07350015%2E2014%2E902236

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TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 20:42

Journal of Business & Economic Statistics

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

Rejoinder

Yu-Pin Hu & Ruey S. Tsay

To cite this article: Yu-Pin Hu & Ruey S. Tsay (2014) Rejoinder, Journal of Business & Economic Statistics, 32:2, 176-177, DOI: 10.1080/07350015.2014.902236

To link to this article: http://dx.doi.org/10.1080/07350015.2014.902236

Published online: 16 May 2014.

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176 Journal of Business & Economic Statistics, April 2014

Rejoinder: Principal Volatility Component

Analysis

Yu-Pin H

U

Department of International Business Studies, National Chi Nan University, Taiwan ([email protected])

Ruey S. T

SAY

Booth School of Business, University of Chicago, 5807 South Woodlawn Avenue, Chicago, IL 60637 ([email protected])

We express our sincere thanks to all the discussants for their constructive discussions and encouragement. In particular, we appreciate their proposals of alternative approaches to finding common volatility factors and their suggestions for improving the estimation of the generalized kurtosis matrix. For obvious reasons, we cannot answer satisfactorily every point or question raised in the discussions. We shall address some common and important issues of the discussions. We use the same notation as that in the article.

1. ROBUSTNESS AND MOMENT CONDITIONS

We share the discussants’ concerns about the higher-order moment condition imposed in the article. Different conditions are used in estimating the generalized kurtosis matrix and in testing the ARCH effects. The test statistics require existence of the sixth moments, whereas estimation uses the fourth moment. Our use of the Huber transform is just a first attempt to relax the moment condition. Thus, we are encouraged by the suggestion of Professors Franke and Ling on using a bounded transforma-tion and intend to investigate further this important issue. We are also glad to see the expanded simulation study by Professors Andreou and Ghysels. Their use of integrated volatility series, heavy-tailed innovations, and GJR-ARCH models is informa-tive. As expected, the performance of the proposed PVC analy-sis deteriorates when the moment condition is violated. On the other hand, their simulation result also provides encouragement for further investigation in using a more robust transformation in estimating the generalized kurtosis matrix.

Following the suggestion of Professor Franke, we consider the Hampel filter apply the PVC analysis to the seven exchange series consid-ered in the article and obtain similar results.Figure 1shows the time plots of the seventh principal volatility component of the exchange rate series with and without the Hampel filter. From the plots, the impact of using the filter on the estimation of no-ARCH linear combination is negligible As a matter of fact, the correlation between the two sample principal volatility compo-nents ofFigure 1is 0.992. The filter, however, affects some of the PVC associated with large eigenvalues. Since the Hampel filter is bounded, we only need the finite fourth moments for the existence of the generalized kurtosis matrixŴ∞. Note that the Hampel filter or the Huber transform is applied toxij,t−ℓ

only in estimating the generalized kurtosis matrix; otherwise, the sufficient part of Theorem 1 does not hold. In general, we agree with the discussants that additional research similar to the robustification of the classical principal component analysis is needed.

2. ALTERNATIVE APPROACHES

Let yt be a k-dimensional financial time series. We agree

with Professors Yao, H¨ardle, and Wang that the conditional heteroscedasticity of yt implies that yty′t is correlated with its

lagged values yt−ℓy′t−ℓ for some ℓ >0. Our use of Equation

(1) is simply a parametric form to show the stated linear depen-dence. Alternative formulations are possible. The matrixM of Professor Yao is more general and interesting. One can indeed define

γ_x=cov(y_ty′_t, x)=E[(y_ty′_t−)x], (2)

where x is a univariate random variable generated by Ft−1,

and consider the null space of γx. In particular, let E be the

intersection of the null space ofγx for allx ⊆Ft−1. Then,E

is spanned by a full-rankk×(k−r) matrix M1 if and only if the transformed series M′

1yt has no ARCH effects. Using this

result and the suggestion of Professor Yao, one can define an

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Hu and Tsay: Rejoinder 177

Figure 1. Time plots of the 7th sample principal volatility compo-nents for the weekly log returns of seven foreign exchange rates with and without using the Hampel filter. The upper plot is with the Hampel filter.

alternative generalized kurtosis matrix as

Ŵ∗_∞=

x∈Ft−1

γ_xγ′_x.

In practice, one may choose xto represent a proper partition of the data to obtain a sample estimate ofŴ∗_∞for eigenvalue– eigenvector analysis. The efficacy of this alternative approach deserves a careful study. Like Professors H¨ardle and Wang, we have considered the covariances between vec(y_ty′

t) and its

lagged values. However, the resulting linear combinations of-ten involve the cross-product terms of y_ty′

t. As such, these

linear combinations may not form proper portfolios in finan-cial applications. On the other hand, the suggestion of using an asymmetric clip is valuable and deserves a careful investigation as leverage effect is important in financial applications; see the discussion of Professors Andreou and Ghysels.

3. COMMON FEATURES AND FACTOR MODELS

As noted by Professors Yu, Li, and Ling, finding linear com-binations of asset returns that have no conditional heteroscedas-ticity belongs to the general concept of common features or co-movements in multivariate time series analysis. The basic idea is similar to that of co-integration, even though the tools used and the statistical distributions involved are different. The issues of interest are the definition of common features and ways to detect them. In the article, we provide a definition of condi-tional heteroscedasticity and use generalized kurtosis matrix to extract the common feature. The concept of principal volatility component (PVC) analysis is also highly related to common factors. The discussion of Professor Yao concerning the link between PVC and common factors is insightful. Our concern lies in the assumption that the matrix Ais of lower rank, that is,r < k. It seems that some test statistics need to be derived to check the rank of the sample estimate ofA.

4. INDEPENDENT COMPONENTS ANALYSIS (ICA) AND JOINT APPROXIMATE DIAGONALIZATION OF

EIGEN-MATRICES (JADE)

Professors H¨ardle, Wang, Yu, and Li all mentioned the use of independent components in multivariate volatility modeling. This is indeed an interesting idea and several related approaches have been proposed in the literature. We have pursued related research in recent years; see Matteson and Tsay (2011). Our limited experience shows that it is important in practice to check the existence of independent components before applying most of the available ICA-based methods. We thank Professors Yu and Li for pointing out the link between PVC and JADE. The connection would be useful in understanding properties of the proposed PVC analysis.

5. MULTIVARIATE VOLATILITY MODELS

We thank Professor McAleer for his pursuit of perfection in multivariate volatility models. His points should be taken seri-ously. Indeed, many properties of some widely used multivari-ate volatility models are yet to be developed. From a practical viewpoint, one often makes a compromise between simplic-ity and theoretical completeness. Our use of the dynamic (or time-varying) conditional correlation models in the empirical demonstration is driven by its simplicity. We believe that the theory for useful multivariate volatility models will be fully de-veloped in due course. On the other hand, his point about the connection between PCA for unit-root nonstationary time se-ries and PVC for weakly stationary time sese-ries is well taken. Note that one can study properties of the subspace of no ARCH effects in y_t, if exists, without investigating properties of the subspace of y_t that has ARCH effects. In spirit, this is similar to studying the co-integrating series, which is stationary, of a co-integrated system.

6. USE OF ADDITIONAL DATA

Professors Andreou, Gyhsels, and Yao all mentioned using high-frequency data to estimate common volatility factors. This is yet another sensible approach. It would be interesting to com-pare various approaches to extracting common volatility factors and to quantify the contributions of realized volatility and co-variances. When the realized volatility series are nonstationary, the traditional PCA tends to provide a few dominating linear combinations. It is then not surprising to see that a small num-ber of principal components explain a high percentage of the variations in volatility. However, this phenomenon could be misleading in real applications similar to a high R2 _{in linear}

regressions involving unit-root time series.

REFERENCE

Matteson, D. S., and Tsay, R. S. (2011), “Dynamic Orthogonal Components for Multivariate Time Series,”Journal of the American Statistical Association, 106, 1450–1463. [177]