07350015%2E2014%2E898585

(1)

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 RAJA ALI HAJI

TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 20:41

Journal of Business & Economic Statistics

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

Comment

Wolfgang Karl Härdle & Weining Wang

To cite this article: Wolfgang Karl Härdle & Weining Wang (2014) Comment, Journal of Business & Economic Statistics, 32:2, 173-174, DOI: 10.1080/07350015.2014.898585

To link to this article: http://dx.doi.org/10.1080/07350015.2014.898585

Published online: 16 May 2014.

Submit your article to this journal

Article views: 119

View related articles

(2)

H ¨ardle and Wang: Comment 173

Comment

Wolfgang Karl H ¨

ARDLE

Center for Applied Statistics and Economics, Humboldt-Universit ¨at zu Berlin, Unter den Linden 6, 10099 Berlin, Germany; Singapore Management University, 50 Stamford Road, Singapore 178899 ([email protected])

Weining WANG

Center for Applied Statistics & Economics School of Business and Economics, Humboldt-Universit ¨at zu Berlin, Unter den Linden 6, 10099 Berlin, Germany ([email protected])

The authors are to be congratulated for a timely and impor-tant contribution. The article proposes a novel principal volatil-ity component (PVC) technique based on a generalized kurto-sis matrix in a time series context. The proposed test statistics allow deep insight into higher moments and tail behavior of multivariate time series. The article considers a weak stationary multivariate time seriesyt(k×1) with finite fourth moments,

the laglgeneralized kurtosis matrix is defined as

γl

The PVCs are then defined as linear combinationsm⊤

vyt, where

the mv’s are the vth eigenvectors of the cumulative

general-ized kurtosis matrixΓ∞for general multivariate GARCH-type models (Γm for ARCH(m) effects in yt) with Γ∞

The kurtosis matrix indicates the correlations and cross-correlations between the current variance–covariance matrix and its lagged one, and thus would be a four-dimensional object (k×k×k×k). Nevertheless, the authors consider ak×k gen-eralized kurtosis matrix, which sums up all the effects of a lagged variance–covariance matrix. In some cases, one might like to look at the componentwise effects, which requires alternatives of defining a generalized kurtosis matrix. For example, one can analyze the variance–covariance matrix between vec(yty⊤_t ) and vec(yt−ly⊤_t−l), whose dimension isk2×k2 matrix. Moreover,

to generalize the idea of impulse response functions, one can look at the matrix

jcov2(yty⊤_t, xi0j,t−l) to isolate the lagged variablei0’s(i0 =1, . . . , k) contribution.

The article employs Huber’s function which is symmet-ric. One might by an asymmetric clip function address the well-known leverage effect, which means that negative returns increase future volatility by a larger amount than positive re-turns of the same magnitude. In particular, to model asymmetry in the ARCH process, for example, as in GJRGARCH models introduced by Glosten, Jagannathan, and Runkle (1993). For instance, setting xij,t−l =y

−

i,t−ly

−

j,t−l may serve this propose,

where y_j,t−−l=yj,t−l only when yj,t−l <0 (negative part of

yj,t−l).

The idea of PVC is a decomposition of a (mixed) moment matrix. In PCA, one considers the variance–covariance matrix,

which falls short on modeling a nonlinear and asymmetric multivariate distribution. This fact reminds us of a strand of literature on independent component analysis (ICA); see, for example, Chen, H¨ardle, and Spokoiny (2007); Chen et al. (2014). ICA looks for a projection that maximizes a non-Gaussianity measure. Similarly, a generalized kurtosis matrix in PVC is connected to measuring non-Gaussianity. However, kurtosis does not provide the whole picture of a distribution function, and therefore other perspectives of the conditional distribution (e.g., conditional skewness and conditional quan-tile) may also be of interest, see, for example, Lanne and Pentti (2007).

Another issue is possible nonstationarity inyt. In this

situa-tion, the nonstationarity can be modeled via switching parame-ters of a stationary model, see H¨ardle, Herwartz, and Spokoiny (2003). The eigenvectors mv’s would then be time varying

mvt. Accordingly, at timetone can adopt local adaptive

tech-niques (see Spokoiny, Wang, and H¨ardle 2013) to identify a local homogeneous interval [t−_t₀_{, t}_{], in which one may apply}

PVC.

Once more we would like to congratulate the authors for this great advance. We are sure that this work will create a new strand of literature with implications on asset allocation: portfolio choice and factor models. If one is interested in the factors that have no ARCH effects, one can certainly employ the presented technique. The factors isolated can be used as factors in asset pricing model, taking into account of rare events as in Martin (2013).

ACKNOWLEDGMENTS

The financial support From the Deutsche Forschungsge-meinschaft via SFB 649 “ ¨Okonomisches Risiko,” Humboldt-Universit¨at zu Berlin and IRTG 1972 “High Dimensional Non Stationary Time Series” is gratefully acknowledged.

April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2014.898585

(3)

174 Journal of Business & Economic Statistics, April 2014

REFERENCES

Chen, Y., Chen, R.-B., and H¨ardle, W. K. (2014), “TVICA—Time Varying Independent Component Analysis and Its Application to Financial Data,”

Journal of Computational Statistics and Data Analysis, forthcoming, DOI:

http://dx.doi.org/10.1016/j.csda.2014.01.002. [173]

Chen, Y., H¨ardle, W., and Spokoiny, V. (2007), “Portfolio Value at Risk Based on Independent Component Analysis,”Journal of Computational and Applied Mathematics, 205, 594–607. [173]

Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993), “On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks,”The Journal of Finance, 48, 1779–1801. [173]

H¨ardle, W., Herwartz, H., and Spokoiny, V. (2003), “Time Inhomogeneous Multiple Volatility Modeling,”Journal of Financial Econometrics, 1, 55– 95. [173]

Lanne, M., and Pentti, S. (2007), “Modeling Conditional Skewness in Stock Returns,”The European Journal of Finance, 13, 691–704. [173]

Martin, I. W. R. (2013), “Consumption-Based Asset Pricing With Higher Cu-mulants,”Review of Economic Studies, 80, 745–773. [173]

Spokoiny, V., Wang, W., and H¨ardle, W. K. (2013), “Local Quantile Regression” (with discussion),Journal of Statistical Planning and Inference, 143, 1109– 1129. [173]

Comment

Michael MCALEER

Department of Quantitative Finance, College of Technology Management, National Tsing Hua University, Taiwan; Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, The Netherlands;

Tinbergen Institute, The Netherlands;

Department of Quantitative Economics, Complutense University of Madrid, Spain

DISCUSSION

It is a pleasure to provide some comments on the excellent and topical article by Hu and Tsay (2014).

The article extends principal component analysis (PCA) to principal volatility component analysis (PVCA), and should prove to be an invaluable addition to the existing multivariate models for dynamic covariances and correlations that are es-sential for sensible risk and portfolio management, including dynamic hedging.

One of the key obstacles to developing multivariate covari-ance and correlation models is the “curse of dimensionality,” namely the number of underlying parameters to be estimated, the article is concerned with dimension reduction through the use of PCA, which is possible if there are some common volatil-ity components in the time series.

In particular, the method searches for linear combinations of a vector time series for which there are no time-varying conditional variances or covariances, and hence no time-varying conditional correlations.

The authors extend PCA to PCVA in a clear, appealing, and practical manner. Specifically, they use a spectral analysis of a cumulative generalized kurtosis matrix to summarize the volatil-ity dependence of multivariate time series and define the prin-cipal volatility components for dimension reduction.

The technical part of the article starts in Section 2 with a vectorization of the volatility matrix, and a connection to the BEKK model of Engle and Kroner (1995).

However, because a primary purpose of PCVA is to search for the absence of multivariate time-varying conditional het-eroskedasticity in vector time series, it would have been helpful to see how PCVA might be connected to the conditional covari-ances arising from various specializations of BEKK (for further details, see below).

Theorem 1 assumes the existence of fourth moments of a weakly stationary vector time series, but Theorems 2 and

3 assume the existence of sixth moments. The latter two theorems beg the question as to whether the assumption is testable.

Interestingly, in the simulation study, the four simple ARCH models considered are understood to “not satisfy the moment condition of Theorems 2 and 3,” with a reference to Box and Tiao (1977) that traditional PCA works well in finite samples for nonstationary time series.

However, the purported connection between PCA for non-stationary time series, on the one hand, and time-varying con-ditional covariances and correlations for a weakly stationary vector time series, on the other, is not entirely clear.

The empirical analysis considers a dataset of weekly log re-turns of seven exchange rates against the U.S. dollar from March 2000 to October 2011, giving 605 observations, which would be considered a relatively small number of observations for pur-poses of estimating dynamic vector covariance and correlation matrices.

The simple GARCH(1,1) model is used to estimate the con-ditional volatility models for the first six principal volatility components. It would have been useful to compare the GARCH estimates with the univariate asymmetric GJR and (possibly) leverage-based EGARCH alternatives.

A simple comparison is made of the PVCA results with the varying conditional correlation (VCC) model of Tse and Tsui (2002), though VCC is referred to as a “dynamic conditional correlation (DCC) model” (see Engle2002).

As the effect of “news” in the VCC model has an estimated coefficient of 0.013 and a standard error of 0.004, it is stated

April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2014.898584