Inference on the maximal rank of time-varying covariance matrices using high-frequency data

We study the rank of the instantaneous or spot covariance matrix ΣX(t) of a multidimensional continuous semi-martingale X(t). Given highfrequency observations X(i/n), i = 0,...,n, we test the null hypothesis rank (ΣX(t)) <= r for all t against local alternatives where the average (r + 1)st eigenvalue is larger than some signal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and a spectral gap of ΣX(t).We establish explicit matrix perturbation and concentration results that provide non-asymptotic uniform critical values and optimal signal detection rates vn. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.