Principal Component Analysis of High-Frequency Data

ABSTRACT

We develop the necessary methodology to conduct principal component analysis at high frequency. We construct estimators of realized eigenvalues, eigenvectors, and principal components, and provide the asymptotic distribution of these estimators. Empirically, we study the high-frequency covariance structure of the constituents of the S&P 100 Index using as little as one week of high-frequency data at a time, and examines whether it is compatible with the evidence accumulated over decades of lower frequency returns. We find a surprising consistency between the low- and high-frequency structures. During the recent financial crisis, the first principal component becomes increasingly dominant, explaining up to 60% of the variation on its own, while the second principal component drives the common variation of financial sector stocks. Supplementary materials for this article are available online.

KEYWORDS:

• Eigenvalue
• Eigenvector
• High frequency
• Itô semimartingale
• Principal components
• Spectral function

Acknowledgments

The authors thank the Associate Editor and two referees for helpful comments and suggestions. The authors also thank Hristo Sendov for valuable discussions on eigenvalues and spectral functions. In addition, the authors are benefited from comments by Torben Andersen, Tim Bollerslev, Oleg Bondarenko, Marine Carrasco, Gary Chamberlain, Kirill Evdokimov, Jianqing Fan, Christian Hansen, Jerry Hausman, Jean Jacod, Ilze Kalnina, Jia Li, Yingying Li, Oliver Linton, Nour Meddahi, Per Mykland, Ulrich Müller, Andrew Patton, Eric Renault, Jeffrey Russell, Neil Shephard, George Tauchen, Viktor Todorov, Ruey Tsay, and Xinghua Zheng, as well as seminar and conference participants at Brown University, CEMFI, Duke University, Harvard University, MIT, Monash University, Northwestern University, Peking University, Princeton University, Singapore Management University, Stanford University, University of Amsterdam, University of Chicago, University of Illinois at Chicago, University of Tokyo, the 2015 North American Winter Meeting of the Econometric Society, the CEME Young Econometricians Workshop at Cornell, the NBER-NSF Time Series Conference in Vienna, the 10th International Symposium on Econometric Theory and Applications, the 7th Annual SoFiE Conference, the 2015 Financial Econometrics Conference in Toulouse and the 6th French Econometrics Conference.

Notes

1 Using the notation on page 583 of Jacod and Protter (2012), Assumption 1 states that the process X satisfies Assumption (H-1) and that c_t satisfies Assumption (H-2).

2 We prefer this estimator to the alternative one using overlapping windows because the overlapping implementation runs much slower. In a finite sample, both estimators have a decent performance.

3 While the constituents of the OEX Index change over time, we keep track of the changes to ensure that our choice of stocks is always in line with the index constituents.

4 Estimators that are robust to both noise and asynchronicity are available for integrated covariance estimation in Aït-Sahalia, Fan, and Xiu (2010), Christensen, Kinnebrock, and Podolskij (2010), Barndorff-Nielsen et al. (2011), and Shephard and Xiu (2017). Extending the current method to a noisy and asynchronous setting is theoretically interesting but not empirically necessary for the present paper, it is left for future work.