, yj t, ypt, yqt)
= E(yityj typtyqt)−E(yityj t)E(yptyqt)
−E(yitypt)E(yj tyqt)−E(yityqt)E(yj typt),
and the fourth-order cumulant tensor,
Qij(M)=
The independent components can be found by using the eigen-matrices of the above tensor. Both the JADE and the PVC share a few similarities. First of all, both made use of fourth-order mo-ments in their component estimation. Second, for zt =M′yt, we have that ARCH effect in this article, whereM1is ak×smatrix
consist-ing of eigenvectors associated with the ssmallest eigenvalues of Ŵ∞. Denoteεt=e′tV
−1e
t, andxt=g(yt), whereVis the variance matrix, and g(·) is a function. The authors construct two tests for ARCH effects, and they both depend on the sam-ple correlation coefficients betweenεt andxt−j withj >0. It may also be of interest to consider tests based on the autocor-relation function (ACF) of{εt}, that is, the sample correlation coefficients betweenεt andεt−j withj >0. Some commonly used portmanteau tests can then be applied; see Li (2004), Li and Li (2005), etc. On the other hand, some information will be missed when we transform the multivariate sequence{et}into a sequence of scalar variables{εt}, and it may be of interest to consider a multivariate portmanteau test here; see Mahdi and McLeod (2012).
REFERENCES
Alexander, C. O. (2001), “Orthogonal GARCH,” inMastering Risk(Vol. 2), ed. C. O. Alexander, Harlow, UK: Financial Times-Prentice Hall. [166] Cardoso, J. F., and Souloumiac, A. (1993), “Blind Beamforming for
Non-Gaussian Signals,”IEE Proceedings-F, 140, 362–370. [167]
Fan, J., Wang, M., and Yao, Q. (2008), “Modelling Multivariate Volatilties via Conditionally Uncorrelated Components,”Journal of the Royal Statistical Society, Series B, 70, 679–702. [167]
Li, G., and Li, W. K. (2005), “Diagnostic Checking for Time Series Models With Conditional Heteroscedasticity Estimated by the Least Absolute Deviation Approach,”Biometrika, 92, 691–701. [167]
Li, W. K. (2004),Diagnostic Checks in Time Series, Boca Raton, FL: Chapman & Hall. [167]
Mahdi, E., and McLeod, A. I. (2012), “Improved Multivariate Portmanteau Test,”Journal of Time Series Analysis, 33, 211–222. [167]
Matteson, D. S., and Tsay, R. S. (2011), “Dynamic Orthgonal Components for Multivariate Time Series,”Journal of the American Statistical Association, 106, 1450–1463. [167]
Wu, E. H. C., Yu, P. L. H., and Li, W. K. (2006), “Value at Risk Estimation Using Ica-Garch Models,”International Journal of Neural Systems, 16, 371–382. [167]
