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

Abstract

We study the rank of the instantaneous or spot covariance matrix ${\mathrm{\Sigma }}_{\mathit{X}}(\mathit{t})$ of a multidimensional process $\mathit{X}(\mathit{t})$. Given high-frequency observations $\mathit{X}(\mathit{i}/\mathit{n})$, $\mathit{i}=0,\dots ,\mathit{n}$, we test the null hypothesis $rank({\mathrm{\Sigma }}_{\mathit{X}}(\mathit{t}))\le \mathit{r}$ for all t against local alternatives where the average $(\mathit{r}+1)$st eigenvalue is larger than some signal detection rate ${\mathit{v}}_{\mathit{n}}$.

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 spectral gap of ${\mathrm{\Sigma }}_{\mathit{X}}(\mathit{t})$. We establish explicit matrix perturbation and concentration results that provide nonasymptotic uniform critical values and optimal signal detection rates ${\mathit{v}}_{\mathit{n}}$. 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.