07350015%2E2014%2E887016

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TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 20:39

Journal of Business & Economic Statistics

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

Comment

Shiqing Ling

To cite this article: Shiqing Ling (2014) Comment, Journal of Business & Economic Statistics,

32:2, 165-165, DOI: 10.1080/07350015.2014.887016

To link to this article: http://dx.doi.org/10.1080/07350015.2014.887016

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 ([email protected])

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:

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]

Comment

Qiwei Y

AO

Department of Statistics, London School of Economics, Houghton Street WC2A 2AE, London ([email protected])

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