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Journal of Business & Economic Statistics
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Comment
Juergen Franke
To cite this article: Juergen Franke (2014) Comment, Journal of Business & Economic Statistics, 32:2, 171-172, DOI: 10.1080/07350015.2014.903652
To link to this article: http://dx.doi.org/10.1080/07350015.2014.903652
Published online: 16 May 2014.
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Franke: Comment 171
observed over a time spani=1, . . . , T and the sampling fre-quencyh of the data used to compute the volatility estimates which rely on data collected at increasing frequency,h↓0. The continuous record or in-fill asymptotics,h↓0, have a key role in that they can control the cross-sectional and serial correla-tion among the idiosyncratic errors of the panel of volatilities. Therefore under suitable regularity conditions, the traditional principal component analysis yields super-consistent estimates of the common volatility factors at each point in time. The in-tuition behind the super-consistency result is because the high-frequency sampling scheme of the filtered volatilities is tied to the size of the cross-section, boosting the rate of convergence. Consequently, the super-consistency of the common volatility factor extracted from the panel asymptotic arguments can also improve upon the individual volatility estimates.
In Hu and Tsay (2013), the analysis is based on the first PVC. An extension of the current analysis could address the choice of the number of factors. This question has been addressed in the context of linear factor models which do not involve common volatility factors (e.g., Stock and Watson2002; Bai and Ng2002). Namely, in the traditional factor models Bai and Ng (2002) among others proposed various information criteria that depend on the asymptotic rates ofN andT of the panel. In contrast, Ghysels (2013) showed that the standard cross-sectional criteria suffice for consistent estimation of the number of factors in large cross-sections of filtered volatilities, which is different from the traditional panel data results. This result on common volatility factor selection criteria draws from the super-consistency.
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Comment
Juergen F
RANKEUniversit ¨at Kaiserslautern, Kaiserslautern, Germany (franke@mathematik.uni-kl.de)
Financial data, in particular in the context of managing portfolios and quantifying their risk, come in the form of high-dimensional time series. The evolution of the coordinate processes, representing asset prices or returns, all depend on common market factors in addition to specific events concerning the value of the underlying asset only. An important task of financial time series analysis with major practical im-plications is the identification and description of such common factors.
In the present article, Hu and Tsay focus on the volatility aspect of financial time series and proposes an innovative way to structure the interdependencies between the fluctuations of the assets in a portfolio. For the volatility matrices, that is, the
© 2014American Statistical Association Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2014.903652
172 Journal of Business & Economic Statistics, April 2014
conditional covariance matrices given the past, they develop an analog to the principal component analysis for conditional means which allows us to identify dependencies between the volatilities of the single asset time series. In addition to estima-tion procedures for such principal volatility components, they also develop tests for vanishing ARCH effects in linear trans-formations of the time series which may be used for dimension reduction. The estimation and testing procedures can be imple-mented in a rather straightforward manner such that they are very useful for analyzing financial data in practice. Hu and Tsay themselves give a nice illustration of this approach to an FX portfolio of dimension seven. Their PVCA approach is in par-ticular valuable as it is flexible and can be easily modified to cope with specific issues like the two following ones.
MOMENT CONDITIONS AND ROBUSTNESS
The tests developed in Section 4.2 are based on the original Ling–Li statistic. The asymptotic results of Theorems 2 and 3 re-quire the existence of the sixth order moments of the underlying time seriesyt. For financial data with their well-known heavy-tailedness, this may seem to be a rather restrictive assumption. It is needed for Lemma 2 where the existence of the second order moments ofǫtxt−j is used. Here,xtis a standardized version of hy,t having mean 0 and variance 1, where
hy,t =ρ(u) with u2=y′t
(compare (13), where the sample version ˆhy,tis defined). Hence, xtis of the orderyt, whereasǫt is of the orderet2, that is, yt2.
As an alternative, one could use Hampel’sρfunction which has a redescending, piecewise linear and continuous derivative (ψ=ρ′, keeping to the notation of robust M-estimates):
ψ(u)=2u, if 0≤u≤a,
=2a, ifa < u≤b,
=2a(c−x)/(c−b), ifb < u≤c,
=0, ifu > c.
As the corresponding ρ(u) is bounded, xt would be bounded too, and a superficial glance at the proofs of Theorems 2 and 3 indicates that the moment condition onytcould be relaxed to the
existence of fourth moments which one would expect anyhow for an asymptotic analysis of test statistics based on empirical covariances.
Beyond that more theoretically motivated viewpoint, it might be advantageous also from practical considerations to use Hampel’sρ function or another one with redescending deriva-tive like Andrew’s sine wave or Tukey’s biweight (compare Sec. 4.8. of Huber 1981) as this would contribute to the ro-bustness of PVCA against isolated outliers. Of course, for a fully robust version, one would have to take into account the other part of the sample correlations ˆρℓ,s determining the test statisticsTd,s, that is, the ˆǫt’s. Developing a version of PVCA which is robust against isolated extreme events in single coor-dinates ofyt would be interesting as that kind of outlier may mask the heteroscedasticity which is common to all the compo-nent time series and which is a main target of the PVCA. That would require quite some additional research similar to the ro-bustification of classical principal component analysis (compare Hubert, Rousseeuw, and vanden Branden2005, and references therein).
NUMERICAL PROBLEMS IN HIGHER DIMENSIONS
The sample version ˆhy,tofhy,trequires inversion of the sam-ple covariance matrixof the datayt. Additionally, the covari-ance matrixValso has to be inverted for calculating the Ling–Li test statistic. For higher dimensions, for example, around 20, that is a tricky problem as sample covariance matrices, then, tend to be ill-conditioned. Fiecas et al. (2012) gave examples of the rather dramatic effects on inference for simulated data as well as for real financial data caused by that problem. A solution which can be easily implemented in calculating the test statistics of Section 4.2 is shrinkage as discussed, for example, by Ledoit and Wolf (2004).
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Fiecas, M., Franke, J., von Sachs, R., and Tadjuidje Kamgaing, J. (2012), “Shrinkage Estimation for Multivariate Hidden Markov Mixture Models,” ISBA Discussion Paper 16, Universit´e Catholique de Louvain. Available at http://www.uclouvain.be/cps/ucl/doc/stat/documents/DP2012 16.pdf
[172]
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Ledoit, O., and Wolf, M. (2004), “A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices,”Journal of Multivariate Analysis, 88, 365–411. [172]
