07350015%2E2014%2E887014

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TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 20:40

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Comment

Qiwei Yao

To cite this article: Qiwei Yao (2014) Comment, Journal of Business & Economic Statistics, 32:2, 165-166, DOI: 10.1080/07350015.2014.887014

To link to this article: http://dx.doi.org/10.1080/07350015.2014.887014

Published online: 16 May 2014.

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Ling and Yao: Comments 165

Comment

Shiqing L

ING

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,

Hong Kong, China (maling@ust.hk)

Congratulations to the authors for their very interesting work. The authors define a cumulative generalized kurtosis matrix to summarize the volatility dependence of multivariate time series in (2) and (3). It is the first generalization of the autocovariance-matrix in the field of time series. Its importance lies in that it not only can measure the dependence of the ARCH effect but also can reduce the moment condition ofyt. A lot of evidence shows that the fourth moment of financial time series does not exist, see ,for example, Zhu and Ling(2011). The classical statistical inference does not work for the ARCH-type time series if its fourth moment does not exist; see Zhu and Ling(2013). This concept opens a new direction to study the financial time series in the future.

Theorem 1 gives an important fact that the linear combination of several time series may not have the ARCH effect even if each individual time series has an ARCH effect. This means that one can select a stable portfolios in funding management such that its volatilities do not depend on the time horizon. Using this, one can also reduce the dimension of the parameter space in modeling vector ARCH-type time series. Its importance may be comparable to the concept of co-integration in the field of time series.

The authors proposed a principal volatility component ap-proach to identify the linear combination ofyt such that it does not have the ARCH effect. Huber’s function is used in (8) to reduce the moment condition. It turns out that the test statistic in Ling and Li(1997)and its generalization can be used to test if a linear combination ofythas the ARCH effect.

Based on the authors’ idea, I believe that there exist some other approaches to identify the linear combinations or make

a dimension reduction. More research can be done after this excellent work. For example, if we replace the componentyit ofytin (9) with ˆyitdefined as follows:

ˆ

then no moment condition is required, whereais a prespecified constant. Is this possible, and how does it affect the kurtosis matrix ofyt?

ACKNOWLEDGMENT

The author thanks the funding support in part from Hong Kong RGC Grants (numbered HKUST641912 and 603413).

REFERENCES

Ling, S., and Li, W. K. (1997), “Diagnostic Checking of Nonlinear Multivari-ate Time Series With MultivariMultivari-ate ARCH Errors,”Journal of Time Series Analysis, 18, 447–464. [165]

Zhu, K., and Ling, S. (2011), “Global Self-Weighted and Local Quasi-Maximum Exponential Likelihood Estimators for ARMA-GARCH/IGARCH Mod-els,”The Annals of Statistics, 39, 2131–2163. [165]

——— (2013), “Inference for ARMA Models With Unknown-Form and Heavy-Tailed G/ARCH-Type Noises,” Working paper, Department of Mathematics, HKUST. [165]

April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2014.887016

Comment

Qiwei Y

AO

Department of Statistics, London School of Economics, Houghton Street WC2A 2AE, London (q.yao@lse.ac.uk)

The authors are to be congratulated for tackling a challenging statistical problem with important financial applications, that is, modeling multivariate volatility processes via dimension-reduction. By introducing the so-called principal volatility com-ponents (PVC), they are able to identify a lower-dimensional space within which the dynamics of conditional heteroscedas-ticity confines.

Technically the authors look at the correlations between yty′t− and its lagged values in terms of the so-called

gen-eralized kurtosis matrices. To link those correlations to the con-ditional heteroscedasticity, they assume a vectorized ARCH(∞) model (1). The Huber truncation (8) is employed to refrain the moment condition required. The whole approach is simple and

April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2014.887014

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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∈B_t

[E{(yty′t−)I(B)}]

2_,

whereB_t_{is a}_{π-class and the}σ-algebra generated byB_t_{is equal}

to the filtrationF_t₋₁=σ(yt−1,yt−2, . . .). Then for any constant

vectorbsuch thatb′_Mb₌_{0, it holds that}

var(b′yt|F_t₋₁₎=var(b′yt),

that is,b′ytdoes not have conditional heteroscedasticity. Hence, the PVC can be defined asa′₁yt, . . . ,a′_ryt, 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 chooseB_t _{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, andε_t _{is a vector white-noise satisfying}

var(ε_t_|F_t₋₁₎=var(ε_t)₌_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 for_y_{(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

U

and Guodong L

I

Department 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),

April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2014.885436