Testing the eigenvalue structure of spot and integrated covariance?

For vector Ito semimartingale dynamics, we derive the asymptotic distributions of likelihood-ratio-type test statistics for the purpose of identifying the eigenvalue structure of both integrated and spot covariance matrices estimated using high-frequency data. Unlike the existing approaches where the cross-section dimension grows to infinity, our tests do not necessarily require large cross-section and thus allow for a wide range of applications. The tests, however, are based on non-standard asymptotic distributions with many nuisance parameters. Another contribution of this paper consists in proposing a bootstrap method to approximate these asymptotic distributions. While standard bootstrap methods focus on sampling point-wise returns, the proposed method replicates features of the asymptotic approximation of the statistics of interest that guarantee its validity. A Monte Carlo simulation study shows that the bootstrap-based test controls size and has power for even moderate size samples.

Automated summary

In this paper, we derive the asymptotic distributions of likelihood-ratio-type test statistics for identifying the eigenvalue structure of both integrated and spot covariance matrices estimated with high-frequency data. Unlike existing approaches, our tests do not require large cross-section and we propose a bootstrap method to approximate the asymptotic distributions. Monte Carlo simulations show that the bootstrap-based test controls sample size and has good power.