The Five Trolls Under the Bridge: Principal Component Analysis With Asynchronous and Noisy High Frequency Data

Abstract

We develop a principal component analysis (PCA) for high frequency data. As in Northern fairy tales, there are trolls waiting for the explorer. The first three trolls are market microstructure noise, asynchronous sampling times, and edge effects in estimators. To get around these, a robust estimator of the spot covariance matrix is developed based on the smoothed two-scale realized variance (S-TSRV). The fourth troll is how to pass from estimated time-varying covariance matrix to PCA. Under finite dimensionality, we develop this methodology through the estimation of realized spectral functions. Rates of convergence and central limit theory, as well as an estimator of standard error, are established. The fifth troll is high dimension on top of high frequency, where we also develop PCA. With the help of a new identity concerning the spot principal orthogonal complement, the high-dimensional rates of convergence have been studied after eliminating several strong assumptions in classical PCA. As an application, we show that our first principal component (PC) closely matches but potentially outperforms the S&P 100 market index. From a statistical standpoint, the close match between the first PC and the market index also corroborates this PCA procedure and the underlying S-TSRV matrix, in the sense of Karl Popper.

Supplementary materials for this article are available online.

KEYWORDS:

• Asynchronous sampling times
• Factor model
• High dimensionality
• High frequency
• Market microstructure noise
• Principal component analysis

Supplementary Materials

The supplement contains the proofs of the theorems and other mathematical results in the main body of the paper (Appendix A-F), as well as additional simulation results (Appendix G).

Acknowledgments

We would like to thank the editors, associate editor, and the referees, for comments that substantially improved the article.

Additional information

Funding

Financial support from the National Science Foundation under grants DMS 14-07812 and DMS 17-13129 (Mykland), and DMS 14-07820 and DMS 17-13118 (Zhang) is gratefully acknowledged.